. 18 8. 21 48 Issues in Epistemic and Modal Logics and their applications A dissenation submitted to the Faculty o f Philosophy of the University o f Tübingen in partial fulfillment o f the requiremems for the degree o f Doctor o f Philosophy by C ezar A. Mortari Dffl l ü W 1L D from Santa Maria (Brazil) 1991 Hauplberichterstalter: Prof. Dr. Dr. Waltcr Hoering Mitberichlerstatten Prof. Dr. Peler Schroeder-Heisier Dekan: Prof. Dr. Günier Figal Tag der mflndlichen Prtlfung: 28.1.1991 Gedruckt mit Genehmigung der Philosophischen FakulUU der Universílât Tilbingcn This work has been done also lhanks to a scholarshin * by CAPES, Brazil. For Daniela, A c k n o w l e d g e m e n t s Many people coniributed in several ways to make this work possible, so any tentative to write a list would certainly end up being unjust by leaving someone out. To ali of you, my warm apprcciation. To some people, however, I owe a special debt: Many lhanks to Prof. Waller Hoering, my adviser de jure , who was always lhere, ready to help, when I needed him, and who has done his best to keep my way free of troubles. This work would not have been accomplished at ali without lhe constant support and encouragement of Prof. Franz Guenthner, my de facto adviser, who, besides showing me a lot of the logical landscape, also introduced me to computers and to the joys of programming. As lhe former director of the Seminar für NatUrlich-Sprachliche Systeme (SNS) of the Universiiy of TUbingen, Prof. Guenthner provided me with a wonderful working and leaming environment, which I will never forgei. Thank you very, very much! I would like also to thank ali my colleagues and friends, boih at the SNS and at lhe Federal llniversity of Santa Catarina (UFSC), Brazil, for the help along the way. In particular, my lhanks to Prof. Sônia Felipe. Finally, my decpest gralitude lo Daniela, for her love, patíence and undersianding. Contents Chapler 0 : Inlroduclion & Road Map .............................................................. 1 P art I : Mininial Belief States in Epistemic-Doxastic Logics ........................ 17 Chapler 1 : An Overview of EDL-syslems ....................................................... 19 Chapler 2 : Minimal Belief States .................................................................... 42 In term ezzo 1 ................................................................................................................ 69 F art II : Valuation Semantics and G eneralized Truth-Tablcs ......................... 71 Chapler 3 : Valuation Semantics for Normal Modal L ogics............................... 73 Chapler 4 : Valuation Semantics for Classical Modal L ogics............................. 94 Chapler 5 : GTTs for K ...................................................................................... 114 Chapler 6 : The S4 problem .............................................................................. 121 Chapler 7 : Valuations, Possible Worlds, and Tableau System s......................... 128 Chapter 8 : Valuations & GTTs for Z5 ............................................................. 134 In te rm ezz o 2 ..................................................................................................................... 145 P a r t III : Im plem enta tions ......................................................................................... 147 Chapler 9 : Implementing a GTT Builder for ZS ................................................. 149 Chapler 10 : A tableau-like lheorem prover for ZP5 ........................................... 159 Chapter 11 : Implemenlalion of the algorithm .................................................... 169 Final Remarks .......................................................................................................... 175 A ppendices a n d R eferences ...................................................................................... 177 Appendix A : Some derivations ........................................................................... 179 Appendix B : GTT.Z5 Listings ........................................................................... 181 Appendix C : TTP.ZP5 Listings .......................................................................... 192 Appendix D : ALG.ZP5 Listings ........................................................................ 203 Appendix E : References ...................................................................................... 207 @ introduction & Road Map Mihi a docto doctore domandalur causam et rationem quare opium facit dormire? A quoi respondeo Quia est in eo Virtus dormitiva, Cujus est natura Sensus assoupire. M O L IÈ R E , Le Malade Imaginaire. 0.1 Painting the background The tiüe of Ihis work, Issues in epistemic and modal logics and their applications, is obviously a very comprehensive one, lhe reason for this being the fact lhat the contents reflect my multiple interests duríng the time I have been studying and woiking in Tübingen. Such a title doesn't tell us very much about exacüy what the contents are, or which issues are actually going to be considered, so of course 1*11 have to say a few more words introducing lhe work and narrowing its subject mauer. However, before we get down to discussing the specific research problems, I guess it would be nice, and even necessary, to dwell awhile on some preliminaries describing lhe big mosaic of which this work is hopefully going to be a small piece. This surely will give the reader a belter understanding of what I'm up lo here, and why. To begin with, one could ask, why (one more work in) modal logics? Such a question is perhaps to be expected since probably everybody has, at least once, heard about lhese logics, and, if they are not one's working area, one probably has this idea that modal logics only deal with funny concepts like necessity and possibility and contingency; in olher words, lhat they deal wilh a lot of pretty meiaphysical stuff-jusl remember ali that talking about Leibnizian "possible worlds" (of which ours is supposed to be the best one), and worlds being "accessible", and "parallel universes", and so on, unül one is caught discussing 1 C hapter 0 how many possible fat men stand on that doorway.1 One would hardly suspect that modal logics could be of use in this (possible) world and utilitarían times of ours. Now, to tell lhe plain truth, the interests of modal logics do not-at least not only-concern metaphysical talking about possible worlds. Modality is in fact a very broad notion, and considerations about necessity and possibility deal with just one small side of it, namely with what is usually called alethic modality. ("Alethic" comes from the Greek word for "truth".) Necessity and possibility are said to be "modes of truth"; i.e., they refer to lhe way in which a proposition can be true, like "necessarily true", "possibly tnie", "impossibly Inie" (that is, "necessarily false"), and so on. Now alethic modal logic was the first one to get developed: we can trace its beginning down to the beginning of logic itself, namely to Aristotle. In his works De Interpretatione and Analytica Priora he discusses logical interconnections between modal notions-such as necessary, impossible, possible, and permiited-as well as giving some thought to the lheory of modal syllogisms, that is, syllogisms which have modalized premisses and conclusion (cf. [Lem77J, p. 1-2). (As an example, "ali animais are necessarily mortal" and "ali humans are necessarily animais", ergo "ali humans are necessarily mortal".) According to Lemmon, much of Arístolle's discussion is quite confused, but "[its] outcome is a remarkably correct set of implications" ([Lem77], p. 1). From Aristotle's time on, beginning with his own school, not forgetting the Stoics and Megarians, and going until the end of the Middle Ages, there were a lot of people interested in and working with modal notions, with sometimes rather interesting contributions. We could mention, as an illustration, Diodoms Chronus, who gave definitions of necessity and possibility by means of temporal notions ("the possible is that which either is or will be"); or the medieval discussion about de dicto and de re modalitíes; or PseudoScotus , who studied, besides "necessary" and "possible", other modalities such as "it is known that" or "it is believed lhat", thus antecipaling epistemic logics (cf. [Lem77, p. 4J). Afterwards, however, not very much happened, the intenegnum between the end of the Dark Ages and the nineteenth century not being the best possible modal logical world. Thus lhe modem development of modal logic starts only in this century with the work of C.I. Lewis, whose main contríbution, one could say, were the so-called "Lçwis systems", SI-S5, which axiomatize increasingly strong concepüons of necessity.2 To make it short, lhanks to alelhic modal Iogic's early, aristoielic beginning, the term "modal" got stuck with this memberof the logic family-it was the only one around in lown. But as the years went by, alelhic's younger sisters came into exislence and grew inio logics in their own right, and it becamc then usual to employ the expression "modal logic" in a broader sense, which wasn'i only rcsuicted to modes of truth. Thus, loday, we classify as modal logics, beyond lhe alelhic ones, also temporal logics t deontic logics, epistemic logics, and so forth. In a sense, one could labei as "modal logics" ali logical systems in which one extends the language of classical logic by means of adding a certain kind of new operators, the so called intertsional ones. Iniensional operators are those which are not Iruüi-functions of lhe propositions to which they apply. For instance: 1 (Cf. {Qu80J, p. 4 ) B y a ve ry suspiciotis co in d d c n ce , their num ber is exacdy the sam e a* the num ber o f angcls in the ey e o f a needle- o r was It in the h ta d o f a p in .. .7 2 T o tell th e tru th , L ew is' m ain in le rc st w as n o t fo rm a lir in g seve ia l no tions o f necessity and possib ility ; h e w as ac tually w orking on difTerent concep tions o f im p lication , try ing to avo id the paradoxes o f m aterial im plication . In the course o f his investígatíons h e arrived to the s tric t im plication , w hich one can characterize as the necessity o f the conditional- this is w here necessity com es in to picture. R y the w ay, readers w anting to know m ore about the h istorical developm ent o f m odal log ic are refeired to [K K 62, L em 77, IIC72J, w hete additional biM iogniphy can also be found. 2 Iníroduclion A R o a d M ap "It is necessary Ihal..." "It is possible that..." "U will be lhe case that..." "Darth Vader believes Ihat..." "It isobligalory Ihat..." In lhe Customary Way Of Doing Things, one takes the classical logic, say the classical propositional logic, and adds to its language two new operators, (box) and 'o ' (diamond), also introducing some axioms and inference rules involving them. Usually the box gets interpreted as ''necessarily", and the diamond as "possibly" . But someone can choose to say that 'q p ' means "always p" (where p stands for some proposition), so he's doing tense logic. And someone else takes 'q p ' to mean "Yoda knows that p '\ so she's doing epislemic logic. Thus one could veniure that only the way you interpret the box (necessary, knows, always, provable) and lhe diamond will give a cue about which kind of logic you are doing. Of course, depending on lhe different interprelations of the operators, different formulas can or will hold, or not, but very often the same calculus is said to be both lhe.nicest alethic and lhe nicest epislemic logic, for ínstance. As lo the possible worlds we mentioned above, we are going lo find them in the so-called possibleworld semantics for modal logics. In the case of classical logic, e.g. in a semantics for the propositional calculus, lo evaluate a formula we proceed by looking at one model (typically a function assigning trolhvalues lo propositional variables) and lhen computing this formula's value. In lhe case of modal logics, we have to consider more "models" at the same time. If we understand a model to be a kind of "world description", this amounts to say Ihat in the modal logic case more worlds have to come into lhe picture. Thus a proposition is necessary (in some world) not only if it holds wilh respect to this world, but also if it holds in every other possible world (or, at least.in every other possible world which is accessible to lhe one we're in). And a proposition is possible if it holds in some (accessible) world. As one can see, this kind of semantics malches well the old Leibnizian account of necessity and possibility. Before going on, lei me remaik Ihat the above characlerization of modal logics-as exlensions of lhe classical one-is obviously too restriclive. Actually it just applies to what one could labei classical modal logic (see [BS84J). According to this view, modal logics do nol try to subsliiute the classical one, just extend il and make it more powerful. But one could as well take anolher position, choosing as underlying logic a rival of the classical one: intuitionistic logic, for inslance, or relevance logic. If we now extend it by adding modal operators, we'll end up wilh, say, paraconsislenl modal logics, or relevant modal logics, and so on. (For relevant modal logics, see e.g. [AB75, Fu88|.) Thus we have seen that there are many other possibilities besides plain alelhic modal logic, so wilh yet anolher work in modal logics one won't necessaríly end up being a melaphysician.3 Nevertheless, before we jump lo the conclusion that alethic modal logics are prima facie metaphysical and hence uninleresting, let me remind you that this is absolutely nol lhe case: lhere are also severa! other inlerprelalions of "necessity" to choose from. One can of course talk about a metaphysical kind of necessity, conceming possible worlds, but "necessaríly" can also mean "according to the laws of physics" 3 T here is o f course noUiing wrong in doing M etaphysics, bul this w ord is often used as an accusation , th u s ... 3 ChapterO or "after the program Icmiinales", or "according to my beliefs" (cf. [FV85], p. 2; also [Go87), p. 6). We can even lalk about "historical necessily", for lhat mauer. Having thus leamed from lhese general remarks what modal logics are, let us lalk a little bit about their importance. To begin with, surely in philosophy: Some of the problems raised by modal logic seem to us to be among the most important and fundamental in philosophy, but it would require a separate book, and a very difTerent one from ours, to discuss them adequately. In our view there is also a link of a different kind between philosophy and modal logic, in lhat modal logic can be used to clarify a number of philosophical problems themselves (...) (IHC72), p. x) Among the problems raised by modal logics, the first is certainly the one conceming their own status as logics. Seen from the point of view of someone for whom there exists a thing such as The One And Only Tnie Logic, which is the classical, two-valued one-modal logics are no more than mathematical formalisms, maybe nice to play with but without real philosophical importance. Witness for instance criticisms such as Quine's, for whom modal logics, first, were conceived in sin-the sin of confusing use and mention; and second, they are of no use anyway, because everything one does in some modal logic can be somehow translated into the formalism of first-order predicate calculus; and third, there are serious philosophical problems in their interpretation-among which one could mention a controversy over the interpretation of quantifiers (objectual vs. substitutional), as well as an apparent commitment of modal logics to essentialism (i.e., the thesis that objects have some of their properties essentially). But letting aside this dispute, which, however interesting, is out of this work's scope4, modal logic's importance to the philosophical analysis of the notions of necessity and contingency should go without saying. Considering now what is outside philosophy's realm, there is hardly any denying of the essential role played nowadays by logic itself in Computer Science and artificial intelligence (henceforth "AI")-but what about modal logics, particularly temporal and epistemic ones? Since we are going to lalk a lot about epistemic logics in this work, I would like, before shifting attention to them, to say just some words about this other kind of modal logic, i.e., temporal (or tense) logics. There is again probably no need to stress their importance, at least not in philosophy: the theory of temporal logic is an integral concem of philosophical inquiry, and questions of the nature of time and of temporal concepts have preoccupied philosophers since lhe inauguration of the subjecl ((RU71], p. 1). Among lhe contributions temporal logics can offer are formal models of lime, which of course "provides the philosopher... with tools for achieving a bclter understanding of the nature of time itself ([RU7I], p. 1). In other areas, like C o m p u ter Science, the number of papers one can find dealing with, say, temporal logic of programs, is legion. It seems that computers, or at least logic programming, cannol d isp o s e of a temporal logic of a kind-witness the following quotalion (from the introduetion of a paper of James Allen's, in which he presents an interval-based temporal logic): 4 T he reader w anting Io know m ore about criticism s o f modal logics can consult [Ilck78], ch. 10. 4 Inlroduction A R o a d M ap The problem of representing temporal knowledge and lemporal reasoning arises in a wide range of disciplines, including Computer Science, philosophy, psychology, and linguistics. In Computer Science, it is a core problem of information systems, program verification, artificial intelligence, and other areas involving process modelling. ([ A183], p. 832) As an example, if we are concemed with planning the activities of a robot, il is necessary to consider lhe effect of the robot's actions in lhe world, if they are Io be effective. What involves lhe need to take changes into account, and changes obviously involve time. This is also emphasized in e.g. [MB83|: lhe authors S ta te lhat "most work in Al which deals wilh real world problems would require some reasoning with lime and space" (p. 343). Allen himself, in lhe mentioned paper, gives us more examples, such as dalabases which conlain historical dala-for example, if we are interested in modelling facts about lhe history of a person, we are bound to take time inlo accounL And so on. I'm not wanting lo go inlo details at this poinl and on Ihis subject, because, in spite of this work's title being very encompassing, not everything gels in. Temporal logics, for instance, are not mentioned-this work is about something else. The reader interested in this kind of modal logic can take a look at [G087], where more examples are discussed, and whose emphasis is on compulational matters, or at [Pr68] and [RU71], where more philosophical aspecls are considered. So let us getdown to the subarea of modal logics which is of special interest here (it does gel mentioned on lhe title): epistemic logics. 0.2 Getting epistemic First of ali, it goes withoul saying that epistemic logic deals with epistemic notions, namely knowledge, belief, conviction, and other similar proposilional altitudes. To put it in other words, epistemic logic is the kind of logic whose aim is "to explicate epistemic notions and to investigate the laws goveming them" ([Len781, p. 16). Conaretely, it is lhe kind of modal logic in which we inlerpret the box as "A knows that where A refers to some particular agent (which can be a human being, a robot, a knowledge base, a processor...). There is also a second side to this, namely the possibility of interpreting the box as "A believes/is convinced that in whicfy case we'd have a doxastic logic. The term "epistemic", however, usually covers both cases. And, instead of using lhe box, one commonly takes 'K' and 'B' to symbolize the desired operators. Somelimes lhese notational changes are the only ones we have: the axioms and inference rules of some alethic system are kept as paradigms. For instance, very often the modal calculus S5 is taken to be lhe logic of knowledge (e.g. in [HM84J), and weak S5 (a.k.a. KD45) to be the logic of belief. This briefly sketched situation describes only the case in which we consider a single agenL But it is common in AI to have situations in which one must consider a whole lot of interacting agents. So, if one has, say, 1,... jn agents, one has to introduce one operator K, for each of these agents. On lhe semantical side, when we now talk about possible worlds we are no longer having in mind some metaphysical sense of possibility, but rather what the agents think to be possible. The terminology 5 ChapterO "possible worlds" is even replaced by "epistemic altematives", meaning lhe different ways lhe world can be according to the agent. Thus an agent knows some proposition p iff p is true in ali worlds she thinks are possible. In a sense, we still are talking about possibility, but now a subjectíve one. As a side remark, there is a lot of discussion about whether one can believe impossible things, that is, whether only logically possible worlds are allowed to count as sound epistemic altematives, or whether we could, maybe, have some impossible ones, too. Opinions are greaüy divided. (More about this question e.g. in [Len78] or [Hi75].) Now what is, concretely, lhe importance of epistemic (or doxastic) logics, aside from a purely philosophical one? Well, their role is central to research in artificial intelligence, but nol only there: examples in economics, linguistics, Computer science, etc., are easy Io fínd. Let us keep to lhe AI case. According to Stanley Rosenschein, everything in AI has to do with knowledge. For instance, he states that lhe major subareas of AI can be described in a way that highlights the importance of the concept of knowledge. I quote: • Perception has to do with an agenfs acquiring knowledge about its environment by interpreting sensory input. • P lanning has to do with an agenfs acting on the basis of its knowledge of the consequences of its potential actions. • Reasonlng involves an agent's deríving conclusions from facis il already knows. • Learnlng involves incrementing knowledge through experience. • Com m unlcatlon (e.g. in natural fanguage) involves the continuai updaling of mutual knowledge possessed by the speaker and hearer. ([Ro85|, p. 3) So it seems lhat one cannot deny the importance of treating knowledge in AI-Rosenschein even speaks of the existence of a "knowledge indusiry" ((Ro85J, p. 3). Now, since epistemic logic is a topic about which this work is concemed, perhaps we could talk somewhat more about its importance by considering some more concrete examples. In doing this, TH closely follow a nice paper of J. HalpenTs [Ha86b], in which the author addresses these queslions. There he lalks about the importance of reasoning about knowledge in certain areas of research in AI, like distributed systems, logical omniscience, common knowledge, knowledge and 8Ction. (To these I would also add nonmonolonie logics.) 111 try lo characterize briefly lhe importance of epistemic matters in each one of these topics, what shall give us a litüe more of the flavor of this subject. / . Distributed Systems Distributed systems of computers, as one can grasp by taking a look at the spccialized lilerature. are becoming more and more popular and widely applicd. Such systems are used, for instance, to compute a protocol, which is "an algorithm whose execulion is shared by a number of independem participants" ([LR86], p. 208). More precisely, a distributed system can be characterized as follows: A distributed system consist of a colleclion of processors, say 1 connected by a communication network. The processors communicate which each olher over the links in the nclwork. Each processor is a staie machine, which at ali times is in 6 Intraduclion <t Road Map some slate. This State is a funclion of the initial State, the messages il has received, and possibly some internai events (such as lhe licking of a clock). (|Ha86b), p. 5) In other words, we have the dilTerent participants in the system compuling different tasks, and, conuary to sequential or parallel processing (where processors share the same memory), each player doesn't necessarily know what the others are doing, even if, in fact, they are exchanging messages ali the time. (This is for inslance a reason why lhe former logics of programs are inadequale when we reason about lhe behavior of protocols. Cf. [LR86], p. 208.) Now this property- Ihat players are not necessarily aware of what lhe others are doing-characlerizes jusl the lack o f knowledge from each player wilh respect to the lotai state of the system. According to [Ha86b|, the nolion of Icnowledge at slake here is an "externai" one, meaning il is not lhe processor who thinks ("and scratches its head") about whether or not il knows somelhing, but it's rather a programmer, from an ouLsitle point of view, who says that the processor knows, or not, some fact. Even lhough, one cannot dispute Ihat reasoníng about knowledge is a very important characteristic of distributed systems. Quoting from [LR86], "any logic of protocols must include as part of il a logic of knowledge" (p. 208).3 Now talking about knowledge in situaüons involving more than one agent involves a lol of "subüeties" ([HM86], p. 1). The point can be better illustraled by lhe following puzzle o f lhe muddy children ([HM86], p. 2): Imagine n children playing together. The mother of lhese children has told them that if they gel dirty lhere will be severe consequences. So, of course, each child wants to keep clean, bul each would love to see lhe others gel dirty. Now it happens during their play that some of the children, say k of them, gel mud on their foreheads. Each can see lhe mud on others but not on his own forehead. So, of course, no one says a thing. Along comes lhe falher, who says, "At least one of you has mud on your head", Ihus expressing a fact known to each of them before he spoke (if k > 1). The father then asks the following queslion, over and over: "Can any of you prove you have mud on your head?" Assuming lhai ali lhe children are percepüve, intelligent, tmlhful, and thai they answer simultaneously, what will happen? There is a "proof" thal the firsl k 1 limes he asks lhe queslion, they will ali say "no" but lhen the h*1 lime lhe dirty children will answer "yes". I'm nol going lo discuss lhe "proor' here; lhe reader is referred to [HM86], where the problem is examined in deiail. Now one of lhe "sublleties" this puzzle is suppose to illuslrate is lhe following: since what the father said was already known by lhe children, il would seem that his statement wasn't needed at ali. Bul this is not the case, lhe proof won't go withoul it ([HM86], p. 2). Thus, before and afler the father's statement, we have Iwo different situations wilh regard to what the children know. The difference involves the lopic we are going lo mention nexl: after the fa(her's statement, lhe children have common knowledge. 3 By the w ay, they State that a log ic o f tim e is a lso necessary. 7 Chaplr.rO 2. Common Knowledge Anoüier theme that very often appears is discussions of knowledge, in particular in cases (as the one before) where more agents are involved, is the noüon of common knowledge. To put it short, we say that a certain group of agents has common knowledge of a certain fact p not only (as one could think) if every member of the group knows p , bul also if everybody knows lhal everybody knows that p , and if everybody knows that everybody knows that everybody knows that p, and if everybody knows ... That there is a big difference between the two situations is one of the points in the muddy children puzzle above. Without the statement of the father, even if every children knows that at least one has mud on his/her forehead, they don't have common knowledge. Now, is common knowledge interesting? Can we flnd applications of it? Sure. It seems that the nolion of common knowledge is essential to the notion of agreement- " 'agreement* implies common knowledge of lhe agreement" (lHa86b), p. 10). We can see ihis clearly in the next example of the coordinate attack problem ((HM86), p. 6): Two divisions of an army are camped on two hilltops overlooking a common valley. In the valley awaits the enemy. It is clear lhal if both divisions attack the enemy simultaneously, they wiU win the battle, whereas if only one division attacks it will be defeated. The divisions do not initially have plans for launching an attack on the enemy, and the commanding general of the first division wishes to coordinate a simultaneous attack (at some time the next day). Neilhcr general will decide to attack unless he is sure that the other will altack with him. The generais can only communicate by means of a messenger. Normalty, it takes the messenger one hour to get from one encampment to the other. However, it is possible that he will get lost in the dark or, worse yet, be captured by lhe enemy. Fortunately, on this particular night, everything goes smoothly. How long will take them to coordinate an attack? [HM86] show that, dcspite the fact that in the said night everything goes smoothly, it is impossible for the two generais to reach an agreement and coordinate an altack (p. 6). It is not difficull to sce why: the first general will not attack unless he or she knows that lhe message proposing a joint aclion was delivercd, and unless he or she knows that the other general knows that his or her acknowledgement of lhe first message was delivered, and unless ... Well, unless there is common knowledge that an attack is going to happen. As a side remark to this, the aulhors in [HM86] show that "not only is common knowledge not attainable where communication is not guaranteed, it is also not attainable in systems where communicalion is guaranteed, as long as there is some uncertainty in message delivery lime" ([Ha86b], p. 10). This also holds for humans-think for instance of how often, and under which dimcutl conditions, do nations reach agreements... 3. Knowledge and Action It is common, I think, that examples inlending to illustrate some point end up throwing light in more than one. In lhe previous example of lhe coordinate attack, not only common knowledge is at stake, bul it also involves communication and acling upon having knowledge. Knowledge and aclion, for instance, are 8 Iniroduciion <t R o a d M ap cracially intertwined: "Knowledge is necessary lo pcrform actions, and new knowledge is gained as a result of performing actions" ([Ha86b|, p. 11). Witness also lhe quotatíon from [Ro83] above, conceming planning: an agent acLs based on the knowledge of the consequences of its actions. Discussions of this lopic can be found in [McH69], and also in a paper by R.C. Moore ([M08I]), where he introduces a logic combining knowledge and action. The main point is that knowledge alone, or reasoning about knowledge alone, is of liltle value-mostly we are interested in having infonnalion about what we can do with the knowledge we've goL To mention the example discussed in [M08I], if there is a safe that John wants lo open, we mighl make the following inferences: if John knows lhe combinalion, he can immedialely open the safe. Or, if he docsn'1 know the combination, but know where it is writlen, he can read the combination and lhen open lhe safe ([M08I], p. 473). As one can see, if John doesn't have knowledge about the combination, his next preoccupations could well be how to obtain this knowledge, and which actions are necessary for that.6 4. Logical Omniscience After having stressed (I hope succcssfully) lhe importance of lhe notion of knowledge in different topics in AI, one should also mention some problems conceming the way Ihings are being done. 1 said over and over again that epistemic logics are important. Now, are lhe ones we have really good for the uses we have in mind for them? Yes and no. That is, there is a problem with lhe way people model knowledge, and the key words here are "logical omniscience". If one would take a day out to dive in the literature conceming epistemic logic, one would surely notice lhat people have been and are still talking a real lot about this problem. One would also notice lhat most aulhors jusl avoid the problem ("we consider in this paper only agents having very powerful reasoning capabilities..."), or lhey accept logical omniscience as a kind of malum necessarium, that is, somelhing unpleasant you have to live with if you want to have a logical system at ali. Not so often (but it has been changing in lhe last years), people do try 10 ftnd a solution to the problem. To presenl things in an informal way, an agent A is considered to be logically omniscienl if she knows ali logical consequences of her knowledge, among which, by lhe way, are ali "logical irullis". In other words, if A knows that A, and moreover A is a logical consequence of A, lhen A knows that B, too. Or take A to be a logical trulh, e.g. a lautology: lhen A also knows that A. Now this presentalion is surely very rough-lhere are many other, more rigorous formulations of the problem-but it will be enough for our motives here. (A more detailed presentalion of several different "formal encodings" of consequential closure principies can be found in |Len78], pp. 53ff.) More ofien than not, this situalion conceming an agent's reasoning capabilities is considered to be a plague. I am not wanting 10 commit myself here and yet on this lopic-one should, first of ali, betler check out whether this situalion is really a Bad Thing, or somelhing ihat's not really that serious. But anyway, one should at least have the possibility of making choices, lhat is, it would be nice to have different epistemic-doxastic logical systems in which one can have or nol, as one likes, logical omniscienl agenls. This is exaclly the rcason why logical omniscience is seen as a plague for lhe possible-world semantics: ^ O ne o o u )d n 't hav e m ade th ii p o in l m c n precisely Uian " S lippery i im " d iG ríz, lhe S ta in le s i S teel Rat: "M oney w u w hat I w anted . O th e r pcopte*i m oney. M oney is lockcd aw ay , so lhe m ore I knew abou t locks lhe m ore I w ould be ab le to gel this m oney." (H arry H arríson , A S ta in less S tee l R ai is B o m . N ey W ork, B aniam Books, 1988, p. 12.) ChapterO you don't have a choice. Because of the way this semantics is built up, agenls end up being logically omniscient-or so they seem. Remember above, where we said ihat A knows p iff p is true in every world (epislemic aftemative) Ihat A ihinks lo be possible? Well, worlds are supposed to be logically consistent- they are logically possible worlds-so tautologies are bound to be true in every conceivable one. Hence, modelling an agent's knowledge in this way is lo assert, right from lhe beginning, that she's going to be logically omniscient. There's simply no world in which fl taulology A will be false, so that we can falsify K \A loo. There are of course tentatíve solulions, even if nol very satisfactory ones-sometimes one ends up wilh mixings of syntax and semantics; or one finds out that agents are no more logically omniscient with regard to the classical logic, but they are, say, in some relevance logic, what doesn*t looks much betier and also doesn't seem lo agree wilh our intuitions. But see for instance (Hi75, Lev84, Va86) on the problem. And more about this on [Ha86b], which also mentions additional bibliography. 5. Nonmonoíonic Logics To close this general introduction and background painting, I have to say a few words aboul nonmonoíonic logics (henceforlh NMLs), not only because one of the approaches to the formalization of nonmonoíonic reasoning makes use of epislemic logics, bul also because our work touches marginally upon these mailers, or betier this work's stariing point arose from some problems in epislemic logics that deal wilh lhe formalization of nonmonoíonic inferences. But what exactly are NMLs? Or, for that maller, what is monolonicity? À characteristic of classical logic-to tell lhe truth, also of a lot of its rivais-is the following one: once you have carried out a valid inference, nothing you may possibly add to your premisses will ever change lhe validity of this inference (even if you add lhe negation of the conclusion, because in this case lhe premisses would be inconsistent and anything goes). Now this is what monotonicity is about: by adding premisses you can gain information (in the form of new conclusions), but you lose nothing. A litile more formally, if p follows from some set Y of proposiüons, monolonicity guarantees that it also follows from Y augmented by whatever set of new proposilions you like. Well, if one now ihinks about how things should ideally be, inferences being monoionic seems to be something desirable in logic. However, more often than not, we humans are confronied wilh siluaiions in which we have to refrain from previously derived conclusions. Consider lhe following proposition ([Gi87], p. 2): birds íly. It is obviously not the case that ali birds fly, but they normally (typically) do. Now if someone tells you that Tweety is a bird, and you know nolhing else about Tweety, you'II gladly jump (or even fly) to the conclusion that Tweety ílies. Bul suppose aflerwards you leam that Tweety has its feei set in concreto, old Chicago style: thcn you are not anymore ready to assert or believc that Twecly flies. So lhe inference from "Birds fly" and "Tweety is a bird'* lo "Tweety flies" was a nonmonoíonic one: upon leaming new information, we have lo reiract ihis conclusion. Aclually lhe inference relied more on the fact that typical birds fly, and, in lhe absence of conlrary information, you assumcd that Tweety was a typical bird, from what you refrained upon leaming of its predicament. Now there are severa! differenl ways of doing nonmonotonic reasoning, or, lo pui it belter, of trying lo formally capture such inferences-like default logics, or circumscriplion. I won't discuss ali them 10 Introduction A Road Map possibiliües here, but I have to mention lhe one among these which is of some interest to us here: the socalled "modal approaches" (cf., also what follows, [Gi87], pp. 8-9). According to Ginsbcrg, the first ones who tried to use a modal logic to model nonmonotonic reasoning were McDermott and Doyle [McD80], in which they used a first-order logic augmented by a modal operator M, which should mean "maybe" or "is consisient with everything else that is known". If we take ' b ' to symbolize 'Birds fly \ and */ to 'Tweety flies\ our example inference could be then formalized as 6 a A / / - > / . This M operator, however, is not entirely without problem s. According to R.C. Moore, it characterizes so weak a "notion of consistency that, as [McDermott and Doyle] point out, MP is not inconsisient with -«/*" ([Mo83], p. 128). Moore set out to change this himself, what he accomplished by introducing an autoepistemic logic, in which he changed this weak consistency operator into an epistemic necessity operator ("it is known" or "it is believed")So the example inference would now be formulated as b a -rL-i f - > f . With such an approach we reach then a State of things where also epistemic logics can make an important conlribuüon to the formalizations of nonmonotonic reasoning. But this is ali what I wanted to say about NMLs here. I hope I could have made clear the importance of epistemic notions in Al and Computer Science, and also that I succeeded in giving an idea of where our work is going to fit in. So now let us get down to our specific research problems here. 0.3 Down to specifics The main interest of this work-its leading thread, or, at least, where things begin-comes from epistemic logic, more precisely, from the lentative of characterizing minimal belief siates. The problem can be traced back to a paper of J. Halpem and Y. Moses* called "Towards a theory of knowledge and ignorance" ([HM84]). In said paper the authors (henceforth HM) consider the problem of characterizing the knowledge State of an agent A in situations where A has only partial In fo rm a tio n about some domain, that is, when A knows only some formula a . In lheir paper they assume that reasoners are logically omniscient, 1.e., that they are perfect reasoners conceming propositional logic: they know ali logical consequences of what they know. Besides, reasoners are also thought to have perfect introspective knowledge about their own knowledge or ignorance: they are completely in clear about what they know and what they don't know. As a consequence of these assumplions, the characterízation of a knowledge State is a non-trivial matter. Suppose, for example, that A(ngela)7 knows only p\ she can discover by introspection that she doesn't know q and thus she knows she doesn'i know it. This entails that something more than just the logical consequences of knowing a belongs to an agent's knowledge State, and so any attempt to 7 It s a litü c b it boring and cum bersom e to speak ali the tim e abou t "an agen t A ", so 1 p rc fer to give her a nam e, like A ngela. T h ings look n ice r th is w ay, too. F o r th e reason w hy w e 're ta lk ing abou t fem ale agents, c f L azarus L ong: " M en are m ore sen tim ental than wocnen. It bturs their th in ldng ." 11 ChaplerO characterize such a State will have to take this fact into consideralion. To cope with the problem, HM present different characterizalion methods, making use for instance of Kripke models, stable seis, and so on. They also introduce the notíon of an "honest" formula, namely of a formula that uniquely characterizes Angela's knowledge S ta te , when this formula is everything lhat she knows. As an example, or rather as a counter-example, lhe formula a = Kp v Kq is not honest, because an agent cannot know a without knowing eilher p or q. On the other hand, Kp a Kg is honest. HM present several ways of defining honesty, and they are ali proved to be equivalenL An algorithm for deciding about the honesty of a formula is also given. So far so good, but the logic used in [HM84] is propositional S5, and, as I mentioned, this implies that agents are supposed to be logically omniscienl and fully introspeclive. What can be just fine for a lot of 8pplications, but using S5 as a logic of knowledge will give us some problems the moment we try to formalize in it nonmonotonic inferences, for instance, default reasoning. Loosely speaking, a default rule could be descríbed as follows: q (which is called lhe default) is true. unless one knows that p is true (cf. [HM84J, p. 17). Formally, -»Ka/> -> q, where 'Kaj>' is to be underslood as 'Angela knows lhat p \ Now according to HM the formula -iK \ p -* q is itself not honest, if p and q are propositional variables; further, the authors State that such a formula doesn'i behave at ali like a default nile: In fact, for an honest a , [-»KaP -> <7] is a consequence of 'knowing only cr' exactly if one of p or q is flHM84J, p. 17). As a consequence one should, in their opinion, eilher give up on the hope of having consistent nonmonotonic default mies, or else give up on SS as an adcquate logic for modclling knowledge. HM, however, would like to preserve both, so they suggest, as a possible way of escaping this dilemma, lhat default mies could perhaps be better formalized in an epistemic-doxastic logic, namely as formulas of lhe form -» q), where 'Ba* slands for 'Angela believes lhat They justify this suggestion by saying that it is not our knowledge or ignorance of p that makes q true, but it is our information regarding our knowledge-gathering capabilities that leads us 10 believe q in lhe absence of our knowledge of p ([HM84], p. 17). (Under "epistemic-doxastic logic"-EDL for short-we undersland a logic system in which both concepLs, knowledge and belief, are contemplated.) So, to begin this work, and to set its main and more importam goal, I will follow their suggestion an d try to characterize "minimal epistemic statcs" in different EDL-calculi. In other words, I'll be looking for ways of describing Angela's epistemic (i.e., knowledge or belief-we'll decide it laler) stale under lhe supposition that she knows or believes only some formula a. In doing this I won't stay resiricted to the only EDL-system proposed by HM (since this logic's "knowledge branch" is S5, and I am not lhat convinced that S5 is lhe best option in formalizing knowledge), but 1*11 ralher try to work with several calculi of different strength. Thus, in Chapter I, we ll have an overview of some epistemic-doxastic logics. I will introduce several systems which we'll be working with, giving for each one an axiomatic presentalion. This will be accompanied by a small discussion about lhe tenability of the various epistemic-doxastical principies involved. Also in the syntactical part are includcd some results about lhe number of modalilies and the 12 Introduction & Road Map existence or not of reductíon laws wilh respecl to each system, results which will prove to be useful later on. In the semantica) part we'll introduce a possible-world semantics for our logics, after what correctness and completeness lheorems will be proved. Chapter 2 will be devoted to lhe characterization of minimal epislemic states, in which I'll try the different approaches already employed by HM. A first trial employs stable sets, which we'll use to represent epislemic staies. A next, short section will esiabtish some relation between slable and salurated sets (which are maximal consistent), giving an altemative to our characterization problem. A third approach will rely on Kripke models-that is, on possible-world models; and last (but not least) an algorithmic approach. We'll see that each of lhese meihods yield different results, depending on the logic being considered-sometimes they work, and sometimes don't. Nicely enough, the algorithmic one will prove to be the most general of them. Now, to talk a little bit more about the importance of this enterprise, I would like to mention that HM's original motivation arose from the queslion of "how communication in a distributed system changes the state of knowledge of the processors in the system" ([HM84], p. 1). I already remarked above, speaking of distributed systems, that the players in the system suffer under a lack of knowledge conceming what the remaining players are doing. This is an example of a situation where agents only have partiat information at their disposal. In other words, everylhing they know (or believe) can be described by some formula ol The question of how to characterize an agent's knowledge state in such a situation comes from our intuition that there must be one, and only one, of such states, which fully describes whai lhe agem knows (or believes). In our case here, where we consider both knowledge and belief, we have the additional motivation that results could also be of use for the formalization of nonmonoíonic reasoning. As I was saying, then, the algorithmic method is the one which will prove to be the most fruitfiil. In the case of HM's paper, the algorilhm relies in a decision procedure for S5, which was the knowledge logic assumed there. Here we'll have obviously to examine decision procedures for each of the epistemicdoxastic logics we are considering. This new goal makes the connection to the second part of the present work, in which I consider valuation semantics and generalized truih tables for alethic modal logics. I hope I have already made a case conceming lhe importance of alethic modal logics in the preceding sections of this chapter. At lhe risk of repealing myself, the structure of modal logics and the EDLs considered here is very similar; thus we can adapt results in alethic modal logics to lhe epislemic case. And 1*11 be taking a look at valuation semantics because they easily yield decision procedures. We*|l thus begin Chapier 3 wilh a short and informal introduction to valuation semantics, trying to give the reader a first, iniuilive idea of what they are about wilhout jumping immediately to the deOnitions. In the following scciions of this chapter we'll be examining the construction of such semantics for normal modal logics. These are, so to say, the mosily known among the modal logical systems, including landmarks such as T and lhe Lewis systems S4 and S5. We ll see that, for some of these logics, valuation semantics are (somewhat) easy to find, whtlst for others we slill are confronled wilh open problems. In Chapier 4 will take care of classical modal logics, where here "classical" is not being employed in lhe sense I did some pages before, lhal is, meaning ali modal logics which extend lhe classical one. In the sense of Chapter 4, classical modal logics are certain subsystems of lhe weakesl normal one (which is K). These logics are, in a sense, of no greater importance to lhe main goal of this work, since the EDLs we'll be considering are ali normal, but it is nice to see in which way valuation semantics can be defined for other kinds of modal logic as well. 13 C hapter 0 In Chapter 5 we will take a look at the main byproduct of valuation semantics, which are generalized trulh-lables. These are, similar to the trulh-lables for the classical proposilional calculus, constructs which allow us to decide on the validity of a formula by examining the value it gets on different assignment of truth-values to its proposilional variables and, depending on this, to ils modalized subformulas. In other words, they are a melhod ofhaving truth-tables-sort of-for modal logics. We'll show how by means of an example logic, K. And in the next chapter, which is number 6 , we'U briefly look at how lo obtain generalized truth-tables for one of lhe "problematic" normal logics of chapler 3, namely S4. Having worked with generalized trulh-lables, one has lhe feeling that there are some similitudes to another decision procedure for modal logics, the tableau systems. So in Chapter 7 we'll have a small comparison between lhe two methods, showing lhal, lo use a metaphor, they are lhe Iwo sides of a coin- as are truth-tables and tableau systems for the classical proposilional logic. Finally, in Chapter 8, which closes the second part, we'll retura to our main interest and show how to adapt valuations semantics, and hence the construction of generalized tmth-tables, to the epistemicdoxastic logical case. We will take then an EDL as example, and adapt for it the whole procedure. As we'U see, this can be done in a more or less straightforward way: some semanlic condilions are automatically given, because now ali we have is for instance "Angela knows" inslead of "necessarily". The condilions, however, lhal take care of lhe validity of "mixed" axioms-the ones involving knowledge and belief-are patticular lo EDLs, thus posing new problems. The reason is lhat in EDLs we have, so to say, two strong operators-i.e., which behave like necessity-and no weak operators. The opposite of your run-off-the» mill alethic modal logic. The Ihird part of this work concems the "and their applications" part of lhe title, what redects my interesls in programming issues, in particular the implementation of theorem provers for modal logics. By "theorem proving" I mean of course automaied theorem proving (henceforth ATP), which also goes by the name of "automaied deduetion". The big interest in ATP developed only in this century, with the Age of lhe Digital Computer, what gave researchers means of Irying oul their lheorelical considerations. The ideas behind the ATP enterprise, however, are quite old, lhe automalion of reasoning, or mechanizing or thought, being something many a philosopher or scientist dreamt about. Following M. Davis, we could say lhal the fundamental stone in lhe history of lhe mechanization of human thought was laid by Descartes with his employing of algebraic methods to develop classical Greek geometry: "what had seemcd in Euclid lo be lhe result of cunning and mathematical ingenuily was now revealed as being accessible to relatively mechanical treatment" ([Da83J, p. I). Descartes himself seemed lo be quite aware of this, bul the dream of doing for ali deduetive reasoning what he did for geometry was really bom in the works of Leibniz, with his ambitious projects of a calculus of reason (calculus ratwcinalor) and of a universal language (characteristica universalis). These projects, unfortunately, were never actually developed, for Leibniz had also many other interesls: from lhe calculus of reason we have some fragments, bul lhe universal language remained really a dream (cf. (Da83), p. 3). As I said, lhe real "boom" of interest in automaied deduetion really began in Ihis century* Speaking of applications, there is of course, if one can say so, a more "thcory orientei!" side of this research: one would desire powerful ATPers in order to gain more knowledge in mathematics-either by obtaining new, maybe shorter proofs of known theorems, or even by proving proposilions which now slill * A short bu t ctearly arranged h istory o f au tom aied deduetion- at least untit the end o f the 6 0 's - can be found in D av is 's paper (D a83|. 14 introduction A R o a d M ap have lhe staius of conjecturcs. ATPers would be ihus helping the progress of Science. On a more "praclical" side, again if one can say so, good ATP techniques can be used by Computer scientisls to prove properties of programs working on axiomatized structures (cf. [Ga86], p. 3). Nol to forget applicalions in logic programming'. let us consider the declarative language Prolog, for instance. A Prolog program consists in a sei of facts and mies, that is, in a set of assertions, and a Prolog computalion is in fact a proof, from which a p ro g ra m 's output is to be extracted. It goes withoul saying that effícient proof techniques are vital to efficient implementaiions of Prolog. Speaking of this, there are extensions of the Prolog language which introduce modal or temporal features, in which case ATPers for modal logics also play an essenlial role when ít comes to implementaiions. Besides this, ATP techniques are of importance also in database management. It should be obvious that one cannot explicitly encode in a database ali possible facts: a lot of them will have to be impliciL Take for instance the true proposition "the Earth has only one sun". It is also true that "the Earth doesn't have two suns", "lhe Earth doesn't have three suns", and so infínitely on. Since il is impossible to store ali these propositions explicitly, one has to make use of inference techniques to deduce such information. This is what also happens in the field of knowledge representation-e.g. conceming the knowledge base of some robot. One just cannot use memory to store every single fact such a robot knows or believes. Because- even if we set aside examples like lhe preceding-the robot is inieracling wilh its ambient and "leaming" new facts. In order to behave inielligcntly, it has to be able lo draw inferences from pieces of information he gathers. So it makes sense to have some mechanism-and an effícient mechanism would be even betier-which allows one to deduce new facts from facts already stored. This is where ATP techniques come tnto picture. Thus, in the third part of this work, we'll try to go lo the practical side of what we have so far discussed. In Chapier 9 we'U examine a program implementing generalized iruth-tables for the example EDL of Chapter 8-a rather "naive" program, but reflecting wilh fídelity the definitions. The next Chapter, Chapter 10, will try to optimize this situation presenting a theorem prover which is an improved version of our fírst program, by using tableau proof lechniques. As I remember mentioning before, generalized trulhtables and tableau systems can be seen as the two sides of a coin, so it is nol surprísing that, once we have GTT definitions for some logic, we can tum them around and generate tableau systems. I cannot go into much details here, because first we will have to see how, exacily, do our generalized truth-tables function. Finally, in Chapter 11, we'll implement, using the ATPer from Chapter 10, the algorithm used to characteríze mini mal belief states. 15 I Minimal Belief States in Epistemic-Doxastic Logics n An Overview of EDL-Systems / will now show offalmost ali the Greek / know: "epistemic" has to do with knowledge; "doxastic", with belief. So in what follows we shall have to do with logics o f knowledge and belief. D. IsRAEL, A Weak Logic o f Knowledge and Belief. In this chapier I make a presentalion of lhe epistemic-doxastic logics we are going to work with. We consider modalilies and reduction laws, a possible-world semantics, and prove correctness and completeness. 1.1 Enter the logics We'll use in this Part 1 a propositional language L which also includes operators for knowing that and believing that. Small lellers (p, q, r,...) will be used as propositional variables, whereas capital letters, italicized, (A, B, C,...) will stand for syntactical variables for formulas. 1*11 also be occasionally using small greek letters as metavariables in some special cases, namely for formulas which denote everything that Angela knows or believes (like in "suppose a is everything that Angela believes"). Since we are going to keep to the one-agent case, 'KA1 and 'BA' will be used as abbreviations for 'K*A' (lhat is, 'Angela knows that A') and 'B\A ' (lhat is, 'Angela believes that A'), r e s p e c tiv e ly .a n d *->' are introduced as primitive; the other boolean operators 'a ', 'v ' and *«-»* are defined in tenns of these in the usual way. 'FOR' stands for the set of ali formulas of L. We begin by considering a basic EDL9 (at least in the bounds of this dissertation), which we will call Z. Actually, if we would follow lhe (more or less) standard way of christening modal logics (like in [Ch80]), this system should be named somelhing like 'K T4K bDb4b5bM \ where the 'K T 4 ' part refers to lhe knowledge branch, 'KbDb4b5b' lo the belief one, and 'M ' to one "mixed" axiom (as one can see 9 I II use lhe expressions " E D L " . "E D l.og ic" , "liD L -ca lcu lus" and '*HDI.-ayslcm" as synonym ous Ihioughout Ihia worlc. 19 C hapler I from lhe axiom listing below). However, since we are here also going lo consider several extensions of Z, names would be growing and growing, so lei us agrce {non surti prolonganda nomina protter necessitatem, or words lo lhal effecl) that 'Z ' stands for KT4KbDb4*,5bM. Thus, in this work, Zxyz will denote the extension of Z by adding schcmas x, y, and z as axioms. Now Z has the following axioms: pc. Ali tautologies of the classical proposilional logic. *. K(A B) -> (KA -> KB) t. KA A 4. KA -* KKA (positive K-introspection) k*>. B(A -> B) (BA -> BB) df>. BA -» -iB-k4 4b. BA -> BBA (positive B-introspection) 5b. -«BA -> B-iB/4 (negative B-introspection) m. KA -> BA As rules of inference we have MP. tA , tA -> B / hB RK. lA / KA. As deríved inference niles we can also have RB. k A / iBA RKE. A o B I \KA f > KB RBE. i A < - > B / t B A f - > B / ? Let us now talk a bit about this axiomatization. The reader has surely noticed that the schema 5. ~«KA K-.KA (negative K-introspection) doesn't belong lo the axiom basis, as it was perhaps lo expect. There are two reasons for this, namely: (1) if we are not considcrihg ideal agents (with regard to their introspective powers), 5 is clearly not valid (at least for human agents it is not, on what everybody seems to agree); and (2) if we put 5 togethcr with some harmless-looking, acceptable EDL-principles, we get as a consequence lots of trouble (in the form of some nasty theorems, what l 'lt be showing soon enough). But o f course we can take ideal agents in consideration, and thus extend Z by adding 5 as a new axiom. This resulting extension of Z we will call Z5 .10 Z5 is actually the system HM suggested, but they mcntion it in a slighüy different axiomatizaüon- instead of m we would have the following axiom schema: W I am not go ing here and yel lo en te r the discussion Ideal vs. N ot Ideal agen ts, even i f l 'd like to d o th a t H ence w e 'll be suffering in this woric o f logical om nisc ience and sim ilar troubles. 20 An Ovtrview o f EDL-Systems m*. KA -> bKA. We could call this other system Z5*. but it is not difficult to prove that both axiomatizalions are equivalent, that is, they axiomatize lhe same logic. (See Appendix Al for a proof of this claim.) Now, lest the reader Ihink lhat 5 is the villain of this story, it should be remarked loud and clear that not ali the other axioms are accepted as evidenL In fact, it seems that not a single one of them is free of criticisms for one or anolher reason. For the sake of completeness I am going to list some arguments againsl the different axioms. Let us begin for instance with k and its belief instance ifi: these formulas can be said to embody one version of the infamous principie o f consequenlial closure, which is just another way of spelling "logical omniscience" (or "logical omnibelief").1 1 Letting these consideralkms apart, since we are taking for granted that Angela is omniscienl, there are still other reasons-or what some researchers Ihink to be reasons-why these principies shouldn't hold. t*. for instance, should be plain false, if we interpret 'B' not as conviction, bul as a general, weaker kind of belief. in this interpretation, Angela believes some proposition p if she thinks p is more probable than its negation (cf. [Len78], p. 36). The argument againsl the validity of kP is based on the louery paradox, because we can show that kb is equivalent to BA a BB -> B(A a B), againsl which principie this paradox is directed. Suppose we have a loltery with, say, 1000 tickets, and le t4 W(n)' stand for "Lottery tickel number n is the wining one". Suppose also lhat Angela is buying a ticket. Since it is obvious that each ticket has only a very small probability of being the winning one, we can say Angela believes, for any n, that not-lV(n). More formally: (*) B-ilV(i) a B-»W(2) a . . . a B-, W(1000). On lhe other hand, it would be false to State that Angela believes lhe conjunction of these negated proposilions: (*♦) B(->IV(/) a -.W(2) a . . . a -^WVOOO)), and this because she's buying a ticket; she is quile sure that some ticket musi win (assuming it is a fair lottery, of course). We thus have a situation where Angela believes several proposilions taken isolatedly, but not their conjunction. This only holds, of course, because we are here talking about weak belief- Angela is far from being convinced, of each ticket, lhat it won'i win. Were ihis the case, she wouldn't obviously buy one. So the conviction analog of kP holds, the same for k, lhe knowledge version. However, it seems that one still could make a case for the validity of £*, lottery paradox just lhe same. A possible way out of the predicament would be to say lhat even if we can believe, for any n, lhat ticket n won't win, this is not lhe same as believing it for ali n (cf. [Har86], p. 71). There are also some other lentatives of rejecting these principies, most running along lhe lines that someone knows some facts, and fails lo draw its consequences-for instance with one of Ihcse logic puzzles that usually come in magazines. A reader can be said to know ali lhe premisses, bul mostly he or she needs a lol of lime lo arrive at the solution-if at ali. This kind of example is actually nol so good, because probably lhe agent doesn't know ("see") lhat A -» S, so it would be improper to assert K(A -» B). ' * T he ten n in o lo g y " p rincip ie o f consenquential c lo su re" is due lo K. K onolige (cf. [K o 8 ó |, p. 242). O n logical om niscience, sec lhe rem akrs on C hap ler 0 o f th is w ork. T here are o f course m any o lh e r princip ies w hich e n u i l log ical om nisc ience , bul they are som ew hat beside the po in t here. A good discussion can be found in both [Len78] and [I.cnSOJ. 21 C hapter l Thus, everything considered, it seems hard to deny that some agent knows Ihat p, and that p implies q, and nevertheless "fails to apply modus ponens" (cf. [Len78], p. 65). Against 4 and 4b there are also some proposed counlerexamples. In lhe case of belief, lhey moslly refer to phenomena like subconscious beliefs and someone's not recognizing or even repressing such. (4b was even acused of being "a short rejection of Freud". Cf. [Len78], p. 71.) Thus we have lhe atheist bishop example, which concems a bishop who losl his failh. He believes Ihat God doesn't exist, bul cannot admit il to himself-Ihus he doesn't believe ihat he believes that God doesn't exist. The problem wilh this "counterexample" is, of course, the confusion between subcounscious beliefs and conscious ones. You can nalurally choose which kind of belief 'B' is going to formalize, but lhen you've got lo be consistent, whal is not the case in lhe proposed coulerexample. Anolher proposed counter-argutnent states that, if I know Ihat p, lhen I know that I know that I know that I know Ihat I know that p , which is certainly somewhat unnatural. Bul the argument misses the point, because unnatural doesn'l mean logically wrong. Besides, in certain systems, where we have reduclion laws, such long iterations of epislemic operators can be proved to be equivalent lo shorter, "natural" ones. Similar attempts have been made to refute 5b, moslly making the same mislakes (cf. [Len78], pp. 77ff). Its knowledge version, 5, is, as I said, false, for lhe very simple reason Ihat we commonly believe, or are even convinced that we know something, when it's nol true. 5 would imply that we always can tell when we don't know something, and this is of course desirable, but highly unrealistic. Against db it is said that il rules oul incoherent or impossible beliefs, which many people (me for instance) seem to find desirable in certain contexts. Not in lhe sense Ihat someone would or could believe a downright conlradiction, like A a -A , but maybe believe a set of facls, which, in lhe long run and in a non-obvious way, proves to be inconsistenL The less dispuled of lhe axioms is f, but even so lhere are some people who think it lo be false, i.e., lhey defend one can know falsities (in which case I am prepared lo concede Ihat lhey know Ihat t is false...) Moslly lhe arguments use the facl Ihat we oflen "know" things that prove afterwards 10 be untrue, and generally lhere is some confusion between knowledge and knowledge claims. And last a remark conceming lhe schema m (Kj4 *B,4). Roughly speaking, this means that whal Angela knows makes part of her beliefs (she believes whal she knows). Putling things this way does lead to some confusion, and it is exactly on this confusion ihat some arguments against this principie are based. For instance, "I don't believe I'm married; 1 know il!" is lhe classical example. What is here at slake is, of course, a merely believing-I don'l merely believe Ihat I'm married, but of course I believe it. Bul one perhaps would like-and as a logician one should certainly try-lo keep both conccpts separaled, in the sense that when one knows something, one doesn'1 actually believe il-one already knows it! We can introduce this concepl without any problem in lhe calculi Ihrough lhe following definition: B*A =df B/l a - .M . But lhere are still other lentatives of rejccting this principie. Some of them (which in |Len78) are called "linguistic") concem lhe dirferent uses of know and believe (|Len78), p. 24). For instance, it is entirely appropriate lo say Ihat I know Frankfurt, I know lhe name of the Bundeskanzler, and so on, but it doesn't make much sense to say thal I believe Frankfurt, or Kohl's name. So m should be rejecled. Bul m 22 A n O verview o f E D L -S y tíem s actually has to do with "knowing that..." and "believing lhal..." Finding out whether things like "I know Frankfurt" can be reduced to that-ciauses is an open question.12 Another proposed argument against m considers an examinee who, being asked for certain historical dates, such as when did James I die, and being unsure about his knowledge, answers with " 1635" while believing it is wrong date. It tums out, however, that " 163S" is the correct answer. So the examinee knew the correcl answer wilhout believing it was correct. But of course the "argument" forgets that there is a diíTerence between "he knew the correct answer" and "he knew that the answer was correct". But let us leave the examples aside, at least for a while, and go back to axiomatizations. There are of course still other ways of extending Z besides Z5. We could for instance introduce one or more of the following "mixed" formulas as axioms: p. Bi4 -> KBA p*. -.BA -» K-»BA c. BA -» BIC4. '3 Some words conceming them. First of ali, p can be accepted without much problem: if Angela believes that A, then it is reasonable to suppose that she knows that she believes iL This can be justified by saying that Angela has a unique ("privileged") access lo her own internai (epistemic) states. (But, as usual, there is a lot of discussion about this and relaled principies in the philosophical literature, mostly variations on the themes we've been discussing above.) p* should be accepted on the same ieasons. On the other hand, we can accept c only if we interpret 'B' not just as a kind of "weak belief'-like "I believe moming it's going to iam, bul l'm really not that sure of it"-bul as convictUm. (It is a normal situation that there are f lo t of propositions we believe, and nevertheless we are not willing to assert that we know them.) Another possibility would be to make lhe knowledge branch stronger: not so much as S5, but, as many people like, as S4.2. We could do that by introducing the next formula as an axiom: g. -.K-iK/4 -> K-,K-»4. If we now consider ali lhe possible extensions of Z by means of p ,p* , c and g, it seems on a First look that we would end up having somelhing like 32 different logics. Bul this won't be lhe case, since, for inslance, p and p* are actually equivalent in Z (see Appendix A2); meaning it is enough lo add one of these formulas as a new axiom in order to get lhe other one as a lheorem. 14 We also get g as a Iheorem, if we add J or p as an axiom. So we gel only the 9 following calculi: Z: (lhe basic syslem) ZP: Z + p = Z + p* ZC: Z + c ZG: z + g ZCG: ZC + g = ZG + c 12 W c «rc here, o f course, m oving in lhe realm o f the s o ca llrd "R eccived V icw ", w here know ledge is ac tually (o r reducib le lo) k now ledge o f fa cu : lo say tha t I know an ob jec l is lo say lha l I know facls abou l it. C f. [Sa87], w here this question is adressed and discussed in detail. I hav e been at g rea t pa in s to find na m el for these schem as. So w c ll have m because o f "M ixed": p because o f "jn iv ileg ed know ledge o f in tcm al s u te s " : c because o f "conv iclion", and p* because it is equivalen t lo p. 1** T h is d o esn ' 1 necessarily hold i f wc are using a w eaker be lief logic (for instance w ithout </"). 23 Chúpier l ZCP: ZC + P = VA + p* Z5: Z + 5 ZP5: ZS + P = Z5 + p* = ZP + 5 ZC5: ZS + c ZC + 5 = ZCP + 5 = ZPS + c The following diagram show us how these syslems relate to each other. (An arrow means that the logic on the left is a subsystem of the logic on the right): ZC5 Looking closer at these systems, we see that adding 5 lo ZC, or c to Z5, is enough to gel ZCS (in which case g and both p and p* are derivable, and this explains why this logic is just called ZC5 instead of ZCPG 5). Let us now consider the problems I mentioned regarding axiom 5 (negative K-introspection): they appear in the systems Z5, ZP5 and ZC5, particularly in ZCS. One can show that the formula BKA -> KA is derivable in ZS-and this doesn't sound that reasonable: if Angela believes (or has the conviction) that she knows A, lhen she really knows it! This formula results from J together with db and m. Since we also have KA -» BKA as a lheorem, we can derive in Z5 the equivalente B M <-» KA. Well, this sure looks a good reason lo forget schema 5, or at leasl lo have serious doubts conceming it-but, who knows, maybe for Angela to be convinced of knowing something is really the same as knowing that something.15 The situation is still further complicated in ZC5. In this system we have BA > BKA as an axiom, and, from this schema, together with B M -> KA, we can prove BA -* KA-and, what is (if possible) still worse, BA-»A tool This of course means an equivalence between the knowledge and the belief operators, at the same time entailing that beliefs are infallible. At the risk of repeating myself, such a situation may be admissible if we are exclusively considering agents like Angela, for whom a notion of fallible bleief may make no sense at ali. But if this were the case, we wouldn't like having these additional complications in the language of our systems-we would certainly prefer to make ourselves comfortably at home in a pure knowledge logic. In view of these consideralions, I propose that we forget complelely the unfortunale system ZC5, and work only with the other ones.16 ^ I 'm no l go ing lo Collow th is question here, bul m aybe we can expla in this strangeness. I f A ngela believes p . and d o esn '1 know it, then , by 5 , she know s (and by m she believes) that she d o e s n 't know that p . S o it is not possib le fo r A ngela to believe Ihat sh e know s p , aga inst c. A s a s ide rem ark on the psycho logy o f ideal agen ts, w e have here an in teresting point: it seem s that lhey canno t have b e lie fs , they ju s t know . B ecause it should belong to the nature o f belief, 1 th ink , that it can be defeated , that one isn 't really sure that it h d d s . So ideal agents canno t believe. (A re they thus unable to have fa i íh l ) 24 A n O verView o f E D L -System s We should also notice that there are still other ways of extending Z, ZC, ZP and ZCP, which l 'm not going to considen namely by using instead of 5 or g different characteristic axioms of the calculi between S4 and S5 (systems like S4.3, and so on). To close this section, a small com pari son between the logics I'm talking about in this paper and EDL-systems that were discussed by other logicians. The axioms and rules I have presented are wellknown in the epistemic-logical liierature, but a thorough study of its different combinations (like putting them together the particular way I'm doing here, with several logics of different deduclive strength) seems, as far as I know, to be missing. (People commonly take one of the standard alethic systems and work with it.) Z5, of course, was already mentioned in [HM84]. In [Len80] we fínd the most complete study on epistemic-doxastic logics I know of, but Lenzen's presentalion is somewhat different from my own here. First of ali, he disiinguishes, in lhe syntax of his logics, between weak belief17 and conviction: to each of these concepts corresponds an operator (namely 'G ' and 'Ü'), and there is of course an operator 'W* for knowledge. The principies we are taking here to hold of 'B' correspond to Lenzen's laws for the 'Ü'-operator.18 So he has three operators, for the two in this work, in which I follow what he calls "the anglo-saxon tradilion". In the second place, Lenzen doesn't discuss sysiems of different strength but, in formalizing the logic of each concept, seis for the strongest possible calculus, i.e., a calculus that encompasses ali the principies he considers lo be valid (with regard to each concept). Thus Lenzen give us (mainly) 5 different systems, namely G (a pure logic of weak belief). Ü (a pure logic of conviction), W (a pure logic of knowledge), D (a combination of weak belief and conviction) and E (the strongest of his logics; the one containing the three operators). The logic G is somewhat weaker than KD45 (which plays here the role of lhe belief logic); Ü ís isomorphic to KD45, and W to S4.2.*9 Since l'm not making here a difference between weak belief and conviction, the logic E, if we leave *G' out, corresponds to our system ZCP. We are now ready to get things rolling. We can for instance define the notions of p ro o f and syntactical consequence for our logics. We say that a sequence/4/,...r/ln of wffs is a proofm some logic L if, for 1 £ i £ n , (i) Aj is an axiom of L\ or (ii) there is j < i and k < i such that Ak = A j-J> A /; or (iii) there is j < i such lhat At = KAj. If A = A„, we say that this sequence is a proof o f A in L (and A is said to be a theorem of L, what we denote by tj. A). If now V is a sei of wffs, we say that A is a syniaçíical consequence of F in L (and we write F t~ iA ) if there is a sequence of wffs such that, for / < í < n, (i) A ,e V, or (ii) Ai is an axiom of L\ or (iii) there is j < i and k < i such that A t = Aj -> A í; or (iv) there is j < i such that Ai = KAj and some subsequence of is a proof of Aj. (Of course, A and r A mean that A is not a theorem of L, and is not deducible in L from T, respectively.) Now, before we go into the next seciion, it is worth mentioning (later also worth using) that the following proposition holds of ali our logics here: Theorem T l. (Deducíion Theorem) r u ( A ) h f l i j f T hA B. Proof In the usual deduction-theoremic way. ■ ^ M aybe " w ider b e l ie f ' w ould be a betler nam e, since lo L enzen this notion ranges from a "m ere surm ise" (blofie V erm tdung) to • " thorough conviction*' (fesíe Ü b c n cu g u n g ). (C f. [Len801, p . 34) ^ A s I m entioned , L enzen says th a t fo r instance kb doesn 't hold if 'B ' is t&ken to be * w eak be lie f operator. L enzen argues, by the w ay, that this ca lcu lus should be considered the log ic o f knowledge. 25 C hapter I M Modalities and reductkm laws In this section, as well as in lhe following one, I will be trying to obtain some results about the EDL-calculi which will be needed by the (tentative) characterization of epistemic states. These results concem primarily the number of modalities in each logic, and whether reduction laws are available. By a modality we understand any (finite) sequence of lhe operators -i, K and B-this including the empty sequence \ which is called the improper modality. The first notion which is of importance here is the modal degree (dg) of a formula A, which we define as follows: if A is a proposilional variable, then dg{A) = 0. If A = S , dg{A) = dg(B). If A = (B # C), for# e ( ' a \ *v'f dg{A) = max[dg{B), dg(C)). If A ~ KB orA = B0, dg(A) = dg{B) + 1. Next comes the defintion of a modal conjunclive normal form (MCNF): a wff A is in MCNF iff (a) the only operators that occur in A are *K\ 'B \ 'a ' and *v'; (b) A = D j a ... a D„ is in conjunctive normal form (like in classical proposilional logic) and, for each disjunct D„ either (i) dg{Di) = 0, or (ii) Dt = #B, where dg(B) = 0 and # e {K, B, ^K , -,B). We begin by examining each logic and trying to determine how many distinct modalities are there in it. In order to make things clearer to grasp, I am going to introduce here two abbreviations (in lhe same way as ' o ' abbreviates 4-o -V in alethic modal logic): PA ->K-u4 CA =yj -iB-w4 Actually there seems lo be no correct semantical inlerpretation for P and C ,20 but I think it is nice to use them as abbreviations, else one gets lost on a forest of negations. Of course our definition of a modality must be extended to coniemplate this abbreviations too. Lei us now examine the different logics. 1. Modalities in Z In Z we have a very little number of reduction laws. Since the knowledge branch is S4, we know that we have at most 14 pure knowledge modalities, and since the belief branch is KD45, we also know that we have at most 6 pure belief modalilies. (Cf. [Ch80], p. 149, 154) But what happens with mixed sequences, like --KB-iK-r-iK-.BK-.B-. (or rather: -.KBPPBKC)? Mixed reduction laws are not legion in Z. In spite of this, there is a finite number of distinct modalilies in this logic, as we can see on the next theorem, even if it is very large: Theorem T2. In Z there are at most 84 distinct modalities, namely *, K, B, C, P, KB, KC, KP, BK/CK, PK, PB, PÇ, BP/CP, KBK/KCK, KPK, KPB, KPC, BKB/CKB, BKC/CKC, BKP/CKP, BPK/CPK, BPB/CPB, BPC/CPC, PKB, PKC, PKP, PBP/PCP, KPKB, KPKC, PKPB, PKPC, BKPK/CKPK, I f w e in terp ret *B' u conv ic tion , 'C ' w ould m ean (see |L en 8 0 ], p. 16) som eth ing like " to think it possib le th a t..." (fü r m õglich halten , dq/J...). But there seem s to be no correspondem in the case o f w eak belief, o r know ledge, fo r lhat m atter. But see [M i62], w here *P' seem s to m ean som eth ing like " for ali that one know s, . . . " F o r an op in ion against the ex istence o f natural duals to *K' and *B', sec (Isr). espccially footnote 8 on p. 3. 26 A n O verView o f E D L-System s BPKB/CPKB, BKPB/CKPB, BPKC/CPKC, BKPC/CKPC, BPKP/CPKP, BKPKB/CKPKB, BPKPB/CPKPB, BKPKC/CKPKC, BPKPC/CPKPC, and o f course lhe negalions ofthem ali: -,K, -.B, and so on. Proof. The proof is quite long, but relatively straightforward. It relies on the fact that the following equivalences (reduction laws) hold in Z: (1) KKA <-> KA (13) BKBKA <-> BKA (25) KPKPA <-> KPA (2) PPA <-> PA (14) C K C K A oC K A (26) KBKPA <-> KPA (3) BBA o BA (15) KBKBA <-> KBA (27) KCKPA <-> KPA (4) CBA «-> B/t (16) KCKCA «-> KCA (28) KPBPA <-> KPA (5) BG4 <-> CA (17) CPCPA <-> CPA (29) KPCPA ♦-> KPA (6) CCA <-> CA (18) BPBPA t-> BPA (30) PKPKA <-> PKA (7) BKA <-> CKA (19) PCPCA <-► PCA (31) PKBKA <-» PKA (8) BPA <-> CPA (20) PBPBA (-> PBA (32) PKCKA <-» PKA (9) KBP/t <-> KPA (21) KCKBA <-> KBA (33) PBPKA o PKA (10) KCPA <-> KP/l (22) KBKCA <-> KCA (34) PCPKA <-» PKA (11) PBKA <-* PKA (23) PB PCA <-> PCA (12) PCKA i-> PKA (24) PCPBA <-> PBA Now we have obviously only one zero-lenght modality, which is •. We consider the other lengths, bul just the positive cases: (i) There are four modálities of lenglh 1, K, B, C. and P, which we obviously cannot reduce further. (ii) If we now add K, or B, or C, or P lo the modálities of length 1, we gel the following 16 modálities of lenglh 2: KK, KB. KC, KP, BK, BB, BC, BP, CK, CB, CC, CP, PK, PB. PC and PP. Many of them, like KK, PP, BB, and so on, can be reduced using the equivalences (1) through (6). Moreover, with (7) and (8) we see that some other pairs are equivalenl, even if not reducible. So we end wilh the following 8 modálities of length 2: KB, KC, KP, BK/CK, BP/CP, PK, PB, PC. (iii) If we now repeat the procedure, we'll gel, first, a lol of reducible modálities, like adding K lo KB and obtaining KKB, which is immediately reducible lo KB again. Trivial cases apart, we have equivalences (9) through (12), which also allow some further reductions. Thus we end wilh the following modálities of lenglh 3: KBK/KCK, KPK, KPB, KPC, BKB/CKB, BKC/CKC, BKP/CKP, BPK/CPK, BPB/CPB, BPC/CPC, PKB, PKC, PKP, PBP/PCP, which we cannot further reduce. (iv) There are 256 modálities of the lenghl 4, bul using the equivalences, now also the laws (13)-(34), we arrive at the following distinct 10 modálities: KPKB, KPKC, PKPB, PKPC, BKPK/CKPK, BPKB/CPKB, BKPB/CKPB, BPKC/CPKC, BKPC/CKPC. BPKP/CPKP. (v) We repeat lhe procedure and obtain 4 modálities with length 5: BKPKB/CKPKB, BPKPB/CPKPB, BKPKC/CKPKC, BPKPC/CPKPC. (vi) There are no irreducible modálities of the length 6. If we trying to expand lhe preceding modálities, trivial cases apart, we obtain things like KBKPKB, which, using (26), is eqüivalem to KPKB, which is of smaller lenglh. And so wilh the other ones. Now lhe negative cases are treated in a similar way, so lhe Iheorem is provcn. ■ 27 C hapter t This list of ali them modálities is everything we get here: I ve becn uniil now unable to draw a simpte, easily understandable diagram depicting the reladons between them. Some of the modálities, as you have noticed, come in pairs, like BK/CK: this means that they are equivalent, so I'm counting them as one. We can of course eliminatc one of lhe pair, bul we are unable to reduce them any further. On the olher hand, one could take a different approach and say that modálities in a pair, even if equivalent, are (syntactically) disdncL In this case, Z would have 124 modálities. 2. Modálities in ZG In ZG we have the additional axiom -»K-ifC/4 -> K-»K-%4 (or PK/l -» KPA), what allows us to reduce a little bit the number of modálities. As a result we end up with the following: Theorem T3. In ZG there are at most 46 distinct modálities, namety •, K, B, C, P, KB, KC, KP, BK/CK, PK* PB, PC, BP/CP, KBK/KCK, KPB, KPC, BKB/CKB, BKC/CKC, BPB/CPB, BPC/CPC, PKB, PKC, PBP/PCP, and o f course their negations. Proof. The proof is similar to that of T2, with the difference that we now also have the following reduction laws: In the next picture we have an idea of the relations among ZG modálities. In lhe diagram, oniy the positive modálities are shown. To obtain the relation among the negative ones, just put a negation sign in front of each modality and then revert the direction of the arrows. (1) KPK/4 f-> PKA (2) BPKi4 o PKA (3) PKPA KPA (4) CKPA KPA. U RBK/KCK KB B K ÍC K KC Brn/rvn PK BKC/CI \ C PB BP/CP PC ? 28 A n O verview o f ED L -System s 3. Modalities in ZC In ZC things begin lo get beuer. We have now as an axiom BA-»BKA, and this allows us, since BKA-)BA is already a theorem of Z, as well as BKA <-» CKA, lo substitute BK and CK everywhere for B. Actually we gel the following new reduction laws: (1) BKA <-» BA (3) PKA PBA (2) CPA CA (4) KCA <-> KPA which allow us to make big cuts in lhe number of Z modalilies. Theorem T4. In ZC there are at most 18 distinct modalities, namely •, K, B, C, P, KB, KC/KP, PB/PK, PC, and their negations. Proof. As in T2. ■ Of these 18 modalities, the 9 positive ones are in the following piclure: I 7 \ I c \ / fig. 3 4. Modalities in ZCG In ZCG we have g as an extra axiom; however, this doesn't allows us to reduce the number of modalilies: Theorem T5. In ZCG there are at most 18 distinct modalities, namely •, K, B, C, P, KB, KC/KP, PB/PK, PC, and their negations. Proof. As in T2. ■ * Now, even if lhe number of modalities is the same of ZC, lhe relations between them are other, what allows us to make the diagram simplen 29 C hopter } K -----------------------------► KB fig. 4 5. Modalities in ZCP In ZCP things improve more. The situalion is, by lhe way, very interesting here. Adding p as an axiom lo ZC, or c lo ZP, allows us now lo derive B/4 <-* -.K-»K/4 as a iheorem, whai means lhat one could just do by introducing belief as a derived concept in a pure knowledge logic. Another point is that lhe knowledge branch is no more S4t since the axioms for belief (wílh 'B' replaced by 'X J Í ') now give us as theorem the S4.2 characteristic axiom as well. So we are back to S4.2, and in this logic the results on number of modalities are well known (see e.g. [Ch80], p. 156 or [HC72J, p. 261): there are ten of lhem, namely •, K, P, PK, KP and their negations. But since we want to keep belief in the piciure, theorems like BA <-* P M allow us to reduce the S4.2 modalities even more (i.e., to only one epistemic operator each). We arrive at the end to the following result: Theorem T6. In ZCP there are at most 10 distinct modalities, namely •, K, B, C, P, and their negations. Proof. As in T2. ■ How they are related can be seen on (he following picture: B -------C / \ K ---------fig. 5 6. Modalities in ZP In ZP we of course don't have everything as in ZCP, only part of it. Thus: Theorem T 7 .In ZP there are at most 14 distinct modalities, namely •, K, B, P, C, BK/CK/PK, KP/BP/CP, and their negations. Proof. As in T2. ■ The relatíons between the positive modalities are the following: 30 A n O verview o f ED L-SysU m s B K /CK /PKÎ B I C l ---------------------K P/BP/CP fig6 7. Modalities in Z5 Theorem T8. In Z5 there are at most 18 distinct modalities, namely *, K, B, P, Ct KB, PB, KC, PC, and their negations. Proof. As in T2. We now the reduction laws which hold in systems containing 5, namely PKA KA and KPA <-> PA. ■ The relations among these (positive) modalities can be seen in the following diagram: fig. 7 8. Modalities in ZP5 ZP5, which is one of the strongest of our systems (the other being ZCP), will of course have very few distinct modalites. Theorem T9. In ZP5 there are at most 10 distinct modalities, namely *, K, B, P, C, and their negations. Proof. As in T2. ■ Now these are exactly the same modalities of ZCP, so ali you have tq do is to look at fig. 3 again! 31 C hapter I Thus ali of our systems have a finile number of distinct modálities. But as we know, this is not suffícíent to guarantee that there also is only a finile number of different modal fiinctions of one variable in each of the logics. In other words, this does not guarantee that we are ablc to reduce formulas, first, to the MCNF, and second, to a first degree one, what would be very nice. The modal logician rcader certainly suspects that we won't fínd ali we need in some systems, but quite probably in ZP5, which seems to be strong enough. We can in fact prove that in ZP5 it is possible to reduce a formula from any degree whatsoever to anolher one of the first degree. Proposition P i. In ZP5 we can reduce every formula o f a degree higher than one to a first degree one. Proof. The proof of this proposition is somewhat long, but actually not difficult. (1*11 sketch it here, details can be found in [Len80], p. 152ff, or in [HC72], p. 53ff.) The important point for lhe proof is the fact that in ZP5 the following reduction laws are derivable, laws which allow us to eliminate iterated modálities: ( 1) KKA <-> KA (19) C(A v B) <-> CA v CB (2) PKA o K A (20) P(A v B) <-> PA v PB (3) BKA <-» KA (2 1 ) K(A v KB) H KA v KB (4) CKA <-> KA (22) K(A v BB) <-> KA v BB (5) KPA <-» PA (23) K(A v C Í J h K A v CB (6) BPA PA (24) K(A v PB) <-> KA v PB (7) CPA o PA (25) B(A v KB) <-> BA v KB (8) PPA <-> PA (26) B(A v BB) <-> BA v BB (9) KBA <-> BA (27) B(A v CB) <-» BA v Cfl ( 10) BBA <-> BA (28) B(A v PB) <-> BA v PB ( 1 1 ) CBA BA (29) C(A A KB) <-> CA a KB ( 12 ) PBA <-> BA (30) C(A a BB) <-> CA a BB (13) KCA o CA (31) C(A a CB) <-> CA a CB (14) BCA <-> CA (32) C(A a PB) <-> CA a PB (15) CCA <-> CA (33) P(A a KB) <-> PA a KB (16) PCA <-> CA (34) P(A a BB) « PA a BB (17) K(A a J J h KA a K Í (35) P(A a CB) <-> PA a CB (18) B(A a B J h BAa BB (36) P(A a PB) o PA a PB It will be enough to show that we can reduce a second-degree formula to an equivalent first-dcgree one. The procedure that we use to accomplish this has four steps: (1) We eliminate (by means of the definitions) the operators '-»* and *<->*. (2) Negation signs are pulled inside with lhe help of the DeMorgan and the reduction laws. At the end negations will occur just immediately before propositional variables. (3) We reduce ali iterated modálities, using the reduction laws, to a single modal operator. 7 (4) If the formula still is one of lhe second degree, lhe reason is that the formula itself, or one of its parts, is of the form W , where # is a modal operator and B a conjunction or disjunction of the first degree. Using the laws (17) - (36) we can distribute and "absorbe" the # operator, so that at lhe end the result is a formula of the first degree. ■ 32 A n O verView o f ED L-System s Proposition VI. In ZP5 there is fo r every formula A an equivalent formula A* such that A* is a conjunction Dj a ... a Dn, and each £), = KB j v ... v KBm v -\KBm+j v ... v -JCfy v BC/ v ... v BC* V -iBCa+v v ... V S C j V E, where dg(B;) = ... = dg(Bp) = dg(C/) = ... = dg(Ct) = dg(E) = 0 (i.e„ A* is in MCNF). Proof. First we eliminate from A implication and equivalence operators using the definitions. Then we examine the possible cases: (a) If A is a zero-degree formula, we can simply reduce it according to PC-laws to the conjunctive normal fomn. It will be then automaticaliy in MCNF (with the indices p and j being equal to zero in every case). (b) Suppose now lhat A is a first-degree formula. Then it is a bulh-function of wffs each of which is eilher a wff of PC or a wff of lhe form KB, -.KB, BB or -iBB, where B is a zero-degree formula. Trealing each of these formulas as if il were an alom, we reduce the whole formula lo the conjunctive normal form by PC melhods. The resulting formula is in MCNF. (c) Suppose A is of a degree higher than one. Then we simply reduce it to a fírst-degree wff A \ according to Proposition PI, and apply slep (b) lo ihis wff. ■ Now lo Z5. We don't have in this system ali the reduction laws from ZP5, bul most of them. However, in ZS it is not possible to reduce every formula to a first-degree one. In spite of lhat, we get somelhing which is almost as good for our purposes. We say lhat a wff A is a fP-formula iff (i) dg(A) = 0; or (ii) for some B, such lhat dg(B) = 0, A BB or A = -iBB. We can then prove the following proposition: Proposition P3. In ZS there is for every w ff A an equivalent w ff A* such that A* is a conjunction D i a ... a Dn, and each Dj = KAi v ... v v v ... v *KAp v KB; v ... v KB, v ->KBr+i v ... v -.KB? v BC/ v ... v BC* v -.BC*+/ v ... v S C j v E, where dg(A;) = ... = dg(Ap) = dg(C/) = ... = dg(Cj) = dg(E) = 0, and Bt,...Jiq are fP-formulas. Proof. Similar lo P2. ■ This of course amounts to saying that we can reduce a formula lo one of the second degree. The other six logics are complicated cases: we also don't have ali of lhe ZP5 reduction laws, just some, very few o f them. We could now be hoping, since lhe number of, for inslance, ZP-modalities is finiie, that il would be possible, like we did in ZS, to reduce any ZP-formula lo a second degree one. Actually, this is not the case. Remember, lhe "knowledge-branch" of ZP (ali lhe formulas in which no 'B' occurs) is S4, and Makinson (see [Ma66]) has proved, for a supersystem of S4 called D, lhat this system contains an infinite number of modal functions of one variable. Makinson's proof can be without much difficully adapted for ZP and the olher five logics, and so we come lo lhe next proposition: Proposition P4. In Z , Z P , Z C , Z G , ZCG and Z C P there are infmitely many different modal functions o f one variable. Proof. Using semantic melhods; as il is in |Ma66], or in [Len80] pp. 241-43. ■ 33 C hapter 1 This result is also interesling with respect to ZCP: one could have hoped, because the modalities of ZCP and ZPS are lhe same, that the reduction laws would also be present. What is not the case, tinfortunately. 1.3 A semantics for the EDL-systems The goal of this third section is to provide each system with a possible world semantics, what we'll also be needing later. 1 will first define models for the basic system Z, and will show soon afterwards how to change the defínition to obtain models for the other systems as well. Definltion D t. A Kripke model íWfor Z is a triple <W,R,S>f where: a. W * 0; b. each w; € W is an assignment of troth-values to each atomic formula; c. RÇL 5 C W x IV; d. S is reflexive and transitive; e. R is transitive, serial and euclidean.21 The set W can be seen as a non-empty set of "worlds", or "points", or "epistemic states'*. To simplify things a bit I will consider them to be assignments of truth-values to propositional variables. R is the belief accessibility relation, and 5 the knowledge one. We can now define, for each formula A, what it means for A to be true in a model and in a siate: Defínition D2. Lei íW= <WJR,S> be a Kripke model, and w, v elements of W: a. "M , w N A iff >v(/4) = 1, if A is a propositional variable; b. 'M , w t A iff M , w & A ; c. 'M , w t= A-*B iff íM, w A or B; •d. M , w N KA iff for every v, such that wSv, 'M , v N <4; e. M , w »= VA iff for every v, such that wRv, 'M, vfA. Now to obtain models for the other systems we need, as usual in possible-world semantics, to introduce some restrictions in the accessibility relations. To each new axiom there is a corresponding condition in the semantics that must be fulfillcd: $: S is incestual22; 5: S is euclidean; p : l-mixed transitivity, that is: VxVyVz(x$y a yRz -> A re la tion R is ser ia l iíT V x3 y (xR y). R is euc lidean iff Vx Vy V i(x R y a x R i -> yR z). 22 A b inary re la tion R is said lo be in cestua l i f f Vx V y Vz{xRy a xR z -> 3>v(yRw a z^w )). 34 A n O verView o f E D L -System s p * : mixed euclideanily, Ihat is: Vx Vy Vz(xRy a x S z > z/iy);2i c. 2-mixed Iransitivity, that is: Vx Vy Vi(xRy a ySz -> xRz). We now obtain models for the other logics just by reslrictíng lhe accessibility relalions S and R of lhe definition Dl in lhe following way: ZG: S is also incestual; ZP: 1-mixed transilivity (or mixed euclideanily); ZC: 2-mixed transitivily; ZCG: 2-mixed transilivity, 5 incestual; ZS: S is also symmetric (or S is reflexive and euclidean); ZCP: 1and 2-mixed Iransitivity; ZP5: S is symmetric, 1-mixed transitivily. Based on this ali we can now give the usual semantical definitions: a formula A is true in a model for an EDL-calculus L ( 'M A) if, for every tv in 'M . 'M\w A . A is L-valid {i-L A) if, for every /. íikhícI 'M , 'M )-l A . A is in L a semantical consequence of a set T of formulas if, for every (M and every C e T such that C, we have M/, A. Il is now relatively easy to prove correctness and completeness theorems, as well as (but we won't do Ihat here) the decidability of ali systems. We begin by introducing the notion of a saturated set (what we'll also need laler for lhe characterizalion of minimal states): a set £ is an A-saiurated sei24, for some formula A, if E b* A and, for ali B t E, E u (B) H A.25 Of course, a set £ is saturated if, for some wff A, £ is A-saturated. Proposition P5. Let 2. be a C-saturated set. fo r some wff C. Then: (a) A e E i f f E l-A ; (b) -v4 e E i f f A t E; (c) A->B e E . i f f A e E or B e E. Proof. (a) In one direction, if A e E then obviously E IA. In the other direction, suppose that E t A. If now A t. E, lhen by definition l u (A) h C, bul lhen il follows (by Cut) that E H C, against lhe hypolhesis that E js C-saturaled. Hence A e E. (b) Suppose -«4 e E. If we also have that A e E, then E is inconsistent and is not C-saturated. So A t E. In the other direction, suppose that A t E. By definition, then, E u M ) fC, and, by the deduclion theorem, Z h A -> C. If now A é I , we also have E u (-w4 ] iC, and, again by lhe deduclion lheorem, £ \-----À C. Bul lhen I h A v A -> C; and, since obviously X h A v A , we have E hC, againsi the hypolhesis that £ is C-saturated. Hence A e E. ^ S ince p and p * a re equ ivalen t- a t lea st in the system * considered here- it w ill be enough to add ju s t o n e o f the restric tions to the sem antics. ^ T he notion o f an <4-saturated set w as fírsl used, as far as I know , by A. Loparió ({Lo77|). ^ W hen there is no risk o f confusion , l 'm go ing lo use V and V w ithou t subscrípis. 35 Chapier 1 (c) Suppose A->B e £. If A t X, there is nolhing lo prove. So lei us suppose that A e E. By (a), E >A. E H Ã*fí; so obviously E1B and, again by (a), B e I . In the other direction, suppose first that A f? I . So, by (b), -iA e E, and, since E i-----A-> (A->B), we have E iA->B, and A-»B e E. Suppose then that B e E. Since E lB-> (4 -»B), we have E > A >B, and A >B e E. ■ Proposition P6. / / IVA, f/ien there is an A-saturaled sei E, juc/i l/wl I L E Proof. By a standard Lindenbaum argument. ■ To make life easier, let us introduce some abbrevialions lo talk about saturated sets. Let T be a set of formulas. We then define lhe subsets of T consisting of Kand B-formulas as follows: rK (A e T: thereisB .A = KB); TB =df (A g T: there is B, A BBJ. Next we define, for each of these sets, its scope sei: f(rK) =df (A: KA e rj; f(rB) =df (A: BA e T). Lemma L l . / / n A lhen #n-#A , where # e (K, B) and # r = (#B : B e T). Proof. By induction on theorems. Suppose r i A. We have four cases: (1) A e r . Then HA e # r and, obviously, #ñ#A. (2) A is an axiom. Then t A and, by RK or (derived rale) RB, i-#A, so #TV#A. (3) A was obtained by MP from B and B -> A. By the induction hypothesis, #1V #B and «D~#(B ->A). Since H#(B ->A) -> (#B ->#A) ( t or *<'), #fV#B ->#A, and hence #n-#A . (4) A = #B and was oblained by RH. If T f A, lhen lhere is a proof of A. By Rtt, i-#A, so #IV#A. ■ Proposition P7. //IV # A , # e (K, B ), there is an A-saluraied sei E jucA lhal r (I *) C E. Proof. If rV #A , lhe» obviously [ * \/UA. By L l, f (!'**) ^ A and, by P6, there is an A-saturated sei E such that e(r*) C E . • Before we go to the next bunch of properties lhat saturated sets have, let us define two binary relations p and ft over lhem. So let T and E be saturated sets; we say that TpE ifí e (r» ) C E; r/iE íit e ( r K) c e . Proposition P8. Let T, E and A be saturated sets. We have: (a) in ali logics: 36 A n O verview o f E D L -Sysiem s i . r>r; i i . r/il a I/iA T/iA; i i i . 30: Fp6 ; i v . TpZ a rpA = > LpA; v . VpL a EpA => rpA; v i . rpL => T/iL; (b) in logics which have g a sa theorem vii. I /íL a I /iA ? IB: L/iB a A/iB; (c)in logics which have p a sa theorem: viii. T/íX a XpA => TpA; ix. T/íZ a PpA =» XpA; (d) in logics which have c a sa theorem: x. Ip L a í/lA I pA; fe) m logics which have 5 as a theorem: xi. I'/iX a TpA => ÎíA. Proof. (i) We have lo show lhal e(rK) C T. Lei A e e(rK); so KA e I . Since HlCA->A, T H A, and A e r. (ii) Suppose P/jE and X/iA, and let A e e(Tk ); ihus KA e T and, since t-KA -» KKA, KKA e r , and we gel KA e X. Since I/lA , A e A; ihus e(rK) C A and T/iA. (iii) Suppose there is no 6 such that Tp6 ; thus, if 6 is a saturaled set, there is some wff A such that BA e T and A t 8 . But then, since i-BA -> ->B-v4 e T; B-«4 t T and r B -A . By P7, there is a -iA-saturated set 8 such that e(rB) C 9 . It follows that Tp0 . (iv) Suppose rp j. and I pA, and suppose it is not the case that XpA. Then there is some A e e(Xb) such that A e A. Since TpA, it cannot be that BA e t , else A would be in A because e(rB) C A. So -.BA e T and, since 1- .BA -> B-.BA, B *IM e I ', and we get -BA e X; but then X would be inconsistent. Hence XpA. (v) Like in (ii), using BA -> BBA. (vi) Suppose TpE, and let A e e ( rK); thus KA e T and, since t-KA -» BA, BA 6 T, and we get A 6 X. Thus £(rK) Q X and T/iX. (vii) Suppose T/iX, F/iA, and there is no 8 such that E/iB and ApB. That is, there is no saturaled set 8 such that e(XK) u e(AK) C 8 . By P6, then, (or every formula A, e(XK) u e(AK) vA; i.e., e(XK) u £(Ak) is inconsistent. So lhere is a B such that, say, B e e(EK) and B e e(Ak). Thus we have: KB e X and K-J? e A; -.KB t X and -.K-J3 t A; K-.KB í T and K -.K -J) e T; thus -.K-.KB e T. But 1- .K-.KB -» K -.K -.fl, so K .K Ji e T, a contradiction. Hence there is a saturaled set 9 such lhal E(XK) u e(Ak) C 8 . (viii) Suppose TftX and XpA, and let A e e(rB); thus BA e P and, since hBA -» KBA, KBA 6 I , and we get BA e X. Since XpA, A € A; thus t'(í rí) Q A and I/iA. (ix) Like in (iv), using 1- .BA -» K-.BA. 37 C hapter / (x) Like in (ii), using BA -» BKA (xi) Like in (iv), using i- .KA -> K-iK>4. ■ Well, it certainly jumps to lhe eyes lhat these properties we've just proven saturated seis have are exacily the ones we require of the accessibility relalions in lhe Kripke models for the different logics. We'll use ali this later in the completeness proof. Theorem TIO. (Correctness) I fW -A thenTt=A. Proof. Let us suppose lhat 0 -/4 . (A) A e r . Then, for ali such lhat íMi=r, ító=A. (B) A is an axiom. We examine each case: (pc) That is, A is tautology. Trivial. (Jk) A is of the form K(p->q)-*(Kp-*Kq). Let us suppose lhat A is not valid. Then there is a model 9Á= <W,R,S> and w € W, such that <M,w \=* Kp, w *= K(p-*q) and M,w & Kq. It follows that there is v e W such lhat >vSv and M,v * q. But it also follows that íW,v n= p and M ,v n p-> q, which is impossible. (0 A is of the form Kp~*p. !f A is not valid, lhen there is a model íM = <WJi,S> and w e W, such that ${,w *= Kp and b* p. However, since S is reflexive, wSw, and then it is not possible lhat íM,w b* P- (4) A is of the form Kp-*KKp. If A is not valid lhen ihere is a model - <WJt,S> and w ç W, such that ¥= Kp and íW,w h* KK/7. Now it follows from D2.d that ihere is v e W such that wSv and ^M,v & Kp. Again from D2.d we have a t € W, such that vSt and tr* p. However, since S is transitive, wSt, so it cannot be that & p. (bP) A is of the form b{p->q)->(Bp~*Bq). Proof like in (fc). (4b) A is of the form Proof like in (4). (5b) A is of the form ~.Bp-*B-.Bp, If A is not valid, lhen there is a model 94 - <Wft,S> and w e W, such that t= -»Bp and ^ B-»Bp; thus M fw & Bp. From D2.e it follows lhat lhere is a v € W such that wRv and íW,v n* p. From D2.e again we have a í e IV such that wRt and & -.Bp, hence 9Á,w \= Bp. R is however euclidean, so we have lhat tRv, and lhen íKftv n p-a contradiction. (db) A is of lhe form Bp->->B-p. If A is not valid, lhen lhere is a model <WJl,S> and w e W, such that 9*(,w ¥= Bp and (M,w -iB- thus %f,w *= B~<p. Since R is a serial relalion, lhere is a v e W, such that wRv and íŴ v >= p. However, it follows from D2.b lhat 9Á,v »= ->p , í*f,v & p, what cannot be. (m) A is of the form Kp->Bp. If A is not valid, lhen lhere is a model íAÍ = <WJi,S> and w e W, such that %{,w »= Kp and ft£w v B p. From D2.e lhere is lhen a v e IV, such that wRv and "My & p. However, since R C S, it follows from D2.d that wSv and thus Í*í,v n p-a coniradiciion. (g) A is of lhe form -.K->K/?~»K-.K-/>. If A is not valid, lhen there is a ZG -model tM = <WJl,S> and w g IV, such that n ->K-.Kp and M yv & K -.K -^; thus fM,w K-iKp. From D2.d it follows lhat 38 A n O verview o f E D L -System s there is a v e W such that wSv and 94,v fc* -iKp, hence 94,v »= Kp. From D2.d again we have a t e W such that wSt and 9{,i w -iK-y>, hence 94,w *= K -p . 5 is however incestual, so we have a u e W such that vSu and tSu. It follows that 94,u N ~ ç and 94tu ►" P-a contradiction.26 (5) A is of the form -iKp-»K-iKp. Proof like in (5*), using now lhe fact that S is euclidean for ZS and ZP5 models. (p ) A is of lhe form Bp-»KBp. If A is not valid, then there is a ZP-model 94 = <WJt,S> and w € W, such that 94,v/ n= B p and íMtw & KB p. From D2.d it follows that there is a v € W such that wSv and 94,v & Bp. From D2.e it follows now that there is a t e IV such ihat vRt and 94,t ** p. Now in ZPmodels the 1-mixed transilivity holds, so we get that w Rt, and consequently, that 94tt\= p - a contradiction.27 (c) A is of lhe form Bp-»BKp. If A is not valid, then lhere is a ZC-model = <W,R,S> and w e W, such ihat 9í,w n Bp and 94,w & BKp. From D2.e it follows that there is a v 6 W such that wRv and 94,v t?* Kp. From D2.d it follows now that there is a t e W such ihat vSt and 94,t t f p, Now in ZCmodels the 2-mixed transilivity holds, so we get wRt, and consequenüy, that 94,i p-a contradiclion.28 Thus, in ali cases and ali logics L, A is L-valid, and so, for ali 94, such that 9ít=r, 9é=A. (C) A was obtained by using MP from B and B-*A. Induclion hypolhesis: for ali 94, such that íM kr, 9Í\=B and (M \-B -+ A . If 94 & A , lhere is a w e W such that 94 tw & A . But 94,w n= fl, 94,w t= B->A, and this is contradictory. Thus, for ali íM such that íM*=r, ító=A. (D) A = KB was obtained from B using RK. Induclion hypolhesis: for ali 9 f, such that 9ít=*B. Now if íMV A, lhere is a w e W such that 94,w & A, i.e., 9i,w ** KB. From D2.d it follows then that lhere is a / € W such lhal wSi and 94jt h* B-and this cannot obviously be the case. Thus, for ali í tf such ihat íWnT, íWi=A. ■ To prove now the completeness theorem we need first to esiablish some relations between saturated sets and Kripke models. If and 9£ase Kripke models, we say that 94**9t {94 ^ a r e equívalent) if, for every A, 94*= A iff 9(j=A. is clearly an equivalence relation. Now let 94= <WJi,S> be a model. For each w e W, let [9(,w] = [A: 94,w t=A). Let W = { P C FOR: V - [9(,w), for some 94, some w), and let now S be the class of ali sets £, such that, for some formula A, £ is A-saturated. Now we can prove the following: Lenuma L2. If [íM^vl H A then 94, w **= A. Proof. If A 6 {9f,w], then 9(, w i= A by definition. If A is a theorem, then A is valid, hence 9{, w i= A. If A was obtained through uses of MP or RK, then 94, w A, because these rules (see proof of TIO) are validity preserving. ■ Lemma L3. W = S. 26 T h is a lso holds fo r Z C G . 27 T h is a lso holds fo r Z C P and Z P 5 . 28 This a lso holds fo r Z C G and Z C P . 39 C hapter I Proof. (=í) Let us suppose lhal Í F W: thus, for some íW = <WM,S> some w G W, T = (i) First of ali, iM,w k* -,(A->A), for ali wffs A, since 'M.w t= A-»A, and then, because of L2, VM.w] v -i(A-»A). (ii) Now we have the following: for ali B, if B e T, then T u (H) i-----.(A->A), because B e [flí.w], hence fM.w & B , !M,w N -B and, since i- .fl-»(B -*-i(A ->A)), 'M .w t= -.B -» (B -»-i(A -»A)), -tB->(B-»-<A-»A)) g ^B->(B->-,(A->A)), so T u (B) H ->(A->A). From (i) and (ii), [íW tvJ is a -i(A-»A)-salurated set, for ali A. Thus Vhí,w\ e S, T e S. N T e S. (a) We construct a model = <S,p,/í>, such that 'M X n= A iff A e I , for every wff A and every L f S From P8 .vi, we have that, if Tp£ then I~fiL. so p C p. It is now easy to prove, using P8 , that p and fl satisfy, for each logic, the required properties of the accessibility relations. (b) We prove now lhe following: for every L f S, e(XB) = n [ 0 E S: e(XB) C 8 ). It is clear that e(XB) C n ( 8 f S: e(EB) c 8 ). On the other direclion, let A be such lhat A ( e(Eb); then BA í X. Since X G S, X v BA, so by P7 there is an A-saturated set X* such that e(XB) Q L*. Then A e X*. and X* e (E e S: e(XB) C P ) . From this it follows that A « o ( 8 e S :t(X B) C 8 ). (c) In a similar way, for every I e 8 , e ( Ik ) = n (8 g S: e(XK) C 8 ). (d) Hence, from (a) and (b) we have that A g e(XB) iff A g n (8 g S: e.(IB) C 8 ) iff A g n {8 g S: Xp8 ). From (a) and (c), A g e(Xk) iff A e n ( 8 s S: e(Xk) C 8 ) iff A e r >(8 e S: X//8 ). (e) We prove now lhat 'M ~ <S ,p.p> is a Kripke model. S is a non-emply set, and p C / l c S x S . p and fl also have the desired properties. We show now lhat 9 4 fulfills the conditions of definition D2. For ali X G S. (i) t= -B iff g X iff B e X iff <MX * B; (ii) !*f,X •= B-*C iff B-»C G X iff B t X or C g X iff M X v B or "M X N C; (iii) !W,X t= KB iff KB e X iff B e e(XK) iff B G o (8 G S: E/l8 ) iff for ali 8 g S such that Tfj8 , B e 8 ; (iv) íW,X BB iff BB € X iff B e e(£B) iff B e n (8 g S: Xp8 ) iff for ali 8 e S such that Xp8 , B g 8 . (1) Now T g 8 , hence T is one of lhe worlds in 'M. Now we define Ajif p = (A: íW.P t= A), ihus [ 'M, T] = r , so r g w. ■ Theorem T l l . (Completeness) T h A iff Ft=A. Proof. One direction is TIO. To prove the other direclion, let us suppose lhat IVA. From P6 there is an Asaturated set X such lhat P C X . From L2 there is a Kripke model J Í E K and w in 'Aí such that X = [!W,tv]. Thus for ali C g T, 9>í.w i C, and "M,v/ ^ A. It thus follows lhal \ A. ■ * Now that we have presented a semantics for the EDLs, and have proven correctness and completeness, it is lime that we tie some loose ends from seclion 1 .2 , where we discusses modalities. Theorems T2 to T9 are slaled in lhe form "there are aí m ost... modalilies". What we should do now is to show that lhe numbers mentioned are exact. Thus: 40 A n O v trv ie w c f E D L -System s Theorem T12. In Z, ZG, ZC, ZCG, ZCP, ZP, ZS and ZP5, the number o f distinct modalities is, respectively, 84, 46,18, 18,10, 14, 18 and 10. Proof Since Theorems T2 through T9 siate thai the logics have at most the mentioned modalities, we need to show lhat one cannot reduce them funhe r. For instance, we affirmed that, in Z, K and KBK are distinct modalities. Since KA -> KBKA is a theorem of Z, we have to show lhat KBKA -> KA doesn'l hold (else they are equivalent). Consider the following model *\( = <(x, y , r ) , {<x,z>, <y,z>, <z,i>), {<Jc,y>, <y,z>, <x,z>, <XyX>t <y,y>t <z,í>}>, such that x(p) = 1 ,y(p ) = 0 and z(p) = 1 . li is easy to check out that R is serial, transitive and euclidean; that S is reflexive and transitive; and that R is contained in S. It is also easy to see, in lhe next picture, that this model falsifies KBKp -) Kp. Here the smooth arrows represent lhe S relation, and the other ones, the R relation. (2R2 was represented by a thicker outline, as you may notice. S reflexivity was left out.) Thus KBKA KA doesn't hokl in Z. In a similar way, we have to show, for each pair of modalities, that they are not equivalent-what I won't do here for reasons of space. ■ 41 Minimal Belief States I f you think the problem is bad now, just wait until we've solved it. A . K A S S P E Clysterium donare, Postea seignare, Ensuita purgar e. M O L IÈ R E , Le Malade Imaginaire. Now lhat wc are done with this overview of the EDL-systems, and are hopefully more in clear about the properties of the logics lhat we are using, we can move to considering our main problem, namely, how to characterize Angela's epistemic states, in the cases where she knows or believes only some formula a. We should maybe begin by asking what does this actually mean. By staling, for instance, lhat "Angela believes only ot' we are surely not pretending lo asscrt lhat a is the only one proposition Angela believes- just remember, she believes ali tautologies, that is, ali tautologies, which are quite a lot, are already contained in her belief State, "believes only" could then be better undersiood as meaning that the formula a should be some kind of information sufficient lo "reconstrucl" or "characterize" or "determine" Angela's belief state; in other words, wilh a in our hands we should be able to know what is in Angela's belief stale. a would be in this sense more a kind of "minimal description", or a "key". This naturally leads to the question of what kind of formula can a be? We don't really want to narrow our choices just to propositional variables: Angela can, for instance, only believe lhat "if p then q'\ Even if she doesn't believe eilher p or q, this situation is of course different of believing just lautologies, because "if p then q" really teils us something about the world. So we should allow a to be at least any propositional (i.e., zerodegree ) formula whatsoever. But why exclude modalized ones? Prima facie there is nothing which speaks against them: some of them will certainly show themselves to be "dishonest" (lhe preferred example in (HM] is the formula a = Kp v Kç), but others won'i (like for instance Bp). So let us agree lhat a can be any formula of L, modalized or not. We should next decide which kind of stale we would actually like to characterize: a knowledge state, a belief stale, or both of them? Well, in ali our systems we have the formula KA -» B/4 as an axiom, and this means lhat Angela believes every proposition she knows. If we now consider epistemic states as being sets of formulas, lhen this would intuitively mean that a knowledge State is always a (probably proper) 42 M in im a l B e lie f S ta tes subset of a belief stale. Thus belief states are more comprehcnsive-and since HM understand default rules any way as "rales of conjecture", like B(-iKp -* q), we could then concentrate mainly on belief states. An additional reason is that agents, in order lo act, normally also take into account what they believe, not only whal lhey know. So this should settle lhe queslion. In the following sections, then, we consider different ways of characterizing belief states. 2.1 Stable sets A first tenlative of characterizing belief states uses the notion of "stable sets", a denomination lhal was introduced by Stalnaker.2? Of course, in lhe original discussion this notion referred only to "knowledge" seis, so we have to adapt il here. Well, in ali EDLs lhe "belief branch" (i.e., the sei of ali formulas in which no K-operator occurs) is as strong as the modal calculus KD45.30 For this reason I suggest for stable sets the following definition (essenlially the same Ihat was already used in [HM84] for knowledge stable sets, with lhe difference that we now work with beliefs): Definition D3. Let L be any EDL-logic. A set S of formulas is an L-slable set if: (stl) S is closed under logical consequences; (st2) A e S iff B/í e S; (st3) A t S iff -B A e S; (st4) S is consistent. This is a general definition, and can be used in every EDL-system, but of course each system will determine in its own way which formulas should belong to the stable set. I'U try to make this clear wilh an example. Suppose we have a situation in which Angela knows and believes p; believes, but doesn't know, q; and neilher believes nor knows r. That is, we have: Kp, Bp, Bq, -iKq, J í r , iBr. Now, in qach of the different logics, Angela's belief state would look like lhe following (where 'bsL abbreviates the belief State in logic L): bs z'. ( p, q, Kp, Bp, B^, KKp, BBp, BBi;, KBp, ->Br,... ) b s z r : b s z u ( K B q , K - . B r , ... ) ; b s z c : b s z u ( K q , K K q . K B 4 , BK<7, b s z c r b s z r o b s z c ' . b s z s : b s z u ( K - .K 17, K i K r , ... ) ; b s z r s 'b s z s u b s z r ■ 29 C f. [IIM 84], p4. Ju st lo rem em bcr, K D 4 5 is a lso know n as " w eak S 5 '\ that is, S 5 w ilh o a - » o a in slead o f the re n ex iv ity ax iom o a - * a . 43 C hap itr 2 Supposing furthcr lhat she doesn't know lhat she doesn't know r (-.K-iKr), we would have: b s z G b s z u ( iK -v , ... ); b s z c G b s z c <-» ( J C v , ... ) . In this example I have included in the feíi-scts only lhe formulas that should necessarily be on them. If we had for instance that Angela also knows that she believes q, lhen KBq would of course also belong to bsz. But in Z Bif->KB? is not a theorem, so KBq would hold for anolher reasons, and we could enlirely as well have a different situation in which KB</ is not true. This is in Z still possible. Bul KB? must be in bsz Pt because Bq-»KB? is an axiom of ZP, hence believed by ideal user Angela. Thus there are no ZP-stable sets containing Bq and not containing KB<j as well. Similar holds of the other logics. Notice also that the additional hypothesis conceming the logics using axiom g wouldn't hold in ZS and ZPS, because in this logics Angela is fully instrospective, hence she knows when she doesn't know somelhing. But let us proceed. Suppose now that Angela believes only the formula a (which can be of course a conjunction of other formulas). How can we characterize Angela's belief State? It is clear lhat this State muüt contain a -but we have obviously a lot of states to which a belongs. This particular belief State should lhen be lhe "minimar', whatever we choose here "minimal" to mean. The easiest and most e le g a n t solution would be to use lhe nolion of sei inclusion: let us lake ali stables sets conlaining a. an d the sm a lle s t of them is now Angela's belief State when she believes only a. But nothing in life is lhat easy, as we can see with lhe next proposition. Proposition P9. [HM84] No stable sei is a proper subsel o f anolher slable sei. Proof. As in (HM84J, p. 5: suppose there is two stable sets S and T such lhat S c T. Hence lhere is A r S , A í T. From definilion D3 we have BA e S and -.BA e T. But BA f S implies BA e T, in which case T would be inconsistent, and this cannot be, by definilion. ■ Oh well, lhere must be olher ways of killing this cat. HM's solution goes as follows: A possible candidate for the 'minimal' [belief] State containing a is lhe stable set containing a whose propositional subsel31 is minimum (w.r.t inclusion). ([HM84], p. 5, italies mine.) ( That this solution works in lhe pure knowledge logic ariscs from lhe fact lhat in (HM84] a stable set is uniquely determined by its propositional subformulas. But here this is not always the case, as we can see on lhe following example.32 Let us suppose that we have two different situations (call them a and b) such that in a Angela knows that p, but doesn't know that q\ and the other way round in b. In both cases she believes that p and that q. So: a = ( Kp, Bp t ~>Kq, Bq, ...) 31 U nder "propositional subsel" should be understood lhe set o f ali form ulas in w hich no Ko r B -operator occurs- o r w hose modal degree is zero (see defin ilion in C hapter I). 32 W ith excep tion o f the system s Z C , Z C P , Z C G and, in certain aspects, o f ZS and ZPS. 44 M inim al B e lie f S ta tes b = { - . K p , B p , Kq, Bqt . . . ) The correspondi/ig (Z-)slable seis would then be: bst = ( p, q , Kp , - , Bp, B r, BKp, bs* = ( p, q, - , Kq, Bp , B^, - , BKq , ... ) As we can see, bsM * bst,, even if their propositional subsets are the same, namely the set (p, 4). In spite of Angela's believing the same "Cacts" (p and 4) both situations, what she believes about her own internai states is different in each of them. I'd like to remark here that this is only so because the known facts are different in lhe two worlds, that is, because the "pure knowledge" propositional subsel (zerodegree wffs lhat Angela knows) is not the same-thafs lhe reason for our problem. (Remember we couldn' 1 reduce formulas to first-one degree ones? Here is where we are going to miss that.) So we are bound to run into trouble with some logics. Bul before we dive into these waters, let us examine closer the cases in which things work. What we will be trying to do is lo find under which condilions two stable seis are the same, i.e., which kind of subsets uniquely determine a stable set. If we have this, we can, as it will be shown later, define a kind of smaller-than relation, and then identify the minimal belief slate we are looking for. 1. The ZC/ZCG/ZCP solutíon ln ZCP we can easily prove that BA «-» ->K-iKA is derivable, or equivalenlly, that one can define the operator 'B' in terms of 'K' (cf. Appendix A3). We can hence consider ZCP, in lhe end, as being a pure knowledge logic, as strong as the modal system S4.2. With this ZCP loses some of its interest to U9, because the characterization problem reduces itself 10 the levei of the pure knowledge logic. Anyway, we can find for it, and for ZC/ZCG as well, a method of characterizing minimal belief states. Since c (BA->BKA) is an axiom of these systems, we have B(BA-»KA) as theorem (see Appendix A4). (Just to remember, in these calculi we should betler interpret B' as conviction.) A natural language rendering of this formula could be: Angela is convinced that, when she is convinced of A, then she knows that A. Now this intereslingly means that, in Angela's belief stales, conviction and knowledge are equivalent. Il is easy lo see why: hB(BA-*KA) entails lhat BA->KA belongs 10 lhe belief state. This is also the case for KA->BA, which it is an axiom. Hence KA <-» BA will also be in Angela's belief state, and the direct consequence of this is that ZC/ZCG/ZCP-stable sets have the same characteristics of lhe knowledge stable sets in [HM84J. Let us call these K-stable sets, and they are defined as follows: Definition D4. A set S of formulas is a K-stable set iff: (ic-stl) S is closed under logical consequences; (K-SI2) A e S iff KA e S; (K-SI3) A t S iff -iKA e S; (K SI4) S is consislenl 45 Chapter 2 Proposition PIO. Let S be a 7.(7Z('G /ZCP stable sei. Then S ij K-stable. Proof. K-stl and K-sl4 follow immediately from the definition of stable sets. The other two conditions follow as easily: (K-sl2) A e S iff BA e S (D3) irr BKA e S (tBKA <-> BA) iff KA e S (D3). (K-st3) A t S irr -.BA e S (D3) iff B-.KA e S (t- ->BA « B-.KA) irr -,KA e S (D3). ■ As a consequence of this proposition we can consider ZC/ZCG/ZCP-stable sets as sets in which one reasons with the rules of the pure knowledge logic. But the most inteiesting in this story is the fact that K -stab le sets a re uniquely determined by their propositional subsels-what was already proved in [HM84] (p. 4, Proposition 1: result due to Moore [Mo83]). To prove it here we need the following definitions. Where S is a s ta b le set, w e say that prg(S) = ( í e S : dg(A) = 0) is the propositional belief subset of S. Lemma L4. ([HM 84], Lemma 1) Let S be a ZC/7.CG/7.('P stable set. Then: (a) K A v í c S i f f A e S or B & S; (b) -.KA v B e S i f f A í S f f B e S . Proof. Exactly like in [HM84]-or similar to the one of Lemma L5 below. The next theorem establishes lhen that ZCVZCG/ZCP-stable seis are uniquely determined by their prg-subsets. Theorem T13. Let S and T be 7Â'f7A'2f7A'Y stable sets such ihat prg(S) = prn(Y). Then S = T. Proof.33 If A is a formula, let A* be a formula obtained from A where ali B-operators were replaced by K. It is firsl of ali easy lo prove that there is for S and T corresponding sets S* and T*, such that A e S (T) iff A* e S* (T*). It is also easy to prove Ihat S* und T* are closed under S5-consequences. We lhen prove that S* = T*. that is, for ali formulas B, B e S* iff B e T*. If dg(B) = 0, we don't need to prove anything, since S and T agree on propositional formulas, and hence S* and T* too. Let us then suppose that dg(B) = 1. Since both sets are closed under S5-consequences, B is equivalent lo a first degree conjunclion B* of disjunctions D,such that, for each D,-, £>; = KC/ v ... v KCm v -iKCm+/ v ... v -.KC* v E, where E is a propositional formula, the same as each Cj, since dg(B*) 1. Now B* e S* iff each conjunct D,e S*. and this holds, by Lemma L4, iff one of lhe following holds: C; e S*,..., Cm e S*, Cm+í * S*.....C* e S*. E e S*. A similar property holds for T*, and, since S* and T* agree on propositional formulas, B* e S* iff B* e T*, B e S* iff B € T*. Hence A e S iff A e T. ■ 33 T he p ro o f o f this theorem , as well as o f T14 and T 15 , Is adapted from (IIM84], P roposition 1, p. 4. 46 M in im a l B e lie f S ta tes The consequence of this ali is that now we have, when we are working with ZC, ZCG and ZCP, a method of characterizing Angela's belief state when she believes only some formula a -it will be the stable set whose prg-set is the "smallest"-in a sense of "small" we are going to define later on. Let us first look at the olher logics. 2. The ZPS solution Anolher among lhe above menlioned nice cases-and this only wilh some restriclions-is ZPS. one of the strongest EDL-systems. To prove lhat ZPS-slable sets are uniquely determined by lheir propositional (belief and also knowledge) subsets we need first some definitions and lemmas. If S is a stable set, we say lhat prx(S) = IA e S: dg(A) = 0 a KA e S ) is its propositional knowledge subset. Of course prjt(S) C prg(S). As we'll see, the pr/c-sels musl also play a role. I.emma L5. Let S be a ZVS-stable set. Then: (a) BA v B e S i f f A e S or B e S; (b) -.BA v B e S i f f ^ í S or 8 E S; (c) KA v S e S i f f A e prK(S) or B e S, ifdg(A) = 0; (d) -.KA v B e S i f f A t prx( S) or B e S, ifdg(A) = 0. Proof. We prove the cases (c) and (d). (c=») KA v B e S. Thus KA e S or B e S. If B e S we are done, so let us consider KA e S. From axiom k it follows lhat A e S and, since dg(A) = 0, that A e prn(S). (c<=) A e prx(S) or B e S. If B e S lhen KA v B e S. If A e pr*( S) lhen by definilion KA e S, and KA v B e S. (d=>) -JCA v B e S. If A e prjf(S), by definilion KA e S, and it follows from lhe hypothesis lhat B e S. If A t pr/c(S) lhen it's alright. (d<=) A t. prn(S) or B e S. If B e S lhen -JCA v B e S and we are finished. So let us suppose that B t S and -.KA v B <t S. Then -.KA t S and B e S. If -.KA t S then -.B-.KA e S (st4). Since i-----.KA -> B-.KA, -I-.KA e S, KA e S and, since dg(A) = 0, A e prjf(S), which is a contradiction. Hence -.KA v B e S. For (a) and (b) the proof is similar and even simpler. ■ The next theorem now shows that ZPS-stable set are uniquely determined by both their propositional subsets (prg and prg). Theorem T14. Let S and T be ZPS-stable sets, such lhat prif(S) = prjf(T), and prg(S) = praÇT). Then S = T. Proof. We prove lhat for every formula A, A e S iff A e T. By Proposilions PI and P2 we have that A «-* A*, where A* is at most of lhe first degree and is in MCNF. If dg(A*) = 0, we don't need lo prove anything, since S andT agrcc on propositional formulas. Let us thus suppose lhat dg(A*) = 1. Since it is in MCNF, A* is a conjunction of disjunctions £>i, such thal each £>, = KB; v ... v KBm v -.KBm+; v ... v 47 C hapter 2 -iKBit v BC/ v ... v BCr v -JBCr+/ v ... v -.BC, v E, where E is a propositional forrnula, same as each Bj, Cj, since dg(A*) = 1. Now A* e S iff each conjunct Di g S, and this holds, by Lemma L5, iff one of the following holds: 8 ; e prjt(S).....Bm e prK(S), Bm+; t prK(S)......B* t prx(S), C i g S......Cr g S, Cr+/ í S.... C , t S, E g S. A similar property holds for T, and, since S and T agree on propositional formulas, and since pr/ríS) = prjf(T), A* e S iff A* g T, A g S iff A g T. ■ We can see in this proof the importance of lhe restriction pr/ç(S) = pr^(T): in order to show lhat A G S iff A g T we need to suppose lhat both sets agree on some "basic" formulas-in this case not only the believed zero-degree formulas, bul also the known ones. This would seem lo imply that, in order to characterize a minimal belief state, we not only need to know that a everything Angela believes, bul also lhat some p is everything she knows. On the other hand, betwen a stale in which she believes a and knows P, and another tn which she believes a and knows nolhing, the second is clearly the smaller-the smallest of ali, thus, being lhe one in which Angeias has only beliefs and no knowledge. 3. The Z5 solution , With respect to this logic lhe silualion is somewhat more complicated, but anyway not beyond salvation. In the first place, it is easy to see lhal stable sets are not uniquely determíned by their propositional subsets alone. Let us imagine two different situations (call them again a and b, and let bs* and frij, be stable sets denoling Angela's belief state in each case) such lhat prn(bst ) = prx(bsb), and PrB(bst ) = prii(frjb). Let »s further suppose that in a Angela knows lhal she believes A (KBA), while in case b she doesn't (-.KBA). We would consequently have KBA g bst , -.KBA g bsb, and of course bs, *■ bst,. We must hence introduce further restrictions in order to characterize lhe desired belief states. This is so, of course, because in ZS we can't reduce every formula lo a first-degree one, only to the second degree. So we need one construction more, namely of a BT-sel. For any stable set S, we define BT(S) as (A g S: A = KB andB is a /3°-formula). We can then go to the next theorem, which proves lhal Z5-slable sets are uniquely determined by their pr/f-, prgand BT-sets. Theorem TIS. Let S and T be ZS-stable sets, such lhat pr*(S) = pr*r(T), prg(S) = prg(T) and BT(S) = BT(T). Then S = T. Proof. Similar lo theorem T14, with now the BT-sets also playing the important role. ■ 4. The lack o fa Z, ZG and ZP solulion Now these are the complicated cases-these logics are going to stay for the prcsent as opcn problems, at least in what concems lhe characterization of belief States through stable sets the way we are trying it here. 48 M inim al B e lie f S ta tes 5. Defining a smcúler-than relation Now wilh respect to the other systems for which we had a solution, we are able to give for them a definition of "honesly" of a formula. We must first only define a kind of "smaller lhan" relation between stable sets, something corresponding lo the notion of set tnclusion. In the logics ZCt ZCG and ZCP, this is no big problem. In Theorem T13 we have proved that ZC/ZCG/ZCP-stable sets are uniquely determined by their pr^-subsets. So it is enough that prg(S) is a proper subset of prs(T) to characterize S as a state in which Angela believes less lhan in T. The following table show the possible relalions that obtain between the prg-subsets of two stable sets S and T (where *S <b T \ or 'S =b T ' mean that in S Angela believes less than, or the same as, in T): i f . . . we have ... P'B(S) c prfl(T) S < b T P''«(S) = pí'fl(T) S =bT pfB(S) 3 prfl(T) T > b S This could lead us then to the following definition: Definition D5a. Let S and T be stable seis. We say that S is ZCfLCGfLCY-smalUr than T iff prg{S) c prB<J)In ZP5 things aren'i that easy, because here we have to consider the two propositional subsets of a stable set (cf. T14). Let us examine lhe possible confígurations: * i f . . . and ... we have ... p rs(S )C p rB(T) P'K(S) c prjciT) S < b T P 'b(S) = pra(T) P'K( S) c prirfT) S <b T prs (S)3 prfl(T) pr/CÍS) C prK(T) 7? Pffl(S)C prfl(T) pridS) = prK<J) S < b T PTBÍ S) = prfl(T) prjríS) = pr*<T) S =b T prÁ S) => prB<J) Pr*(S) = pr*(T) T >b S P'B(S) c P^flfT) prKiS) 3 pr/dJ) 77 prg(S) = prg(T) prtfS ) 3 pridJ) T = b S prB<S) => preO t pr/d S) 3 pr/d T) T >b S As one can see, in ihree of the lines S is smaller lhan T; in other three T is smaller than S; in one lhey are the same and, in two lines (marked with *??'), lhere is no comparison possible. We arrive then to lhe following definition: Dermition D5b. Let S and T be stable sets. We say Ihat S is ZP5-smaller than T iff prg(S) c prg(T) and prjríS) C pr/dT), or if prB(S) = prB(T) and pr/dS) C prK(]). 49 Ckapter 2 ln the case of ZS, now, things are going lo gel really tough, because here we have to consider three different subsets of a stable set (cf. T15). Let us try to make some sense of ali lhe possible combination in lhe following table: i f . . . and ... and ... ive have ... prs( S )c prfl(T) PríKS) c pr#(T) BT(S) c BT(J) S <b T prg(S) = prgÇT) PrK<S) <= pr*(T) BT{S) C B7(T) S <b T prg(S) 3 prgÇT) P'jríS) c pr/cÇT) flr(S) C B7(T) 77 prg(S) C prgÇT) Pr*(S) = pr*(T) B7(S) c f l/(T ) S < b T pre(S) = prgÇY) pr/rtS) = prjrtT) BT{S) C B7XT) S < b T pr/KS) 3 prg(T) PrAT(S) = pr*(T) Br(S) c BT(T) 77 prg(S) C prgÇT) pr*(S)3prjrtT ) BT(S) C B7'(T) 71 P'B(S) = pra(T) pr«(S )3pr*(T ) BT{S) C B7XT) 77 prg(S) 3 prgfD Pr*(S) prfc(T) B7"(S) C BT(T) 77 prB(S) C prgÇT) PrK(S) C prjrtT) BT(S) = BTfT) S < bT prg(S) = prgÇT) PrAr(S) C pr*r(T) Br(S) = BT(T) S < b T prg(S) 13 prgÇT) PrAr(S) c prK(T) B7(S) = B7(T) 7? prg(S) C prgÇT) PrK(S) = Prrc(X) B7XS) = BT(T) S <b T P'B(S) = prgÇT) prfdS) prtfT ) B7(S) = B7(T) S =b T prg(S) 3 prgÇT) pr/dS) prKCD Br(S) = B7'(T) T < b S prg(S) C prgÇT) prjrtS)=3pnríT) B7(S) = B7(T) 77 PrBÍ^) Prfi(T) pr*(S)=>pr^T) B7(S) = BT(T) T <b S prg(S) 3 prg(T) pr*r(S) 3 p rj^D BT( S) = B7XT) T < b S PrB(S) c prgÇT) P'K(S) c pnríT) B7'(S) 3 f l /'(T) 77 prB(S) = pr/KT) Pr*r(S) <= pr/cÇT) BT(S) 3 B7(T) 7? prg(S) 3 prg<J) pr/ciS) c pr^CO BT(S) 3 B7(T) 77 prs(S)C prB (T) prjtfS) = pr/cÇT) BT(S) 3 B7"(T) 7? P'B(S) = prgCH pr/dS) = pr/f(T) B7'(S) 3 B7'(T) T <b S prg<S) 3 prg<T) P^aKS) = prjç(T) B7'(S) 3 B7XT) T <b S prg(S) C prgÇT) P^íKS) 3 pr/(iT) flr(S) 3 BT(T) V. P^BiS) = prg(T) prrfS) => prríT ) BT(S) 3 BT(T) T <b S prfl(S) 3 prgÇT) ( prrfS) => prx(T) BT(S) 3 B7"(T) T <b S One can see lhat we have more undecided cases as in the logic before. Anyway, summing up what this table tells us, we arrive at lhe following Definition DSc. Let S and T be stable sets. Then S is ZS-smaller than T iff (i) prg(S) c prp(T), pr*(S) C pr*r(T) and BT(S) C B7'(T); or (ii) prg(S) = prgÇT), prK(S) c pruÇT) and BT(S) C BT(T); or (iii) P'fl(S) = prg(T), pr/c(S) = prKÇT) and B7(S) C BT(T). We can now characteriíe Angela's belief State , in which she believes only a, as the "l-smallest" stable set containing a , for L e (ZC, ZC(í, ZCP, Z5, ZPS). There are of course lot.s of formulas a for 50 M inim al B e lie f S ta tes which there is no such a State, for instance let a = Bp v Bç. This leads us lo the following definition of honesly: for L fc (ZC, ZCG, ZCP, Z5, ZP5), a formula a is L-honesls iff there is an /.sniallest stable set S containing cc We can see at once that the formula a = Bp v Bq is not Z C /Z C G /Z C P -h o n e s tj. Ali Z C /Z C G /Z C P-stab le sets which contain a must also contain either p or q. Further, lhere is a ZC/ZCG/ZCP-stable set Sp which contains a and p, but not q, and another set Sq which conlains a and q, but not p. Neither Sp nor S9 are ZC/ZCG/ZCP-smallest, and lhe intersection Sp n contains neilher p nor q. Hence lhere is no ZC/ZCG/ZCP-smallest stable set T containing a , such that prgÇT) c prg(Sp) and prg(T) c prs(Sq). Thus a is not ZC/ZCG/ZCP-honestj. In a similar way we can show lhat a is not honesty in other systems as well. 2.2 A sldestep: saturaled sets An allemative way of characterizing Angela's knowledge slate using these smaller-than relatiuns just defíned concems saturaled sets. We begin by establishing some relations between stable and saturaled sets. The reader has surely noticed lhal a saturaled set can be seen as a world-or, to put il belter, as a world description: this description tells us what is true in lhe world, also including what Angela knows or believes-these are facts, too. So the following should be tme: to each saturaled set (world) £ conesponds a stable sei, namely the set of lhe formulas believed (in Ihis world) by Angela, and this set is no other than e(Eb). As we prove in the next two propositions: Proposition P l l . Let 2.be a C-saturaled set. Then e(Eb) is a stable set.3* Proof. We prove that e(Eb) fulfills lhe condilions of definition D3. (stl) Let A be a PC-tautology; thus h-A, hBA (by RB), so 1 1BA and BA € E, A e e(Eb). Let us now suppose that A, A-»B e E(EB). Thus we have BA, B(A-»B) 6 E. From Ifi and MP it follows lhat BB e E and fmally fl e e(Eb). (sl2) Let us suppose that A e c(EB). So BA e E and, since we have BA->BBA as an axiom, BBA C: E, BA e e(EB). O r the olher direclion, if BA e e(EB), then BBA 6 E. But t-BBA-»BA, s o BA e E, A e e(EB). (st3) If A t e(EB) then BA e E, -.BA e E and, from J», B-.BA e E, -,BA e e(EB). On lhe other direclion lei us suppose that -.BA e e(EB), A e e(EB). From -.BA e e(EB) we get BA t e(EB), BBA t E. From A e e(Eb) il follows that BA e E, and, through 4b, BBA e E- a contradiction. Thus A í e(EB). (st4) To prove lhat E(Eb) is consistent we have, since E is C-saturated, that for some wff C, it holds E ^ C, so E is consistent. Let us now suppose e(Eb) is inconsistent. Then lhere is an A such that A and A e ^ When 1 talk about "suble" aetj without spcdfying some EDL-systcm I im of course meaning üuu what is being s&id holds for ali systems wc are considering hcrc. 51 C hapter 2 e(£B). From this fact it follows that BA, B-A € £. But BA-»-iB-tA is an axiom, and thus -»B-iA g E, and £ is in this case inconsistent, what cannot be. Hence e (IB) is consistent. ■ Proposition P12. Let T be a stable set. Then, for some C-saturated set I , T = e(XB). Proof Since T is stable, we know that T is consistent, so there is a formula C such lhat Tb* C. From proposition P6 it follows that there is a C-saturated set I , such that T C l We have now to prove lhat T = (A) Let us suppose, for some A, that A e T. Then BA e T (st2) and, since T c X, BA e I , A e e ( IB). (B) Let us now have A e c(EB), A í T. Then BA e L. However, if A tf T, then -.BA e T, -.BA e Z, and this is a contradiciton. Hence A e T. ■ By now, it jumps to the eyes, since stable sets are the bs-sets of some saturated set, that there is (sort of) a way of defining honesty using saturated sets: a is L-honestw iff lhere is a A-saturated set E containing B a such lhat e(Eb) I s /--smallesl (for L e (ZC, ZCG, ZCP, Z5, ZP51). This is of course just another way of making use of stable sets. A second method employed by HM in the characterization of knowledge states uses Kripke, or possible-world, models. Basically, the procedure is: (i) define, for each model the set of known formulas in ^ (nam ely the wffs that are true in every state of the model); (ii) show that this set of known formulas is a stable set; (iii) show that the model in which Angela knows only a is the union of ali models in which she knows Well, in [HM84] this task is easily accomplished, and again this is so because the knowledge logic this almost amounts to saying that each world is accessible to every world, which fact has as a consequence that one can complelely delete lhe accessiblity relation from the picture: Ihus KA is tnie in a model if A is true in every world of the model. But I said almost: in fact, we could have a model like the one in lhe following picture: e(XB). 22 Kripke models a . they used is S5. In models for this system, the accessibility relation must be an equivalence relation. Now fig. 9 52 M inim al B e lie f S ta tes As one can see, (he accessibility relation (dcpicted by arrows and black-filled circles in case of reflexivity) is an equivalence relation: it is easy to check visually Ihat it is reflexive, symmetric and transitive. However, not every world is accessible to every world: they are grouped in different "clusters" which have no communication to anolher. As Hughes and Cresswell already pointed out ([HC72], p. 67), this means the same as having two SS-models glued together since no cluster has any influence on lhe other, to evaluate valid formulas we have to get their values separatedly in each cluster-same procedure as looking in two different models. Hence we can in fact use models without lhe accessibility relation for S5-which we can call monocluslered models. Now this has anolher interesting consequence: HM define lhe set K(M) of the known facts in model 94 as (.4 : 'M ,l t= A ,(o t every l in 94) (cf. [HM84], p. 7). It is now easy to show that (1) A e *T(fW) iff ' M j ^ K A for ali t\ and (2) A t K(M ) iff M,t t= -iIG4 for ali l. (I) would give no problem even with standard (multiclustered) models, but (2) would. In a monocluslered model, if A t lhen, for some State w, 9{.w fe* KA. So there is a state v such that % v & A. Since now every world is accessible to every world, for every i there is a world (namely v) where A is false, so KA is false in every world, and -K A is true in every world. That this doesn't work in a multiclustered model can be seen in the next picture: -*KA fi*. 10 tftre we have on the left a cluster where, for every l, 94,1 ► .KA. Bul worlds of this cluster are not accessible lo worlds of lhe second one, so there, on lhe righl, we have iM, w í= KA, for every t». S o K(fM) won't have lhe nice properly (2), and won't be stable. Now gelting rid of this problem is only lhe first advantage of working with monocluslered models. The second concems the method of characterizing lhe model in which Angela knows only a. As I said, a formula K a is then true in a State s if and only if a is true in ali states simpliciler. Now the inluition behind t Kripke models is the following one: states are worlds which Angela thinks are possible relatively lo whal she knows/believes. If now a model í t f contains more states lhan anolher model we can say that in 94 Angela is more ignorant than in This fact implies that lhe model in which Angela knows only a should be the union 94a o f ali models í tf such that JMn Ka, i.e., ali models in which Kaholds. And this works wilh monocluslered models because we can lake any iwo models whaisoever and neverlheless sliU be sure ihat their union will be a model. Bad luck, wilh our EDLs ihis is not always the case. Let us consider the following example: let 94 = <MJIm,Sm> and ?*£= <N,Rn>Sn> be iwo EDL-Kripke models, where M = [a, b), Rm = [<a,b>, <b,b> ), Sm = [<a,a>t <a,b>, <b,b>), N = [a, c ) , R n = [<a,c>t <c,c>], Sn = (<a,a>, <a,c>, <c,c>). Let now l i lhe union of ítf and lhal is, t i = <U,Ru,Su>, where U = M u N, R u = Rm ^ Rn - [<a,b>, <b,b>,<a,c>,<c,c>), S u = Sm u SnThe trouble here is lhal lhe belief 53 C hapter 2 accessibility relation R y in 11 is not euclidean: we have aRub and aRijc, but the pair <b,c> doesn'l belong to Ry. Thus 11 is not a Kripke model. On the other hand there are some cases where the union of two models is still a model, namely, if the two original models don't have any states in common. (States, remember, are considered here to be truth-value assignments to propositional variables.) As follows: Proposition P13. Let - <MJIm ,Sm > and <NJff/,Sn > be two Kripke models such lhat M r tN = 0. Then 11 = < I J • where U = M u N, R y = Rm k j Rn <*nd S y = Sm v Sn , is a Kripke model. Proof. U is obviously a non-empty set. What we must show is that lhe relations R y and S y have the desired properties. (a) R y is serial, i.e., for every u in U there is a v such that uRyv. This is evidenl, because Rm and Rn are serial, and the pairs <u,v> are consequently in Ry. (b) R y is transitive. Suppose not: then there is u, v, w in U such that uRyv, vR yw , but not uRyw. However, since M n N = 0, we have as a consequence either (i) u, v and w are in M, in which case uRmv, vRmw and-since R u is transitive-uRm w , with lhe consequence that uRyw-, or (ii) u, v and w are in N, in hich case the same holds. So R y is transitive. (c) R y is euclidean. Suppose not: then there is u, v, w in U, such lhat uRyv, uRyw, but not vRyw or wRyv. However, since M r i N = 0, we have as a consequence either (i) u, v and w are in M , in which case uRm v , uRm w and-since R u is euclidean-vR uw , and hence vRyw-, or (ii) u, v and w are in N, in which case the same holds. Hence Ru is euclidean. In a similar way we can show that S y has the desired properties. ■ How this fact could help us is still not clear to me. So how can we go on? Well, there should be a way of getting a kind of monoclustered model for knowledge and belief togethcr. Let us see. To begin with, in handling belief, things are likely to be somewhat different from lhe knowledge case. In fact, it is perfectly possible to have íW ►= BA (i.e., BA is true in every state w e M) and nevertheless there could be w* e M such lhat tM, iv* tr' A. Now, if this happens, then w* must be a special kind of world. If for instance lhere were a i e W such that iRw*, then we would have M,i w BA (because there would be an accessible world with A false). So we can conclude the following: if M n BA and lhere is »>* such that 'M, w* ^ A, then lhere is no 1 e W such thal iRw*. If this is so, we say lhal w* is a losl world (or closed, or forbidden-take your choice). Worlds lhat are not lost we will call open, or accessible. The interesling about lost and open worlds seems lo be lhat for KD45 (which is our belief logic here), we can put the open worlds together in the same baskel: in fact, they are ali accessible lo every other world, if in the same clúster. A lypical, mullicluslered KD45-model could look like this: v fig. 11 54 M inim al B e lie f S ta tes Unfílled circles represem the worlds that are not accessible to olhers, not even to themselves. The only thing we need to do, if we drop the belief accessibility relation from the picture, is lo single out lost and open worlds when we define a model. So let us put ali lhese ideas logelher and see what we get. We define a monoclustered EDL-model as follows: Definition D6. A monoclusterd model fh íis a pair <W,0>, where: a. W * 0; b. each tv, e W is an assignment of truth-values to alomic foimulas; c. O C W and O * 0. Again the elements of the set W of worlds are assignments of truth-values to propositional variables. O, of which we require to be non-empty, is the subset of W which contains the open worlds. Of course, the set Wl of lost worlds can be defined as W-O. It should be now obvious, since we dropped the 5-accessibility relation from the picture, that the knowledge branch of this as yet unknown EDL is S5. But do we really get KD45 as belief logic? And which of our EDLs here is characterized by the class of monoclustered models? Probably ZP5, and we'll see that this is the case. A first way to show that is to define for monoclustered models two accessibility relations over W, Rm and Sm, and to prove that they have exactly lhe characteristics of ZPS relations. So let 'M = <W,0> be a monoclustered model. For any two worlds w, v € W, we say lhat wRmv, if v e O; and that wSmv. Now we show that: Lemma L6. (0 Rm is serial, transitive and euclidean. (ii) Sm is an equivalence relation. (iii) Rm C 5m. (iv) 1-mixed transitivity holds. Proof. (i) Since by definition O is non-empty, for every w e W there is a v such lhat wRmv. So Rm is serial. Suppose now wRmv and vRmt. So I e O, hence wRmt, and Rm is transitive. Suppose now wRmv and wRmt. So l e O, hence vRmi, and Rm is euclidean. (ii) Since w5mv, for any two worlds w and v, Sm is obviously an equivalence relation. (iii) Since O C IV, it is trivial that Rm Q Sm. (iv) Suppose now wSmv and vRmt. S o l e O, hence wRmt, and I-mixed Iransitivily holds. ■ ll is also easy to see lhal, for inslance, 2-mixed transitivity does not hold. Suppose wRmv and vSmt. So v e O; however, we have no guaiantee that l also belongs lo O-il could be a lost world. So we just got monoclustered models for at least one logic. Is there any chance of having this kind of model for the other systems as well? Well, the way we defined things entails lhal ali these models validale the schema p. In view of this, it seems lhal if we want the models having this monocluslerdness characteristic-which is importam in order lo have B/l Une in lhe model iff A is true in ali worlds-then we must accepl lhal BA->KB/t shall tum out valid. Else there should be a world where BA is false; and yet another (obviously not in lhe same clusler, then) where A would be false. Thus BA would come oul false 55 C h a p te r2 in lhe model. Well, what we want to have is the following: the worlds that matter to evaluate belief formulas should be kept together in a cluster-this doesn't mean that worlds that matter for the cvaluation of knowledge formulas couldn'1 be arranged in a different way. To put it in anolher way, we could try models with the open-worlds-story inslead of lhe belier accessibility relation, but introducing back again the 5 accessibility relation for knowledge.35 Let us see what we get. Definition D7. A mixed monocluslered model ü f is a triple <W,(),S>, where: a. W * 0; b. each tv, e W is an assignment of truth-values lo atomic formulas; c. O C tV and O * 0; d. S is a binary relation over IV x IV which is at least reflexive and transilive; moreover, if v e O then for every w, wSv. I claim that these just defined models are models for ZP. If we now add the following requirement: e. for every w, v e W, if w e O and wSv then v e O; then we should get models for ZCP. We'll be proving ali that soon. We must now redefine, for each formula A, what it means for A to be true in a (monoclustered) model and in a state. In what follows, l will use the following notation lo denote monoclustered or mixed monoclustered models: !M = <W,0(,SI> . This just means that the relation S only applies, obviously, in the cases of ZP and ZCP. Definition D8. Let = <W,OI£f> be a monoclustered model, and w an element of VV: a. mf, w A iff w(A) = I , if A is a propositional variable; b. -w4 ifT íM ,w if'm A; c. tU , w A->B iff M , w H*m A or *M, w Mm B; d. íW, w t=m KA iff for every v e IV, (M. v t=" A; (ZPS) for every v such that wSv, 'M , v t m A; (ZP/ZCP) e. ÍA{, w t=m BA iff for every v e O .íM .v t=m A. Validity and semantical consequence are defined as before, just with V " 1 inslead of plain 'm'. Now, in the following, let L be one of ZP, ZPS and ZCP. Lemma VJ. I fV is a saturated sei, then, fo r some monoclustered model ?M, T = [ Proof. Let us suppose thal r is saturated. The best way to show the lemma is lo take some subset of lhe set S of saturated sets which includes í as the set of worlds, and show that we can have a model on this-r would be then one of the worlds and we'd be done. Well, for ZP and ZCP we'II use again the relation /i we defined over the set S of ali saturated sets, namely T.fiA iff e(£K) <Z A. As one see from the proofs of P8.i and ii, this relation is reflexive and transitive. 35 T ty ing to k r r p the S 5-characteristjcx and extend / . 1*5 leads only to Z C S , w hich w e already have throw n in to (th is s to ry s) trash can. 56 M in im a l B e lie f S ta tes (a) We first construct a model 9A= <(rjlc. (H ^/. Hl>, where (H * = (8 e S : e ( rK) C 6 ), [I~)P = (6 e S : e ( rB) C 8 ). and, for ZP and ZCP, /i is as above. (b) We first prove (in ZP5) lhat, for every 1 , 8 e (H*. e(LK) = e(8 K). Let A e e(£k). By constmction of [ r ]K. e ( rK) C I . We then have KA e I ; -.KA t £, K-.KA í T, -.K-.KA e T and, since i- .K-iKA -» A, A e r. Hence E(£k ) C I , from what it follows that e(£k ) = £(I K). By a similar reasoning, e(8 K) = e ( rK), from what it follows that e(Xk) = e(8 k). (c) We now prove, still in ZPS, lhat, for every I e (r|lc, e(£k ) = n ( 8 e [P |K) . Well, A e e(£K) iff, from (b), for every 8 e [ r ]K, A e e(8 k) iff for every 8 e (H *. KA e 0 iff for every 8 e (n * . A e 8 iff A e n (8 e [r]* ). (d) We prove now, for ZP and ZCP, lhat for every £ E [H*, e(£k ) = 0 ( 8 £ [H*: í ^ 8 ) . It is clear that e(£k ) C n (8 e [ I ]k: e(£k) Ç. 8 ). On the olher direction, let A be such that A t e(EK); then KA t £. Since £ is a saturated set, £ ts KA, so by P7 there is an A-saluraled set £* such that e(£k) C £*. Then A t £*. Now, since £ e [r]* , e ( r K) C £; and, since /l is transitive, e ( r K) C £*; so £* e (8 e S : e ( rK) C 8 ), i.e., £* e [P]*. From this it follows that A i n (8 e [ r )K: e(£k ) C 8 ). (e) We prove now that, for every £ e [H*, e(Xb) = r >(8 e ( r |P ) . Let A be such lhat A e e(Eb); lhen BA e £; -.BA t £. By the construction of [ r ]K. K-.BA t r , -.BA t r , BA e T, and, finally, A e E(rB). Obviously lhen A e n ( 8 e [f~]P| (by lhe construction of (H^). so e(£b) C n (8 e [FjP). On the olher direction, let A be such that A t £(EB); then BA t £. By lhe construction of [D*. KBA t T, BA t P, and T v BA, so by P7 there is an A-saturated set 8 such that E(rB) C 8 . Then A t 8 and, by the construction of [r]P. 8 e (rjP. Thus A t o (8 e m P )It follows that, for every £ e (D K, e(£b) = n ( 8 e (r)P ). (f) We prove now that [T|P * 0. Since T is a saturated set, there is some wff A for which T is A-saturated. So T iv A, and obviously I ts KA, T ^ BKA. By LI lhere is some saturated set £ such that E(rB) C £. So £ e (DP. and [H^ * 0It follows immediately lhat also (r]K * 0. because clearly [TjP Ç. [H*. (g) We now have lo prove (in ZP and ZCP) that, if 8 e [P]P and £ e [rjK. then £/j8 . Let A be such that A e e(£k ); then KA e £; BA e £, and A E e(£b). By (f) above, A E r i ( 8 E [r]f*), so A 6 8 and e(Ek) C 8 . That is, £ /i8 . (h) We now have to prove (in ZCP) lhat for every £ , 8 e (T)*, if £ e [F]P and £ /i8 then 8 e (r |P . Since £ e [rjP , E(rB) C £. Now, for every wff A such that BA e T, BKA e T, and KA e £. But then A e 8 , so £ (rB) C 8 and 8 e [T] P. (i) We have now lo prove that !\f= <11 ]lc, [I ]P is a monocluslered model. We already proved that (rj* and (T|P are non-empty, and that fi satisfics lhe required conditions. We show now that íWfulfills lhe conditions of definilion D8. We say that 'M X A iff A e £, for every wff A and every £ e [ r jK. Now we have, for ali £ e [rj* . (i) M X f iff e £ iff B t £ iff 'M X * B, (ii) 'M x N B -iC iff B->C e £ iff B t £ or C e Z iff -MX * B or 'M X »= C; (iii) M X •= KB iff KB e £ iff B e e(£*) iff ZP5: B e n (8 e (H*) iff for ali 8 e ( r ) lc, B e 8 ; ZP/ZCP: B E n (8 e [D*: £/i8 ) for ali 8 e [rj* such lhal £ ^ 8 , B e 8 ; (ív)5V Í,£nB B iff B B e £ iff B e e(£b) iff B e n | 8 e ID P) iff fo ra ll8 e [ H M e 8 . 57 Chapter 2 Now obviously V € IH *. hence V is one of lhe worlds in *M. Now we define [!MX] = M X *= A), and obviously enough, T = líMX]- ■ Theorem T I6 . *~ tA i ff *=mA. Proof. ( I) Suppose *~l A. (A) A is an axiom. We examine each case. If A is a tautology, it is evident from clauses D8b and D8c that \=mA. And it is also obvious that (k), (í), (4) and (5), ali valid formulas, are also monoclustered valid. Let us now consider the belief case, and the mixed formulas. (Jfcb) A is of the form B (p -* í/)-» (B p-»B ^). Let us suppose that A is not valid. Then there is a monoclustered model íAf = <W,0[,S]> and w e IV, such that *M,w *=m Bp, CM, w t=m B(p-*q) and M,w \*m B^. It follows that there is v e O such that bem q. But it also follows that íM,v p and M,v Nm p->q, which is impossible. Thus A is valid. (4b) A iso f the form Bp-*BBp. If A is not valid then there is a model !M - <W,0 (£]> and w e W, such that lM,w t=m Bp and *M,w b*m BBp. Now it follows from D8e that lhere is v è O such lhat (M,v Bp. Again from D8e we have a t € O, such that íM,t p. However, i=m Bp implies lhat for every open world f, 9A.,\ Nm p. Thus A is valid. (5*0 A is of the form -iBp-»B-tBp. If A is not valid, then there is a model M - <W,0 l,S]> and w e W, such that íW,w v=m -,Bp and í*f,w B-»Bp; thus lHtw m Bp. From D8e it follows that there is a v e O such that *M,v >*m p. From D8e again we have a / e W such that 9>(,t btm ->Bp, hence 94,t *=m Bp. Now this entails that, for every open world, inclusive w, *AÍ,w t=m p-a contradiction. Thus A is valid. (db) A is of lhe form Bp-»-.B->p. If A is not valid, then there is a model M ~ <W,0 [,SJ> and w e W, such that M,w Mm Bp and -«B- thus *M.,w B-f*. Since O is not empty, lhere is at leasi a v ê O such that 9<i,v Nmp. However, it follows from D8e that i=m p , !M,v & p, what cannot be. Thus A is valid. (m) A is of lhe form Kp-*Bp. If A is not valid, then there is a model M = <W,0[,SJ> and w e IV, such that *=m Kp and t\(,w Bp. From D8e lhere is then a v e O, such that 9á ,v p. However, íM,w >=m Kp entails that p is true in every world: thus Stf.v *=m p-a contradiction. Thus A is valid. (p ) A is of the form Bp->KBp. If A is not valid, then lhere is a <W,Of,SJ> and w e W, such that íW,w »=m Bp and í*f,w fe*m KBp. From D8d it follows that there iá a v e IV such that íM v Bp. From D8e it follows now that lhere is a f e O such that íMj p. But M,w t=m Bp entails lhat for every open world í, M,t p-a contradiction. Thus A is valid. (c) A is of lhe form Bp-»BKp. If A is noi valid, then lhere is a * <W,0 l,SJ> and w e W, such lhat íWfH» Nm Bp and BKp. From D8d it follows lhat there is a v e O such that yem Kp. From D7e and D8e it follows now that there is a / e O such that p. But Bp entails that for every open world í, *=m p -a contradiction. Thus A is valid. 58 M inim al B e lie f S ta tes (B) A was obtained by using MP from B and B *A. Induclion hypolhesis: NmB and *mB M. If there is a monoclustered model M such that M A , there is a w e W such that 9í,v> & A. But 94,w t=m B, 94,w B ->A, and this is contradicloiy. Thus, for ali 94, 94\~mA and A is valid. (C) A = KB was obtained from B using RK. Induclion hypolhesis: l=mB. Now if some A, lhere is a w e W such that 94,vt A, i.e., 9>l,w h*m KB. From D8d it follows then that there is a t e W such that 9 4 , t B -and this cannot obviously be lhe case. Thus, for ali 94, 96^ mA and A is valid. ( II ) Suppose that ,A . From P6 there is an A-saturated set £ such that P C I . From L7 there is a monoclustered model 9 4 and w in W such ihat £ 1 Thus 94,w b'm A. U thus follows lhal A. That, for example, (c) does not comes out valid in ZP5 models we can see at once. Suppose Bp->BKp is nol valid, lhen there is a model 9 4 - <W,()> and w 6 W, such lhal fhí.w t= Bp and íM,w 1/ BKp. From D8e it follows lhal there is a v e O such that 94,v & Kp. From D8d it follows now that there is a I e IV such Ihat 9 4 , t v p . Now we have from D8e and 94,t N Bp that, for every open world s, 94, s N p. But here we don't get a contradiction, because I is not necessarily open. By a similar reasoning we can show Ihat things like Bp-»p and Bp-»Kp are also not valid. Now let Af be an /.-monocluslered model, for some EDL-calculus L e (ZP, ZPS, Z C P). We define the sets K(94) and B(94) of the known and believed facts in 94, respectively, as: Jt(!M )={A : « í n K A ); B(94) = ( A: 94, w N A for every w e O J. It is easy 10 see, from this definition, ihat: (i) A e B('AÍ) iff for ali w in 94, 'M,w N BA iff 1 B.4; (ii) A t B (9t) iff for ali w in 94, 94,w M -,BA iff 94*= -.BA. For inslance (ii): if A t B(9f) lhen there is a w e O such that 9>{, w \r, A. If now there were a l e W such that 94 ,1 1= BA, then we would have that, for every w e O, 9{, w t= A, a contradiction. So for every 1 e W. 9 4 ,1 * BA; 9 4 ,1 1= -.BA, and thus 9 4 M -.BA. Proposition P14. Let 9 4 be a monocluslered L-model. Then B(94) is an L-stable sei. Proof. (stl) For ali tautologies A, 94*^ A; thus 9 4 t= BA, A e B(94). Let us now suppose that A ,A-> B e B(94). I.e., 9 4 1= BA, 9 4 t= B(A-»B). Using ** , 9 4 ^ B B .B e B(íM). Thus B(94) is closed under boolean operations. (st2) A e B(94) iff 9 ( N BA iff 9 4 n BBA (i-BA <-> BBA) iff BA e B(!M). (st3) A t B(94) iff 94 V BA iff DWV -.BA iff 9 4 1= B-.BA (H -.BA «-> B-.BA) iff -.BA e B(94). (st4) Suppose B(94) is inconsislent. Thus A, -A e B('M ), ihus 94*= BA, 9 4 B-*4. But 9>{t B/A implies that 9{t= iB v\ (using d1'), and this is a contradiction. Hence B(íM) is consislent. ■ 59 C hapter 2 L e m m a L 8 . Lei b r a monoclustered model, and w g W. Then lhere is a model 'Msuch that, for every formula A, íM,w M A iff 9(y= A. Proof. Let us have \9(,w ] defined again as (-4: 9 ( ,w <= A ). Now let 9(? = [9(: [9í,w]]. We show that: (i) "X ' t 0. Suppose íY* = 0 . Thus for ali models 'H, 'H y [ 94,w\. From this fact it follows that for ali models 9{, if 90= (!W,w) then 9(y= a a - .a . Thus ( ÍVÍ,w) n o a - .a , and l'M,w] ta a - .a , and it follows that [94w) is inconsistent, what cannot be. Hence * 0. (ii) Let us have 9\£ !Af g 9 £ . We show that 5V= Suppose that 9£. Then there is a wff B, 9{t= B, 9(_ fc* B, 9(_ t= -B . But it is easy to show (like in L3) that [ lM ,w) is a saturated set, thus il cannot be the case that, for bolh models, 90= [9f,w] and 9f_ t= )'M,w). Hence !AÍ" 9(_. It is clear, then, that for some 9{je 9{* and for every formula A, 94,w hA iff 9{_f= A. ■ Proposition PIS. Let S be an L-stable sei. Then lhere is a monoclustered L-model 94$, such lhai S = B(94s). Proof. Let S be L-stable. Thus, for some T G S, S = e(rB). From L7 lhere is 94 and w G W, such that, for ali formulas <4, A g T iff 94,v/ *= <4. From L8 lhere is 9 fs such lhat !Ms1 A iff 94,w n= A. We now show lhat S = B(!tfs)So: A e S iff A g E (rB) iff B/4 g T iff N= BA iff M s t= B/t iff A g B(94s ).m Afler having eslablished which kind of relationship holds between monoclustered models and stable sets, we can now ask ourselves which is lhen lhe model in which Angela believes only a. As I mentíoned, HM's elegant solution in terms of (S5-)Kripke models consists in just taking the union of ali models in which a holds. In ZPS here this is not so problematic, but what about lhe olher cases, in whichwe have to cope wilh an accessibility relation S which is not an equivalence relation? We can easily conslrucl two models such lhat lhe plain union of Sm and Sn is not, for instance, transitive. So what can we do? A first way would be trying to define a stronger union operation, namely one in which additional pairs would be added to the union of lhe S relations, so that properties like Iransitivity and the like could be preserved. We can get Ihis introducing lhe notion of a closed union of two models: if <M,OmI,SmI> and 9C= <N,OnI ^ nI> are two L-monoclustered models, we say that lhe model 1 i= 94& 5V]is lhe closed union of 9 (m á 9{i( V is a triple <V,OulSul>, where: i. t / = M u f í ; ii. O u OM VOff, iii. Su = r> (T C V x {/: Sm u Sn u (<w,v>: w e U and v g O u ) C T and such lhat T is reflexive, transitive and, for ZCP, it holds lhai if w g O u and < m\ v> g T lhen v g O u ) ■ Some words conceming this definition. The set U is lhe union of the universes of the two models; nolhing new here. The idea behind lhe definilion of Su is that Ihis relation should be the smallest subset of U x U containing Sm k i Sn lhai slill fullfils the desired properties of the knowledge acccssiblity relation. It must also contain the set (<w,v>: w g ( / and v g Ou) (which takcs care, in standard model terminology. 60 M in im a l B e lie f S ta tes of R being a subrelation of 5). For ZP5, of course, the clause (iii) doesn'i apply. We can easily prove that the intersection of ali subsets of U x U respecting this condition is the smallest set. This ensures that the (closed) union of two models will still be a model. The set of open worlds, of course, is the union of the open worlds of the two models. A second way of resolving the diffículty would be by means of defining a submodel relation. Let again 94 = <M,Om [.Sm J> and 9{^~ <N,On(,Sn]> be two monoclustered models. We say that íWis a submodel of {94 <, 9fi if M C N t Om £ On , and Sm C Sn - (We also say that 9^is an extension of •M .) Some properties of this submodel relation: first, 'í 1 is clearly reflexive and transilive. is also antisymmetric: if <M<> íÂand 'M holds, then as consequence M = N, Om = Ou. and Sm = Sn Hence íAf and 9{mc the same model. The next proposition shows how the trulh of certain formulas is preserved under submodels or extensions. Proposition P16. Lei (M = <M,Om I,Sm J> and $ í= <N,OnI,S n]> be two Kripke models and A a formula, such lhal 'M < 0\(and dg(A) = 0. Then: (a) i f 9 i ^ K A lhen « Í n K A; (b) i f 7{y* B/t then M t= BA; (c) 1/ íMV -JCA ihen <H)= -JCA; (d) i / -,BA lhen íA£n= -BA. Proof. (a) Suppose Q{*=KA and íW V KA; so there is a w e M such that M .w li' KA. From this fact it follows that there is a v e M such that (h>5mv and] M, v b* A. However, we have that M L N [and Sm £ Saí], so e N [and <w,v> e S/y). Since dg(A) = 0, 'M ,v t= A iff 'H .v t A;36 thus 9{,v fc* A, hence 9{,w ir1 KA and 'M.)'* KA, against the hypolhesis. (b), (c) and (d) are provable in a similar way. ■ As we see, this proposition is provable exactly because dg(A) = 0. That the property doesn't need to hold if lhe formulas are modalized is shown in the next picture. •M B -> B p 0 0 1 0 B -. B p 0 0 1 1 H. B 1 B p 1 1 0 0 B-B p 1 1 0 1 fig. 12 ^ T his o f course holds because A is a propositional form ula , so its evaluatíon is independcn t from the values it m ay g e t in o ther w orlds: w e don'( need to consider anolher world difTcrent from v. 61 C hapter 2 Here we have 94 = <W,Ou 1,Sm I> an<-i <W,On IHnI>Open worlds are thicker oullined; thus O u = (v) and On = (w. v). So Om C On - (The S-relation doesn't malter.) The difference between the two models is that in 9{w is open, but not in 94. As a consequence of w also being open in 9(is lhat 1= B-.Bp, but this is not lhe case in 94. The propositional variable p has in w and v the same value, and this, as the reader can see, doesn't hold anymore for the modalized formulas. We can now use one of these two altematives to characterize the state in which Angela believes only a. We can for instance take the set of ali monoclustered models in which B a holds, and then prove that this set has a biggest element. It is not a suiprise that this set is exaclty lhe closed union ° f >11 models. For a wff A, let mds(A) = l 9 f .^ í* = A ] . Proposition P17. / / 9 fa = ® m ds(Ba), then fo r each fM e mrfj(Ba), 94 <, 94a. Proof. Let 94a = © mrfj(Ba). Obviously for each 9 4 e mrfj(Ba), Wm C WMa, O u C 0 ^ a, and Sm C S<MaThus ÍM"5 94a. Let us now suppose lhal lhere is 94* such lhal for each 9 4 e mdi(Ba), 9 4 < (M*. But then 94a S and 94* S 94a, with lhe consequence lhat 94* = 94a. ■ Thus Angela's belief State, when she believes only « , would be lhe set B(94a). As in [HM84], there are formulas that don't belong to B (9f„j, for instance our old acquainlance a = Bp v Bq. Let us consider the ZP-models 9 4 - <M,Om I £ mI> and 9(= <N,On I,Sn1>, where 9 4 - (fc.v), Sm = (<b,b>, <b,v>, <v,b>, <v,v>); O u = On = N jfc.w), S/v = (<b,b>, <b,w>, <w,b>, ov,w >), and 6(p) b(q) = I, v(p) = 1, v(q) = 0, w(q) = 1, w(p) = 0. Graphically, so that we can understand il better (the relalion S is again not necessary): 94 *í b B p v B q b B p v B q 1 1 1 0 I 0 I 1 1 1 V B p v B q w B p v B q 1 1 I 0 0 0 0 1 1 1 fig 13 As we see, 94*= B a and 9(y= Ba. Let now 94a be the closed union of ali models 9Í+, such lhal 94* t= Ba. Then we have M a k* B a, because lhe set [b,v,w] is contained in 0<Ma and thus there is a world, namely b, such that 94a, b K Bp (since w e O nfa , and 94a , w & p) and 94a , b fc* Bq (since lhere is v e OMa, and 9 ía, v H q). Hence 94a , fr Bp v B?. Since b e 0<M„, 9 fa, b w B(Bp v Bq), that is, 94a, b ir1 Ba, so 94n w B a and consequently a t B (M a). We can now introduce a second defuiition of honesty, based in monoclustered models: a formula a is L-honestu iff a g B (9 fa)We must stress here that, like trying to find the minimal stable set, lhe melhod using monoclustèred Kripke models doesn'l have the advaniage of working equally well for ali EDL-syslems, as one could 62 M in im a l B e lie f S la lej expecl. We don'i have here, in (act, to introduce restrictions conceming the stable sets (like pr/c-, prgand BTsets), but, on lhe other hand, we didn't get monoclustered models for ali logics. 2.4 An algorithmic approach The third approach in trying to characteríze knowledge states is done in HM Ihrough the use of an algorithm. In olher words, the algorilhm decides whelher Angela knows a certain proposition B, given that she knows only a. We try to do the same with the EDLs here. The idea is to generate, for each formula a , a set £>" which is the set of things Angela believes, if she believes only ou HM begin by asking themselves which formulas belong to Da. In their case, since the knowledge logic is S5, any formula B for which Ka -* B holds must belong to D a. Since, however, more than just lhe logical consequences of a should be in D a, the algorilhm ends Up being the following: B e Da ilT *=s5 K o a T a(B )-> B , where f'a(B) is the conjunction of KC, for ali subformulas KC of B for which C e Da, and of -JCC, for ali subformulas KC of B for which C t D" (8 being considered a subformula of itselO. (cf. [HM84], p. 9) The intuilion behind the algorithm is that a formula B belongs to D a iff it is a consequence of knowing a and the K-subformulas of B which have already been decided. So, for instance, a propositional formula C is in Da iff t" ss K a -> C. Afler ali is said and done, one could ihink that D a is a stable set, but Ihis doesn't always happen. Some of them would be inconsistent-lhe ones corresponding lo dishonesl formulas. Well, how can ali this apply to our EDL case here? The answer is: pretty much the same way, but changes are of course due to be made. First of ali, obviously, if B is a consequence of believing a (i.e., B a -> B holds), lhen surely B should be in D a. However, taking the algorithm as it is would imply, for instance, that a propositional wff a, againsl our wishes, would not belong lo Da, because, obviously, Ba -> a doesn't hold. Bul we can solve Ihis by stating the following: if believing B is a consequence of believing a (i.e., Ba -> BB holds), then B shall be in D a. Of course, since Ba - » B enlails lhat B a -> BB, B will also bé in D" , if it is a consequence of believing a Besides, in the same way as in lhe knowledge case, nol only logical consequences of believing only a will be in D " . However, we cannot just take lhe " 4/a(fl)" part of the algorithm as il is, since we are working with logics lhat deal with knowledge and belief. So we should end up with lhe following: B e D a iff B a a VVB) a <MB) -* BB, where 'fcí.B) is the conjunction of KC for ali subformulas KC of B for which C e D a, and -<KC for ali subformulas KC of B for which C t Da\ <í>oífi) is the conjunction of BC for ali subformulas BC of B for which C 6 Da, and -.BC for ali subformulas BC of B for which C l D "; and where L is an EDL. 63 C h a p te r2 Same case as in HM, lhere are formulas a for which D" is nol consistent. For example (HM), o = p a -iBp . a is clearly consistem, but B a , i.e., B(p a -.Bp) implies both Bp and -.Bp, so il is not consistent, and hence Da is also inconsislent. Moreover, even for a consistent a lhe set D " might nol be consistent. Again we consider our preferred example a = Bp v Bq. It is easy lo see that -.Bp e D", -.Bij e D" , because it is not the case that Nj. B a -» Bp, I.e., B(Bp v Bq) -> Bp, thus -.Bp e D a. For the same reason -.Bij e D " , and therefore -.Bp a S q e D a. In view of this, we have that B o a Vo(<*) a <Da(tt) = B(Bp v B17) a -.Bp a - .B ^ , and since B(Bp v Bq) a -.B p a S q -» B(Bp v Bq), we get a e D " . Hence D a is inconsislent. This fact induces HM 10 give anolher definition of honesty based on the algorilhm we have speaking of so far. We say Ihat a formula a is honesto if the set D" is l-consistent, for some EDL-syslem L. As we'II soon be proving, this new notion of honestyo is equivalent to the other two (for the logics to which they apply). We first prove lhe followihg proposition. P ro p o s itio n P 18 . I f a is honesto l^en D ° « stable set. Proof. Let us suppose that a is honesto. First, it is easy to see from the examples above, that (st2) and (st3) are satisfied, that is, B e Da iff BB e Da and B t Da iff -.BB e D a. By the definition of honestyo, Da is consistent, so we have (st4). If then B is some propositional tautology, we have immediately that B a -* BB, so B E D " . Suppose now Ihat B - * C € D a, B e D ", and C t D a . Since we have already proved that (st2) and (st3) hold, we have B(B -> C) e D " , BB e D a and -.BC r £)" , what implies that Da is not consistent, and a would not be honesto against the hypolhesis of the proposition. So (stl) also holds and we are done. ■ The algorithmic approach, then, seems to be the mosl promising of ali, since it applies to ali the logics considered here. Now I guess the reader is buming to slale an objection-or at least a doubt. Remembcr stable sets, and how we didn't find a solulion for some systems, say, Z 7 Why does it work here? Well, the problem in trying to locate a minimum stable set via some set of propositional formulas had this drawback that, for instance in Z, lhere were many stable sets with this same propositional subset. Here we are not taking a lot of sets and trying to choose one-we are building a stable set from scratch. The way the algorilhm works, it always chooses the path of mosl ignorance-if BA doesn't follow from the already decided formulas, then add -.BA. So it is. 25 Puttlng H ali together After taking a look at ali these dilTerent methods of characterizing minimal belief states, with of course difTerent degrees of success, we can try to sum it ali up and see what we get. The following table, in the first place, gives an overview of lhe differenumelhods we have and which of our EDlogics they apply to. 64 M in im a l B e lie f S ta tes logics stable sets saturated sets monoclustered models algorithm Z - - Z2 - _ * ZC . • _ ♦ ZP _ _ • • ZS • _ • ZC2 * • _ • ZCP • • * ZPS . • * • We can now prove lhe equivalence of ali these definilions of honesty, what we do with lhe following theorem, which we also find in |HM84) (Theorem 2, p. 10). The proof is adapted from there. We prove lhe theorem only for lhe cases where L e (ZPS, Z C P), which are lhe only two logics in which ali methods work. First we will need lhe following lemma: Lemma 1,9. (i) I f 94 < 9{ihen B(9$ is IPS-smaller than fl(íM), (ii) I f fM í 9{and O u c Ou lhen B(!\} is ZCP-smaller than Proof. Suppose 94 < 9^ That means 9 4 í 9{, but ÍMV 9{, Le* A be a zero-degree wff such that A e So BA aTM!. *>y P17, 9 4 i= BA. A e B(94). Thus prg(B(9{)) C prs(8(!M)). Now let B be a zerodegree wff such that B e B(9fi and KB e B(9ÓSo KB and, by P16, Hf t= KB, M t* BKB, B e B(94), KB e B(!M). Thus prn(B{9^) C pr%{B{94)). Now lei us consider lhe two logics separately'. (i) In ZP5, since 94%9(1 'n t have lhat M C N and ()\t C On Since 94 * 9{, we must have eilher M c N, or Om c On Suppose Om cOuSo there is a world w such lhai w t O u and w e Of/. Now it is easy to show that lhere is some zero-degree formula A such lhat w & A, but, for every v e Om , % v N A. H follows that 9 Í n BA, and A e fl(íW), bul 9 f y BA, 9 fj= -.BA and A í B(9(j. So prg(B(9{)) c prg(B{94)), Since we already have that pr/c(B(9^)) C prx(B(9t)), B (!\) is ZPS-smaller than B{94). Now suppose lhai M c N. So lhere is a world w such lhat w t 9 4 and v e 9(, If w 6 On , then Om c I On , and lhe proof goes as before. So suppose Om = On We can easily show lhat lhere is some zerodegree formula A such that m> * A, but, for every v e M, 9Í, v i= A. It follows lhat 'M I KA, 'M í BKA, and A e prK(B(94)). Il also follows lhat KA, -,KA, 90= B-.KA and -JCA e B(!Af), hence A e pr/c(B($()). So pr*(B(!\5) cprK(B(94j). Since we already have that prs(B(!AÓ) C prg(B( 94)), B( ' is ZP5-smaller than B(!M). (ii) In ZCP, since Í*í < we have lhat M Q N and Om C 0/ySince ÍV/ 1 we must have eilher Aí c Aí, or O;# c 6>n, or Sm c If now O u c 0N, we can show as before that prg{B(9()) C prg(B(9C)), and thafs enough lo gel lhatB(5\) is ZCP-smaller than B(9f)M 65 C hapter 2 Theorem T17. [HM84) A formula a is hontstM iff il is honesto iff il is honesis iff il is honesig. Proof. The proof is adapled from HM's one. We do a cycle of implications. (a) honesta => honesto: If a is honest*/ then !Ma is the maximum model that satisfies Ba. We need to show that B(íWa) = D" , what we do by proving, by induction on the structure of a formula R, that B € D " iff B e Let B be a propositional variable, and suppose B e D " : then Nj, B a -> Bfl (because VaCB) and <I>a(8) are obviously empty). Now it follows that M a *=L B a -» BB and, since !Wa i=l B a, it follows that 9 fa BB and so B e B((Ma). So assume that, for proper subformula C of B, that C e D " iff C e fl(!Ma). (=>) Suppose now that B e D ". Then we have (by the definition of D") that 1=1 B a a "Ja(fl) a <Pa(B) -> Bfl. Now for every conjunct of the form BC in 'f/a(fl), we must have by definition that C e D" , and thus by the induction hypothesis, C e B(!Ma), and hence M a,w * /. BC, for every w in 'M. In an analogous way, for every conjunct o f lhe form -.BC in fV B ), we must have C t D " , and thus by lhe induction hypothsis C t B(94a), and hence 'M a, w i=/, -.BC, for every w in So M„,w >=/ <Pa(B). In the very same way we get lhal !Wa ,w 1=1, 'Pa(B)It follows that, for every w in !M«, <M a, w B a a 'foi.B) a <Pa(fl), and Ihus that M a, tv t=t. BB. This also holds for every open world w, so fl e B(!Mo). (<=) Suppose now that B e B (M a), and lhat B « D " . Thus i B « a ¥a(B) a 0a(B) -> BB, and hence B a a Vo(B) a d>a(fl) -* fl. We then must have some model 'M - <W. O lSJ> such lhat !\ít= BaA 'Pa(B) a <t>a(B), and B. So there is some w in M such that w fe* fl; M , w N -^B. Now obviously IMS !Ma, so we have W C and w e Wa. We now prove lhe following: ( t) for any proper subformula C of fl, if w t= ¥ a ( 0 a <t>afC) and íW| w N 'f'a(C) a <T>a(C) then !Wa, w N= C iff tv N C. (i) C is a proposilional variable, so f'a(C ) and &M.B) are obviously empty. Now, since worlds are assumed to be trulh-value assignmenls to propositional variables, it is immediate that Hfa, w ►= C iff w N C. (ii) C = -.D. By the induction hypothesis, fW«, w N D iff >v N= D; and obviously 9 fa, w >C iff 'M, w N C. (iii) C = D -» E. By the induction hypothesis, iWi,, w N D iff !M, w N D; and íMa, w t= £ iff w i= E. Obviously M a. w •= C iff !W, w N C. (iv) C = KD. Then íW«, w N KD iff KD is one of lhe conjuncts of <Pa(KD) (since fMa, *v VtÁKD) by hypothesis, and one of KD and -.KD must be a conjunct of l/ ' í(K/J)) ilf 'M, w f KD. (v) C = BD. Then <M a, w n BD iff KD is one of lhe conjuncts of d>o(BD) (since íMa, *v f d>o(BD) by hypothesis, and one of BD and -.BD must be a conjunct of <Pa(BD)) iff í*( w N BD. It thus follows from (t> that !Wa , w i= B , against the hypothesis lhat fl e fl(!Wa). Hence fl € D" . Now, since O n, D " must be consistent, so a is honesto. (b) honesto => honests: If a is honesto lhen by PI 8 D a is stablé. By the conslniction of D °, a e D " . Moreover, for any rerodegree formula fl, we have that fl e D " iff t=i. B a -» BB. We must now show, for each logic L, that D" is the /.-smallest stable set containing a. 66 M inim al B e lie f S ta tes (i) In ZCP, this means that D " musi be the stable set containing a whose belief propositional subset is minimum. It is easy to see that, for every stable set S containing a, prB(Da ) c prj(S). For suppose there is a stable set S containing a and a propositional wff A t prg(S) súch that A e prg(D "). Then, by construction of £>" , B a -* BA. However, B a -> BAe S too, and, since stable sets are closed under boolean consequences, BA e S, A e S. Suppose there is now a stable set T containing a such that T * D a, but prfl(T) = prfl(D"). In ZCP that cannot be the case, because (by T13) stable sets are uniquely determined by their propositional subsets. (ii) In ZP5, we must show that, for every stable set S, either prg(Da) c prB(S) and pr^(D ") Q pr*-(S), or pra(D ") = prfl(S) and prjf(D°) c prrfS). In the same way as in the ZCP case, we prove that prs(D a) C prg(S). Suppose now there is a propositional wff A t pr*(S) such that A e prjf(D"). So KA e D". Then, by construction of Da, t= t B a -> BKA. However, B a -» BKA e S too, and, since stable sets are closed under boolean consequences, BKAe S . KA e S, A e S, A e pr/(( S). Hence prn(Da) C pr/c( S). If now prg(Da) * prB(S), lhen D a is automatically /--smaller lhan S. Suppose pra(D" ) = prg(S): we then have thatprjt(0") c pr*(S), or else S and £>" are the same (by T14). So again D a is /.-smaller then S. In both cases, we have that £>" is the /.-smallesl stable set containing a Thus a is honestf. (c) honests =» h o n esta Suppose that a is honesta, but not honest*/. Since a is honests, there is an Lsmallest stable set S such Ihat a s S. By P15 there is some model 'M s such that S -£ (% )■ We prove first in ZPS lhal iMs = ® mds(Ba). So suppose there is a model 9{o( B a such lhal 94s < By L9.i, B ( t is ZP5-smaller than S, whal cannot be. So, for every mds(Ba), íMj. By P17, í*ís is lhe closed union !Ma o f ali models in which B a holds. Now, since a is honests, a e S, a e 8(3*0), and hence a is honestM. Now in ZCP, let M* be ®{!W: BCM) = B(íWs)) We prove that M* = f l mds(Ba). So suppose lhere is a model 7{o( B a such that %{* £ lf Ou* c On then by L9.ii we have that BCV5 is ZCP-smaller than S, what cannot be. So suppose that Om* = On Il is easy lo show lhal, in this case, B(!\) = so [!M: O(ÍM) = fl(íMs)). It follows lhal !W*, and íW* = 0{, Thus, for every mds( B a), 'Hí, !M By P17, 0>{* is lhe closed union 'M a of ali models in which B a holds. Now, since a is honests, a e S ,a e and hence a is honest*/.» 67 Intermezzo 1 With the end of Part I we have reached a considerable success conceming our main goal, which was to find a characterization melhod for minimal belief states. It's a pity it didn't happen in ali cases with ali melhods, but, most interesting for us, there is an algorithm that we can use with ali systems. Since one of my interests here are programming issues, we could now consider ways of getting the algorilhm implemented. As we saw, the basis of it consists in having a decision procedure for the corresponding logic, so this is going to be our main concem in the first place. And since alethic and epistemic logics have a very similar structure- sometimes, as I already menlioned once or twice, they are lhe same, differences being found only in lhe way you interpret lhe operators-we could lake a look at proof melhods for modal logics as well. In [Pel89] (Section 3, pp. 18ff) we find a discussion of several types of such proof methods. So we have, among the so-called direct melhods, the rnbleau, resolution, and natural deduction methods, and, among the indirect ones, syntactic and semantic methods. We'll talk a liule about tableau systems later in this work (particularly when implementing one), but what I would like primarily to investigate is the melhod of generalized truth-tables, which, I Ihink, deserves a liule more attention, even if, as we'll see, it is not so as efficient as other possible approaches. So in lhe Part II of Ihis dissertation we are going to lake a look at valuation semantics and generalized Iruth-tables for several modal logics- as well as for an example EDL. After that we'll move on to some programming. E I Valuation Semantics and Generalized Truth-Tables Valuation semantics for normal modal logics "That must be wonderfiü! / dorít understand it at ali." 3.1 An hformal overview The aim of Ihis first section is to make an infonnal presentalion of what is called valuation semantics for some systems of modal logic, and of its main byproducl, lhe generalized truth-tables (GTTs for short). I'd say Ihis is a rather complicated kind of semantics-in comparison with possible-world semanlics perhaps even an unintuitive one-so we'U begin take a look at its main ideas, how it is supposed to work, which are the differences relatively to possible-world semantics, and so on. We'll have afterwards a foimal development of lhe whole. I guess probably few people ever heard about valuation semantics, or still remember what it is, so I'd belter tell what I know of the slory. Valuation semantics were first introduced by Andréa Loparitf, in a 1977 paper, for the modal propositional logic K (see [Lo77]). In order to give a brief description of what valuation semantics is, let us take as a starting point a semantics for the classical propositional logic PL: lhere we see that a model is nolhing more than an assignmeni of truth-values to the propositional variables, since the value of complex formulas can be calculated if the value their subformulas have is known. We could also say, in other words, that a model for PL is a function / from wffs into truth-values obeying certain conditions (like/(-v4) *flA), for instance). If we now consider a possible-world semantics for some intensional logic, we notice that lhe structure of a model undergoes a deep change: one doesn't lalk anymore about only one assignment (which, in a sense, describes a possible world), but about a whole set (a "universe") of them. The value of a formula whose main operator is an intensional one thus also depends on lhe value its subformulas get on various other worlds which are accessible. Here is where the famous accessibility relations come into the picture: formally, a model is now a triple <W, R, V>, where W denotes a set of worlds, R is a binary (acessibility) relation over IV, and V is a function which takes arguments in formulas and worlds and goes into tnith-values. The beauty of this construction is that one can get models for different modal logics by laying different conditions upon the relation R. (For instance, requiring of it to be reflexive singles out a class of models which characterizes lhe logic T.) On the other hand, in spite of models changing in this 73 C hapter 3 way, trath definitions for intensional operators like 'o ' (for "it is possible that...") are still given as usual, namely by means of necessary and sufücient conditions (gf-conditions: "o A is true iff this-or-that holds"). Valuation semantics proceed the other way round: a model, which is called a valuation, is just one "world" (a function from wffs into (0,1) having some special properties); that is, one doesn't have to introduce a set of worlds and an accessibility relation. The change comes with respect lo truth definitions for intensional operators, which now appear in the form "if o A is true then such-and-such conditions hold; and if o A is false then such-and-such other conditions hold". One could argue, of course, about the propriety of the statement "a model is just one world", since, as it will be shown later, to evaluate a formula one also has to take other valuations (i.e.: other models) in consideration. More than that, when ali is said and done a valuation ends up being proved to be the characteristic function of a maximal consistent se t In a sense, then, the whole could be like saying, in the setting of a possible-world semantics, that the only universe (model) you have to consider is the class of ali MCSs and, besides, you don't have to bother about introducing accessibility relations. This can be a question of seeing things this or that way. Later on we'll prove some kind of equivalence between valuation and possible-world semantics-which is not surprising at ali, since the same formulas have to come out as valid. Well, if one asks my opinion, I would say the main difference lies on the fact that valuations are not declared a priori to be characteristic functions of MCSs; unlike possible-world models, they are defined inductively for certain sequences of formulas; it is only afterwards that they are generalized and proved to be characteristic functions of MCSs. And it is exactly because they are so defined that they generate in an easy way decision procedures, namely the GTTs, which allow us lo examine ali relevant models to some formula. Back to historical matters, Loparid and I gave, some years after her original paper, a valuation semantics for lhe minimal tense logic Kt ((LM84J; it was presented in 1980 as a short communication on the 4th Brazitian Conference on Mathematical Logic). In my master dissertation, under her supervision, I extended this semantics to several other tense logics as well, including here some naive logics combining time and modality. ([Mor82a, Mor82b]) In my dissertation lhere were also some problems left open, like to adequately define a valuation semantics for S4, still a tough and open case.37 But let us talk a little bit about GTTs. As we will see, one could argue about the propriety of the name "truth-table". They certainly neither are, nor pretend to be, connective-defining truth-lables-as we have, for instance, the one defining the truth-function "conjunction": A 1 0 1 1 0 0 0 0 fig14 We already know that intensional operators like "it is necessary th a t..." are not truth-functional (where the value a formula gets depends exclusively on lhe values of its subformulas). Thus, if one takes the expression "truth-table" in this narrow sense, as meaning something that defines a trulh íunclion, then GTTs are not truth-tables, but something else ("truth-tableaus", maybe). On the other hand, we also talk 37 T here is a "natural*' defin ition o í valuations for S 4 , but an irrçxntan t re su lt cou !dn't be proved. 74 Valuation sem antics fo r norm al m oda l log ics (perhaps by abuse of the language) about lhe truth-table for some formula A, like lhe following one for a->(fr-xj): a b />->a a-»(fc-»a) 1 1 1 1 0 1 0 1 1 0 1 I 0 0 1 1 fij is If we thus understand "truth-table" as denoting this kind of construction, lhen certainly GTTs deserve lhe name. As we will soon be seeing, with GTTs the procedure is pretty much lhe same as in lhe classical, truth-functional case: we also build, for some wff A, a sequence A/,...Am of its subformulas, where A AH is the last elemeni; next we assign values to lhe propositional variables, and after having done this we compute values for the remaining formulas of the sequence. The difference is that lhe value of a modalized Ai in a certain line j of lhe GTT now depends not only on the value in j of its subformulas, bul also on the values which some olher wffs can take in other lines. It should now not be surprising at ali lhat through this construction one can also determine whether A is valid (meaning il is true on ali lines) or not. Well, one can discuss a lot about whether and in which way valuation semantics (with the corresponding GTTs) are somelhing new, or whether they are just another way of presenting possibleworld semantics, or semantic tableaus-whether they are, so to speak, possible-world semantics disguised in another clothes. Guess l 'll better make my presentalion, and let the reader judge by himor herself. (We'U retiun briefly to this lopic in chapter 6.) 3.2 Noimal modal logics I am going to present, in the remaining of this chapter, valuation semantics for some normal modal logics. The conlents will be, first, resuming Andréa LoparuTs original paper on the subject (for K, see [Lo77]), with small changes of my own, and second, also presenting some results I gol in my master dissertation (for K T, K TB , K T5, see [Mor82a]), as well as, third, presenting some new, even if straightforward, extensions of these (KD, KB, KDB, K45, KD45). I'II begin by introducing some notions that will be of general use here as well as in later chaplers of this Pari II. We'll still be considering a propositional language, which now we'U call Lm. Il is like the language L of lhe first part, bul now, instead of lhe epistemic operators 'K' and 'B', we have the alethic necessity 75 C hapter 3 Wfís are defined in the usual way; and 'FOR* still denotes the set of wffs. We introduce now the weak modal operator with lhe rollowing definition:3* D /o . 0-4 =df - .0 -.A. Now an axiom basis for PL consist of lhe following axioms and rale of inference: A l. A -* (B -» A ) A2. (/I -» (B -» O ) -> ((* - » B) -> (A -* O ) A 3 . (-ifl í -*4) -> ((-tB -> A) -► B) MP. A, A B I B A normal modal logic is then to be defined as an extension of PL which includes at least D/O , the following axiom schema: K. D(í4-»fl)-» (q 4 ->oB) and is closed under the following rule of inference: RN. vA I 0 4 ." Taking K as the minimal normal modal logic (i.e., the smallest extension of PL containing D /o , K and closed under RN), we can now build other systems by adding lo it other axioms. In this chapter we are going to consider only logics which can be obtained by adding to K one or more of the following axiom schemas: O. oA -» o A T. OA -> A 4. DA -> DOA B. A -> DOA 5. OA -> DOA.40 In general, we will have KSi...Sn as lhe extension of K obtained with axiom schemas (in any order). For instance, KTB is K plus schemas T and fl; KT5 (or KST) is K plus T plus 5. It can be proved, for instance, that KTS is the same as KT4B. Taking the equivalences in consideration, we arrive at the following picture (cf. [Lem77], p. 58, or [Ch80], p. 132) of 15 non-equivalent normal systems (an arrow means that the logic on the arrow's left is contained on the one on the right): 38 W ork ing w ith delin itío n s maVes proofs shorter and lifc in general easier. N ow , even i f I d o n 't handle " it is possib te that..." here as a prim itive operato r, 111 co n sid er it so in the sem antics part, to show how things can be done. By the w ay, in [Lo77], a lso in [M or82J, on ly necessity is considered. 39 T h e reader w an ting to know m ore about norm al m odal log ics is kindly Teferred to [ChRO], a very readable book. W ith the exception o f B , these ax iom s (in the ep istenác-doxastic version) are a lready know n from P art 1 w here they have nam es like d , and for D here. ■ 76 V aluation sem antics f o r norm al m oda l log ics Some of these systems also have other names in lhe lilerature. Thus KT, KT4, KTB, KT5, KD, and KD45 are also known as T (or M), S4, B, S5, D, and weak S5, respectively. In a possible-world semantics, models for lhese logics are obtained if we lay some constraints upon the accessibility relation R. For K, R could be any (binary) relation whatsoever, but for the other axioms the following conditions must hold of it: T : reflexivity; 4: transilivity; B : symmetry; 5 : euclideanily; D: seriality. Dermitions of proof, theorem, and syntactical cortsequence, for some normal modal logic L, are the same as in lhe epislemic-doxastic logical case (cf. Chapter 1), with the only care of substituting 'K' for so I won't repeat them here. Il is also woth mentioning that the Deduclion Theorem (see T l) also holds heie. Moreover, the following property-an analog of L l, wilh an (almost) idenlical proof-also hold for ali normal modal logics considered in this section: Proposition P19. I fT tA then o r u -n o r tLlA (where D r = [dB : B e T | and - . o r = (-.OB : B e T}). As I said, valuations are going to be defined induclively over certain sequences of formulas, so we need first lo characterize which sequences we are interesled in. We say that a sequence of formulas is a normal sequence of a logic /. if, for l £ i S n , (a) if B is a subformula of A, then there is j < i such that B Af, and (b) for / £ i £ j , if At = Aj lhen i = j . Condition (a) ensures that, for every formula occurring in a sequence, ali its subformulas occur before il. Condition (b) ensures that we won'l have unnecessary repetitions.41 4 1 A l One can see , norm al sequences are ju s l the plain, o ld sequences o f form ulas one leam s in the school to construet i f one is go ing to build a truth-table. 77 C hapter 3 Now a valuation is supposed to be a function from the set FOR into the set (0,1) of truth-values having ceitain properties and satisfying certain conditions-which conditions exactly will of course depend on the logic being considered. The basis of lhe whole construction are functions which satisfy the classical (extensional) conditions: so a function j is called a semi-valuation if j is a function from FOR into (0,1) such lhat: (a) j ( - ^ ) = l ifT j(A) = 0; (b) i(A -> B) = 1 ifT j(A) = 0 or s(B) = 1. It is now easy to prove that semi-valuations also have lhe following properties: (c) s(A a B) = 1 iff j(A) = j(B )= 1 . (d) s(A v B) = 1 iff s(A) = 1 or s(B) = l. (e) s(A «->B) = 1 iff s(A) = s(B). Thus a semi-valuation is, in fact, a model for the classical propositional logic PL. For the modal logics extending PL we need to add some clause or clauses which will lake care of lhe modal operators. We'II do Ihis in two steps: the first one is to define, for each logic L, the notion of A /, . . .^ n -valuations for L , where A i, . . .^ n is a normal sequence. They form a subset of lhe set of semi-valuations, and are obtained inductively: we define first A /-valualions, and lhen go on by laying upon the newly defined A;,....A,-valualions some constraints each time we find a modalized formula. When the A i,...A ñ valuations are at last defined, we exlend the construction to ali normal sequences, thus getting the vatualions for L. And that's il. Having defined valualions, one can go on doing business as usual: a formula is valid if it gets lhe value 1 in every valuation; the semantics can be proved correct and complete, and so on. As I'I1 show later, valualions happen to be the chaiacteristic functions of MCSs, and one could have of course begun by defining them lo be so, bul doing things the way we do here gives us easily the GTTs and decidability. 3.3 Defining »f,.„/(rvaluations So the main point is to find, for each logic, a nice definition of an A;,...,A„-valuation that suits il. Before we do jusl this, !'ll have to introduce some definitions and abbreviations which will be needed. In the following let us suppose lhat I~ is some sei of formulas, and / a function from FOR into (0,1). In a similar way to what we have done by lhe EDLs, we first define the sets of necessities, possibililies and impossibilities of I as: r ° =df (<4 e T: for some B, A = Clfl); r ° =<lf (A e n for some B, A = OB); r - ° =df (A g T: for some B,A = -iOB) 78 Valuation sem antics fo r norm al m oda l log ics As one see, lhey are the subsets of T containing wffs whose main operator is or 'o ' or the combination '- .o Next we define, for each of these sets, its scope ser. e ( r ° ) =df |A : M E r > ) ; e ( r ° ) =df ( A i O A e r 0 ); £ ( r - ° ) =jf Now we define what it means for a function/to satisfy (rejecl) a set of formulas: « /*=« T =yf for every A e T, and for u e (0 ,1),/(A ) = h. Of course, it only seems to be correct to spcak of satisfiability-like "/satisfies P '-in the case of (that is, if u = 1). So in the case of i=o I decided to say lhat" / rejects r ". Just note, however, that "not satisfying" (w hen/gives 0 to at least one of the wffs in O doesn't mean the same as "rejecting" (when/ gives 0 to ali of them). Next we define lhe subset of T having value u according to /as: f/,n t ií M £ n y(A) - íi ). And at last some abbreviations. First of ali, it's going to be quite a job for me having to type-and for you having to read-things like 'A t....A n every second line. So let us agree on the following convenlion (Abbl): we will use 'a ' as a typographical substitulion for 'A /,...A ', so, when we write 'a*. / ' and 'et),', what we mean is actually 'A i , . . .A k í and 'A i,.. .A n , and so forth. In the second place we have the other abbreviation, which will be meaning different things for the different logics, so please pay attention. Abb2: Let a,, be a normal sequence (i.e.: A],... An) and/ , g two functions from FOR into (0,1). We say that, for 1 £ k i n , (a) for K, KD, KT: f< k > g iff g t i e((a*}Q/,i), and * t=o edaíP/.o); (b) for KB, KDB, KTB: f< k> g iff í t= |E ( ( a » ) ° / , i ) , g t o e ( (a t) ° / .o ) , /•= ! e d a * } 0, ,! ) . and / N e d a j l ^ o ) ; (c) for K45, KD45: f< k > g iff j N i £ ( ( a » ) "/ ,i) , * >=o e({a*)°/,o), ( a * )Q/,i = (a * )a«,l. and {at )° /.o = (« itP í.o ; (d) for KTS: f< k> g iff (a * )Q/,i = (a* )a*,l, and 79 ChapUr3 Some words about this ali. First, this abbreviation (showing only the "k" parameter) can be used without fear of confusion if we are working with a fixed normal sequence. If needed, we can also write f<Oh>g-or even f<Ai,...Ak>g, which is more precise. Now to understand what exactly is at stake here, let us consider the sets involved, beginning with (a). '(a*)Q' denotes the subset of a* consisting of those wffs in this sequence whose main operator is a necessary; the subscript \ ' now forms a new sei by choosing lhose wffs among them which are given 1 by / . We take now the scope of this set, and lo this formulas the function g must give a I , iff<k>g is to obtain. Similarly for the second half of (a), only that we are now dealing with impossibilities which must be rejected. For the other cases, things are pretty much the same: variations on a theme. One can think, if one wishes, of 'f<k>g' as representing a kind of "accessibility relation" between two functions, what is not enlirely wrong, just a tilde bit. The idea is lhal, when f<k>g obtains, for instance in (a), g satisfies the scope set o f / s true necessities, and rejects the scope set of its impossibilities. This is similar lo what happens on possible-world models if we have two worlds x and y such that xRy. The difTerence is, we don't have here an accessibility relation simpliciler, since f<k>g holds between this two functions just for the small set of formulas being part of the normal sequence (which is ali we need to evaluate some wff): this relation may not hold anymore if we consider a longer sequence A i,...A kA k+ l. -Ak*j■ Moreover, contrary lo the possible-world semantics, we don'l introduce worlds and relations as primitive elements of models, so we don't have to bolher about what worlds "really are" and what accessibility "really means". And yet anolher remark: in (a), (b) and (c), for instance, the f<k>g abbreviation is the same for two or three different logics. Thinking of f<k>g as being an accessibility relation would imply that it should mean different things, or have difTerent properlies, for each particular logic, what is not the case here. In fact, only the way in which we compute lhe value of modalized formulas will allow us to make the differences (see below) for logics in which "f<k>g" means the same thing. But let us now proceed and take a look (at lastl) to our main deflnition. I II give first the definitions for K, and then we'll see which changes are needed for the other logics. Definition D9. v is a a„-valualion (for K) if a„ is a normal sequence and: ■ 1) n = 1 and v is a semi-valuation; 2) n > 1, v is an a„./-valuation and, if for some m < n, A) = n 4 m, I) if v(/1n) = 0 theii there is an (^./-valuation v„ such that v„(Am) = 0 and v<n-/>v„; II) if v(/4„) = 1 then for every p, every q ,q < p <n, such that Ap = d /t? and v(Ap) = 0 [Ap o Aq and v(Ap) = 1] there is an a„.;-valuation vp such that vp(Aq) = 0 [vp(Aq) I], vp(A„) = 1 and v<n-l>vp. ®) - <>Amt I) if v(/4„) = 1 lhen there is an a„./-valuation v„ such Ihat v„(Am) = 1 and v<n-/>v„; II) if v(A„) = 0 then for every p, every q ,q < p <n, such \i\alAp = o Aq and v(Ap) = 1 [Ap = and v(Ap) = 0) there is an a„ /-valualion vp such that vp(Aq) = 1 \vp(Aq) = 0 |, vp(Am) = 0 and v<n-l>vp. 80 Valuation sem an tics f o r norm al m oda l log ics This definilion certainly loolcs scaring, so let us go slowly through it. Clause 1) gives the basis for the inductive definilion: we have there a normal sequence with just one element (which must be by definition a propositional variable; else there would have been some subformula of it occumng before), and so everything required of v is that it shall be a semi-valuation. By clause 2), for n > 1, if lhe main operator of A„ is not a modal one, nothing has to be done, because lhe semi-valuation properties already take care of the extensional operators. And so we come to lhe case where An = Q4m. If v gives 0 to it, then we have lo look for another a„./-valualion v„ giving 0 to Am and satisfying/rejecting the scope sets of v. One can draw here a parallel to possible-world semantics, where lhere must be an accessible world falsifying Am. If, on lhe other hand, v gives 1 lo Q4m, we only require of v that it has had a "good behavior" before, i.e., that for every false necessity, or true possibility, the condition corresponding to case I) was satisfied. Here there is a difference in relation to possible-world semantics, where we require, for a true necessity 04 , that its scope A gels truth in ali accessible worlds. With valuation semantics this is not the case: we can have v(Q4ra) = 1 and, nevertheless, it may exist some "accessible" /-valuation v„ with vn(Am) = 0 and v<rt-l>vnOur requirements are thus weaker. The case where A„ = <Mm is similar, only with reversed values. Of course, since possibility is not a primitive operator, we might have not considered it here, what would have made the definition shorter. Having thus defined (V valualions, the resl follows in a more or less straigthforward way. We can now say that a function v from FOR into (0,1) ís a valuation iff for every normal sequence a*, v is an a*valuation . The next steps would now consisl in getling some results about valualions, and lhen taking a look on how to prove correctness and completeness. But I would like first to show which kind of modifications are needed in order to get valuations for the other normal modal logics as well. Actually the changes are not that big. I am not going to repeat lhe whole definition; just the places where changes are needed. It goes as follows: • for KB, K4S: - same as K. ♦for KD, KDB.KD4S: A) A n = DAm, I) (as in K); II) if v(An) = 1 then there is an a„;-valuation v„such that v„(Am) = 1 and v<n-l>v„; moreover, for every p, every q , ... (as in K); B) An - O Am, I) (as in K); II) if v(/4„) = 0 then there is an a„ ;-valuaiion v„ such that v„(/t„) = 0 and v<n-l >v„; moreover, for every p, every q , ... (a t in K). « for K T . KTB, KT5: A) A„ = UAm, I) (as in K); 81 C hapter 3 II) if v(i4*) = 1 then v(/4m) = 1 and for every p, every q t ... {as in K); B) A n = o A m, I) ( t u in K); II) if v(A„) = 0 then v(Am) = 0 and for every p, every q , ... (as in K). Now lhe definition for K, KB and K45 is only superlicially the same: remember, 'f<k>g' abbreviates in each of these logics something different! The same holds for the KD, KDB, KD45, and for the KT, KTB and KT5 definitions. Well, what we did until now was to define an â-valuation for a normal sequence a b u t sure we would like to consider longer sequences and so be able to extend this constniction to an on+j-valuation, an af,.t2-valuation, and so forth. There are ways of doing this, but useful is going to be a particular kind of extension which will be called a canonical extension, and whose definition is the same for ali our logics: Definition D10. Let o» be a normal sequence and v an a„;-valuation. We say that vc is the canonical extension of v to a* if: A) for ali m < n ,A „ * üA m, A„ jf OAm and vc = v; or B) for some m < n,A„ = 04», [A„ = 0Am] and vc is a function from FOR into [0,1) such that, for every formula B, 1) if An is not a subformula of B, then v^B) = v(B); 2) if An is a subformula of B, then a) for B = An, vc(B) = 0 [vc(B) = 1] iff there is an a*./-valuation v+ such that v*(Am) = 0 (v+04m) = 1] and v<n-/>v+; b) for B = -iC, vc(B) = 1 iff vc(C) = 0; c) for B = C -> O, Vc(B) = 1 iff v ^ Q = 0 or vc(D) = 1; d) for B = OC or B = o C. vc(B) = v(B). We have now to show that canonical extensions satisfy the requirements of Definition D9, i.e., that they are on-valuations loo. In this we will use the notion of normality. Let v be an otn-valuation: for / < t < n, we say that v is ntrO t-norm al if for every p, every q ,q < p <k, such lhat Ap = oAq and v(Ap) = 0, there is an a*-valuation vp such that vp(Aq) = 0 and v<k>vp. We say lhat v is o \-<ty-normal if for every p, every q ,q < p S k , such that Ap = oA q and v(Ap) = 1, there is an (X* valuation vp such that Vp(Aq) 1 f and v<k>vp. This definition of normality applies not only to K, but also to every normal modal logic here considered, and it corresponds to the condition required on clause I) of the defintion of an a„-valuation. However, for systems other than K, KB and K45, we also need other kinds of normality, namely those corrcsponding to the special condilions occuring in clause II). So we have, for KD, KDB and KD45, that a valuation v is 11) <1^ -normal if for every p, every q, q < p < k, such that Ap = Q4Ç and v(Ap) = 1 lhere is an a*-valuation vp such lhal vp(Aq) = 1 and v<k>vp, pis o o-at-normal if for every p, every q ,q < p < k, such lhat Àp = o A q and v(Ap) = 0 there is an a*-valuation vp such lhat vp(Aq) = 0 and v<k>vp. As lhe reader has probably grasped by now, this condition is the one required to render axiom schema D valid. 82 Valuation sem antics fo r norm al m oda l log ics In lhe case of KT, KTB and KT5, we have: v is Q i-a t n o r m a l if for every p , every q , q < p £ k , such that Ap = OAq and v(Ap) = 1, v(A4) = 1; v is o ool/t -n o r m a l if for every p , every q , q < p £ k , such lhal Ap = oAq and v(Ap) = 0, v(Af) = 0. (This is lhe condition which takes care of axiom schema T.) Thus one can see ihat, even if some logics use 'f<k>g' lo abbreviale lhe same property (as we had for K, KD, and KT), the O iand Oo-normalily requiremenls are different for each of them. In the following table I try lo give an overall view of ali these difTerences: Logic /<*>* clause 11 K (a) _ KD (a) w KT (a) s KB (b) _ KDB (b> w KTB (b) s K4S (c) KD45 (c) w KT5 (d) s fig. 17 Some explanalions. The lelters (a), (b), (c), and (d) in lhe field '/<*>*' refers to lhe meaning of lhe corresponding abbreviations; lhal is, cases (a) ihrough (d) of Abb2. "Clause II" (of the definition of an a „ - valuation) shows whal kind of □ ) and <>o-normality are required in each logic: namely none ("-"), KD-type (" ifeak") or KT-lype ("jtrong"). Now we are ready to gel some results. Lemma LIO. I fv isan a„.i-valuation and vc is lhe canonical extension from v lo On, lhen vc is a semivalmtion . Proof. Straighlforward: just consider that, for i < n, vc(Aj) = v(A;), and v is a semi-valualion. For i < n, clauses b) and c) of D10 ensure that lhe classical properties are respected. So vc is a semi-valualion. ■ f Proposition P20. Let a* be a normal sequence, v an ctH-j-valuation and vc lhe canonical extension o fv lo a n. Let iis suppose, for ali systems, lhal v is Doand o i~a„.i-normal, and, fo r KD, KDB, KD4S, KT, KTB and KT5, that v is Q iand O(s-an-i-nonruú. In this case, vc is an a n-valuation. Proof. First of ali, vc is an a„./-valuation, because it is a semi-valualion and, by construction, for I S i < n, vc(/4,■) = v(A,). Now, if, for every m < n,A „ * ílA m, A„ OAm, vc fulfills every condition of Definition D9, so it is an a„-valuation. Suppose, lhen, lhal for some m < n,A„ = aAm. We have two cases: (I) vc(i4„) = 0. By D10.B.2.a lhere is an <V/-valualion v+ such ihat v+(Am) = 0 and v<n-l>v*. Since v and vc agree for i < n, vc<n-I >v*. So vc is an a„-valuation. 83 C hfyter 3 (II) Vc(i4h) = 1. We consider separately lhe different systems: a ) K , KB, K45: ( t) By D10.B.2.a, for every a„./-valuation v+ such lhat v<n-l>v*, v+(Am) = 1. Suppose now there is q < p S n such lhai Ap = Q4Í and vc(Ap) = 0 [or Ap = o Aq and vc(Ap) = 1], Then v(Ap) = 0 [v(/4p) = 1] and, since v is Doand o i-a„-/-norm al, lhere is an a„ /-valuation vp such that v<n-l >vp and vp(Aq) = 0 \vp(Aq) =1], Since v and »e agree for f < n, we have lhat vc<n-l >vp. Now, from (t), we have lhat vp(Am) = 1 (eise we would have vc(A„) = 0). It follows, in this case, that vc is an a „ - valuation. P) KD, KDB, KD45: If there is q < p < n such that Ap = OAq and vc(Ap) = 0 [or Ap = OAq and vc(Ap) = 1], we prove as in a) that the conditions are fulfilled. We have now to prove that there is an on-f-valuation v„ such that vn(Am) I and v<n-I>v„. If there is som e? < p S n such lhat Ap = aAq and vc(Ap) = 0, or Ap = o Aq and vc(Ap) = I , then we have already proved i t there is an cr*./-valuation vp such that v<n-/ >vp and vp(Aq) = 0 ívp(Aq) = I] and Vp(Am) = 1 Suppose then that there is no q < p i n such that Ap = 04^ and vc(/tp) = 0, or Ap = 0Aq and v^Ap) 1. We have two possibilities: i) there is some q < p S n such that Ap = oAq and v^Ap) = 1, or Ap = oAq and vc(Ap) = 0. Then v(Ap) = 1 (or 0) and, since v is D iand O o-an./-normal, there is an /-valuation vp such that v<n-I >vp and vp(Aq) = 1 (or 0). Since v and vc agree for / < n, we have that v<n-I>vp\ and it follows from (t) thal vp(Am) - 1. ii) there is no q < p S n such that Ap = aA q and vc(Ap) = 1, orAp = o A q and vc(Ap) = 0. Well, in this case, (a * lD= ( a i ) ° = 0, in which case vc t=t E((an)DV(:-t) and vc n o e ( ( a i ) ° , Cio); so vc< n -/> v c and, from (t), v^Am) = 1. It follows, in this case, thal vc is an â-valuation. T) K T, KTB, KT5: If there is q < p S n such that Ap = o 4 í and vc(/tp) = 0 [or Ap = OAq and vc(Ap) = 1], we prove as in a ) that lhe conditions are fulfilled. We have now to prove that Vc(Am) = 1. Since, for every a,,./-valuation v+ such that v<n-I>v+, v+(<4m) = 1, we only need to prove that vc<n-l >vc. In KTS this is immediate, because (a * )Dvc,i = («*)°»c,i and (a* )°»c,o = Í«*)°vc,oFor K T and KTB, we make use of lhe fact that v is D iand o 00 . /-normal. For every q c p S n such that Ap = aAq and Vc(Ap) = 1 [Ap = oAq and vc(Ap) = 0], we have that v(Ap) = 1 (v(Ap) = 0], and it follows from v's normality lhat v(Aq) = 1 K / t f ) = 0]. So v t=i Ê((a * )uv. 1) and v N=o E((a*| °„,o); it follows lhat v<n-l>v, and, since v and vc agree for i < n, vc<n-I>vc and we are done. Hence vc is an (v-valuation. If now, for some m < n, An = oA m, the proof goes in a similar way. ■ We have thus proved that canonical exlensions are a„-valuations under the assumption lhat lhe c^ ./ valuations they are exlending are normal. With the next lemma, we can show thal (Xn-valuations are normal without restriclions, and thus that they can be exlended as long we we want them to be. Just remember that d | and Oo-normalily doesn'l apply to K, KB and K45, only 10 the other systems. 84 Valuation sem an tics f o r norm al m oda l log ics Lemma L U . (Normality Lemma) Let v be an a„-valuation. Then v is Do-, D j-, o o and o i a » - normal. Proof. By induction on n. For n 1 it holds trivially, so let n > 1 and let us suppose lhat every oc„.j valuation is do-, D i-, <>oand o i-a*./-norm al. It fbllows then from P20 that ( t) The canonical extensions of a„./-valuations lo a,, are â-valuations. We have now three cases: (1) For every m < n ,A R * aAm, A„ * o Am. So v is trivially do-, □ ]- , o o and o i-a„-norm al. (2) Let us suppose lhat, for some m < n,A „ = o 4 m. (I) Let v(An) = 0. We have: D ( a , ,) ° v,i = ( a , . i ) o , ll ; 2) {Gfl)°»+to = (a»i./)0v+,o. for every a„./-valuation v+; 3) E ((a«)Dv,i) = E ((aii-/)Q»,i); 4) £ ((a „ )°w+,o) = e((otii-;)0v+,o). for every a«.;-valuation v+. It follows that, for every a„ /-valuatk>n v+, 5) if v<n-l>v* lhen v<n>v*. From the induction hypothesis, v is Doand o i-on j-nomial, so we have: 6) for every p, every q ,q < p < n sucht lhat Ap = Q4f and v{Ap) = 0 (Ap = o Aq and v(Ap) = 1 ], there is an (Xn-Z-valuation vp such lhat vp(Aq) 0 lvp(Aq) = 1] and v<n-I>vp. Now, for each p, let vp* be the canonical exlension of vp lo ot„. Obviously VfHAq) = vp(Aq), and, from (t), vp• is an ofl-valuation. From this, 5) and 6), lhen: 7) for every p, every q ,q < p < n sucht lhat Ap = aAq and v(Ap) = 0 lAp = OAq and v(Ap) = 1], there is an (V-valuation vp' such lhal vp*(Aq) = 0 [Vf,(Af) = 1] and v<n>vp'. On the other hand, since v is an ctn-valuation, we have: 8) there is an a*./-valuation v„ such lhat v„(A„) = 0 and v<r>-l>v„. Now let v„* be the canonical extension of v„ to oc„. Obviously v„'(Am) = v„(Am), and, from (f), v„* is an (V-valuation. ' Thus we have from this fact, togelher with 5) and 8), and from the fact lhal An * oAm\ 9) forp = n , q m , A p aAq and v(Ap) = 0 l<4p = 0 Aq and v(Ap) = 1], there is an an-valuation vp* such that vp'(Aq) = 0 [vp(Aq) = 1] and v<n>vp'. From 7) and 9), then, v is an Dqand o i-a„-norm al. Now, since v(<4„) = 0, v is trivially □ ]and oo-ot,-normal (for systems olher than K, KB and K45). (II) Let v(A„) = 1. We lhen have: 1) {a„)°,.i = (a„./)av-i u M„); 85 C hapter 3 2) (o<i}°*+.0= ((*»!-/)°»+,0. for every a„_;-valuation v+; 3) E ( (a B)°v ,l) = e ( ( a w.í ) ° v,1) u M m); 4) E((otn)°v+.o) = e({a ,./)%+,<>), for every a*/-valualion v+. Since v(A„) = 1, we have from definition 1 that: 5) for every p, every q ,q < p S n , such that Ap 0 4 , and v(Ap) = 0[AP = o A9 and v(Ap) = 1 ] there is an On-7-vahiation Vp such that vp(Aq) = 0 [vp(Aq) =1], vp(Am) = 1 and v<n-l >vp. For each p, let »p* be the canonical extension of vp to a , . Obviously vp'(Aq) = vp(Aq), and, from (t), vp* is an (v-vaiuatkm. It follows that; 6) for every p, every q ,q < p S n , such that Ap = aAq and v(Ap) = 0 [/tp = 0Aq and v(Ap) = 1 ] there is an dn-valuation vp* such that vp'(Aq) = 0 (vp*(/4,) = 1], vp'(Am) = 1 and v<n-I>vp'. We only need to prove now that v<n>vp-, the Odand o i-a„-normality follows. In order to do so we need to consider some logics separately. a ) K , KD, KT: Since vp*(í4m) = 1, vp« *=t E ((a „ ./)a „ j) o (Am); thus, from 3), vp' n ( E ( |a „ )D„ i). From 4), vp ' *=0 E({ot*) °»,o). Hence v<n>vp», and v is Doand o i-a„-norm al. P) K B , KDB, KTB: Since vp*(Am) = 1, vp' i=) e ( ( a „ . ; ) Qv>i) u (Am); thus, from 3), vp* 1=1 e ( (a „ )Dv,i). From 4), vp* >=o e (( a* }%,())• Since v<n-l>vp*, we have by definition that v »=i e ( (a „ í )a V(,*,i), v t=oE((a,./]°»p.,o). From 4), v •=o £ ( (« ,) ^vp^.o). Now, if vp*(A„) = 0, e ( ( a « ) % . , i ) = e ( ( a B. i ) a v,M ) , so v i=m e ( ( a „ ) ° V;, . ti). if vp*(An) = 1, it follows from the definition of canonical extension that for every a„.;-valuation v*, if vp<n-l>v* then v+(^ m) = 1. But now, since v<n-l >vp*, it follows that vp*<n-l >v, and, since vp and Vp* agree for / < n, vp<n-l>vi Thus v(Am) 1, v 1=1 E ({a„)°Ví,»,i). In any case, v<n>vp '; hence v is Doand o j a B- normal. 1) KTS: From 2), {a»}°v+,o = ( a n-/)°»+,o, for every a ll./-valuation v+; so, since v<n-l>vp' , ( a „ ) ° w,o = (ct«) °vp*,0f A) If, now, vp*(A„) = 1, íotM}a W/, • , i = (a » -í} Dvp*,i u M«}: ( a * ) Dv,i = ( a . l Dv/,M and thus v<n>vp*. Hence v is Doand o i-a„-normal. B) Suppose now vp*(A„) = 0. We define, for every p, a new function vp" from FOR into (0,1) in the following way: for every formula B, 1) if A* is not a subformula of B, then vp*(B) = vJ,*(B); 2) if An is a subformula of B, then a) for B A„, vp*(B) 1; b) for B = -C , vp»(B) = 1 iff v /(C ) = 0; c) for B = C -> D, v / (0 ) = 1 iff v /(C ) = 0 or v /(D ) = 1; 86 Valuation sem an tics fo r norm al m oda l tog ics d) for B ~ DC or B = OC, v /(B ) = V (B ). Il is now easy lo see (wilh lhe same reasoning as in Lemma LIO) lhat vp# is a semi-valuation. Besides, for 1 <i < n, vp*(Aj) = vp'(Ai). Since vp' is an a*_/-valuation, V is an cin-y-valualion. We prove lhat vp* is an afl-valuation for KT5. First, we have lhat vp*(A,„) = 1, so vp*(Am) = 1. Lei us now suppose that there is r, s, s < r S n , such thal A , = a A , and vp*(Ar) = 0. Now, ( a , . / ) 0 ,,* ,! = ( a n-í }°»p#,i; and (a<i / ) 0 w? *.o = (" n-/)°vp#,oSince v< n-l >vp ' , we have that ( a „ í ) Qv,i = (ot„_/)a yp* i ; and )°v,o - (an /P v ^ .o Thus for every r, every i . K r í n , if Ar = 0 4 , and vp*(Ar) = 0 then v(Ar) = 0. From D9 it follows that there is an a„./-valuation vr such that vr(A,) = 0, v^A m) = 1 and v<n-l>vr. Thus vp*<n-l>vr, and vp* is an a„-valuation for KTS. Now, since vp*(A„) = 1, (a»}0,.! = (a „ )avp#,i; and (otn)0v,o = (ait)°vp#.oThus v<n>vp* and it follows that for every p, every q ,q < p S n , such lhat Ap = uAq and v(Ap) = 0 there is an a„-va!ualion vp* such lhat vp*(Aq) = 0, vp*(Am) = 1 and v<n>vp*. If now there is p, q, q < p S n , such lhat Ap = oA q and v(Ap) = 1, the proof is similar. That is, v is Cfrand o 111,,-nonnal. 8) K4S, KD45: We prove as in p) lhat vp*t=i £ ((a „ )aVii), vp* t=o £((<*„) ° v,o)From 2), ( a „ ) ° v+,o= («n-i)°»+,0, for every a„./-valualion v+; so, since v<n-l>vp', ( a „ ) ° w,o = (ot„)0^*,0A) If, now, vp »(An) = 1, {a„}aVp*,i = (<*/!-;)qV/,m (A*); (a * )° v,i = (a » la vpM . Thus v<n>vp*, and v is d o and o i-a„-norm al. B) Suppose now vp*(A„) = 0. We define as in y), for every p, a new function vp* from FOR inlo (0,1}. It is now easy to see (with the same reasoning as in LIO) that vp* is a semi-valuation. Besides, for 1 S i < n, vp*(Aj) = vp»(A,). Since vp* is an a„./-valuation, vp* is an a„./-valuaiion. We prove that vp* is an Onvaluation for K45 and KD45. First, we have that Vp*(Am) = 1, so Vp*(Am) = 1. Let us now suppose that lhere i s r , j , i < r S / i , such that Ar = aA s and vp*(Ar) = 0. Now, ( a „ . / ) a v<,«ti = ( a „ . / ) DVí,* i; and ( a / i - / ) 0 »,,»,o = ( « „ . / ] 0 ŵ iK.oSince v< n-I> vp>, we have that ( a n.í) ° » ,i = ( a , , . i ) a vp*,i; and ( « i t - / ) 0 v ,o= ( « » - ; )°w,#,olt also follows, thus, lhat £ ( ( a „ . i ) D»,i) = e d a , , . / ) ^ # . ! ) ; and £((" »-/) °v,o) = £ ((tt„ - /)°Ví,*,o). Thus for every r, every s .s < r í n , i f Ar = a A , and vp*(Ar) 0 lhen v(Ar) = 0. From definition 1 it follows lhai lhere is an a„;-valualion vr such that vAA,) = 0, v^A m) = I and v<n-l>vr. Thus vp*<n-I>vr, and vp* is an an-valuation for K45 and KD45. Now, since vp*(A„) = 1. ( tt„ )°v,i = ( a „ )avp#,i; and ( a „ ) ° Vio = ( a „ ) ° Vf,#.o. Thus v<n>vp* and it follows that for every p, every q ,q < p S n , such that Ap = UAq and v(Ap) = 0 there is an a„-valuation vp* such that vp, (Aq) = 0, V(<4m) = 1 and1 v<n>vp*. If now there is p ,q ,q < p S n , such that Ap = OAq and v(Ap) = 1, lhe proof is similar. Thal is, v is Ooand o i-a„-norm al. We prove now that v is □ ] and o o-a„-norm al (for lhe systems different from K, KB and K4S, of course). That v is Oo-a,,-noimal follows trivially from the fact lhat it is <>o-a„/-normal, because A„ * OA„. By induction hypothesis, v is ü ia » /-normal, and, from D9, we have that v(Am) = I (for KT, KTB and KTS). That is, v is Di-an-norm al. In lhe case of KD, KDB and KD45, from D9, for p = n, q = m, lhere is an (Vj-valualion vp such thal vp(Am) = 1 and v<n-l>vp. We lake the canonical extension v,,* from vp to a„. It is of course an a,,-valuationt and, since vp*(Am) = 1, it follows from 3) and 4) that v<n>vp". So v is Di-an-normal. (3) Let us suppose lhat, for some m < n,A „ = o Am. Proof as in (2). ■ 87 Chapter 3 As a direct result of combining this Lemma with Proposition P20, we have the following Corollary. Let a n be a normal sequence, v an On-i-valuation and vc the canonical extension o f v lo a*. Then vc is an ô-valualion and, fo r 1 i i i n 1 , = v(A,j. That is, now we are sure, if we have some a„-valua(ion, and if we build a normal sequence Ai,...AnAn+ l- A t, that it is always possible to extend this ota-valuation to the new sequence. The next theorem puts ali these fact together: Theorem T18. v is an ctn-valuation iff: l ) a „ is a normal sequence; 2) v is a semi-valuation; 3) v is a* normai . Proof. Immediate. ■ 3.4Correctness Having thus proved these properties of (otn-)valuations, we are now ready to consider correctness. The notions of satisfiability, validity and semantica! consequence are defined, as one could expect, in the standard way: a formula A is satisfiable if there is some valuation v such that v(A) = 1. A is valid (t=A) if for ali valuations v, v satisfies <4. Last but not least, if T is a set of wffs, we say that A is a semamical consequence o/1~, or lhat T semantically implies A ( r A), if, for every valuation v such that v *= T, v(A) = 1. ("v t= 1" , of course, means that v(B) = 1, for ali B e T. And, needless to say, ali this is relevant to some logic L.) In the following. let L be one of K, KB, K45, KD, KDB, KD45, KT, KTB, KTS. Lemma L lí . Let v be an â-valualion; then, fo r I S i i n , i f A,• is an axiom o fL lhen v(Aj) = I. Proof. If At is an axiom of one of the said logics, then it is either an axiom from PL, and it follows from lhe fact that v is a semi-valuation, that v(A|) = 1, or it is one of the modal axiom schemes. We consider each case. (a) Aj o A <-» -o iA . Suppose v(o A «-* -o -u 4 ) = 0. Then we have, say, v(o A) = 1 and v(-.d-tA) = 0, so v(O-iA) = 1. From the normality lemma it follows that for every p, every q ,q < p < n, such that Ap = OAq and v(Ap) = 1, lhere is an a„-valuation vp such that Vp(A ) = 1 and v<n>vp. Thus vp(A) = 1. Now we consider each logic: a) K, KB, K45, KD, KDB, KD45, KT, KTB: v<n>vp means (among other things) that vp t=\ e ((a „ )° v,i). Hence vp(-A ) = 1, vp(A) = 0, what cannot be, since we already had vp(A) = 1. b) KT5: 88 Valuation sem antics fo r norm al m oda l log ics v<n>vp means (among olher things) that î, t=i ( a „ ) aVii. Hence Vp(a-u4) = 1. Now, vp is P i a „ - normal, so vp(-A ) = 1, Vp(A) = 0, what cannot be. If now v(<M) 0 and v(-o-u4) = 1, the proof goes in a similar way. Hence v(OA -0 -v 4 ) = 1. (b) Ai = n(<4 -» fl) -» (0 4 -» aB ). Suppose v(/4,) = 0. Then we have v(D(A -> B)) = v(Q4) = 1 and v(DB) = 0. From the normality lemma it follows that for every p , every q . q < p S n , such that Ap = 0 4 , and v(Ap) = 0, there is an a„-valuation vp such that vp(Aq) = 0 and v<n>vp. Thus vp(B) = 0. Now we consider the logics in two cases: a) K . K B. K45, K D . KDB. KD45, K T. KTB: v<n>vp means (among other things) that vp t=i E ((a ,)av,i). Hence vp(A) = vp(A -» B) = 1, vp(B) = 0, what cannot be, since vp is also a semi-valualion. b) KT5: v<n>vp means (among other things) that vp i=i (a* )av,i. Hence vp(a(A -* fl)) = vp(DA) = 1 Now, vp is O i-a„-norm al, so vp(A) = vp(A -» B) = 1, vp(B) = 0, what cannot be. Hence v(A,) = 1. We must now consider the special axioms of each system. (c) Ai = 0 4 -* o A. (KD, KDB, KD45) Suppose v(/4,) = 0. Then we have v(Q4) = I and v(OA) = 0. From the normality lemma it follows that for every p, every q , q < p S n , such Ihat Ap = 0 4 , and v(Ap) = 1, there is an cg,-valuation vp such that vp(Aq) = 1 and v<n>vp. Thus vp(A) = I. Now v<n>vp means Ihat vp ►o E((<Xn)°v,o). Hence V p ( A ) = 0, a contradiction. Thus v(A,') = 1. (d) Aj = q 4 -» A. (KT, KTB, KT5) Suppose v(/4,) = 0. Then we have v(Q4) 1 and v(A) = 0. From the normality lemma it follows that for every p , every q , q < p S n , such that Ap = 0 4 , and v(Ap) = 1, v(Aq) = 1. Thus v(A) = 1; a contradiction. Hence v(/4,) = 1. (e) Aj = A -» o o A . (KTB) Suppose v(A,) = 0. Then we have v(A) = 1 and v(DOA) = 0. From the normality lemma it follows Ihat for every p, every q , q < p S n , such that Ap 0 4 , and v(Ap) 0, there is an a„-valuation vp such that vp(Aq) = 0 and v<n>vp. Thus vp(oA ) = 0. Now v<n>vp means (among olher) Ihat v i=q £((an}0VJ>,o). Hence v(A) = 0, whal cannot be; thus v(A,) = 1. (0 Aí = dA -> DOA. (K45, KD45) Suppose v(<4/) = 0. Then we have v(OA) = 1 and v(OOA) = 0. From the normality lemma it follows that for every p,every q ,q < p S n , such that Ap = 0 4 , and v(Ap) = 0, lhere is an oti,-valualion vp such that vp(A,) = 0 and v<n>vp . Thus v^fuA) = 0. Now v<n>vp means (among olher) that (On)0*,! = {otn)°vp,l. Hence v(oA) = 0, what cannot be; thus v(A;) = 1. (g) Ai = oA -» DOA. (K45, KD45, KTS) Suppose v(A,) = 0. Then we have v(OA) 1 and v(QOA) = 0. From the normality lemma il follows Ihat for every p, every q , q < p S n , such that Ap = 0 4 , and v(Ap) = 0, there is an a„-valuation vp such Ihat vp(Aq) = 0 and v<n>vp. Thus vp(oA) = 0. Now v<n>vp means (among olher) Ihat (a n )° Vio= (a „ }0Wpio. Hence v(«A) = 0, whal cannot be; Ihus v(A,) = 1. ■ Theorem T19. I f A is an axiom o f L and v is a valuation, then v(A) = I. Proof. Let A be an axiom of one of the said logics, and v a valuation. Let On be a normal sequence such Ihat, for some i í n, A = A,. By definition, v is an a„-valuation and, from L12, v(A) = l .a 89 Chapler.? Lemma L13. For ali n .a ll i, 1 S i S n . if v is an a„-valualion and i--f. A j , then v(Ai) = I. Proof. By induction on the number r of lines of a proof of Aj in L. A) r = 1. Then Ai is an axiom, and the propeity follows from L12. B) r > 1. If Ai is an axiom, the property follows from LI2; else: (a) Ai was obtained by MP from B and B -* A[. We have that (-8 and i-B -> Ai. Let us form the following set t = fC : C is a subformula of B -» A/ and C t \A i,...A n }} . If t * 0, let us put the elements o f t in a sequence C ;.... C* respecting the length of the formulas. If t = 0, let o = A;,...,A„ and vc = v. Else let o = A i,...A n ,C i .....C*, and let us define a sequence vo.v;.....v* where vo = v and, for 1 S j £ k, lei vj be the canonical extension of vjj . Let us take vc = v*. Obviously o is a normal sequence, and vc a o-valuation. Since iB and t-B -> Ai, we have by the induction hypothesis that vc(B) = vc(B - » Ai) = 1. So vc(A/) = 1. Since v(A,) = vc(A,), v(A,-) = 1. (b) Ai 08 and was obtained by RN from B. Well, in every normal sequence a in which Ai occurs, B occurs too; so, by the induction hypothesis, for every o-valuation v, v(B) = 1. If now v(A,) were to be 0, there should be an a„-valuation v„ such that vn(B) = 0, what cannot be. So v(<4j) = 1. ■ C o ro llary . I f i-A lhen t=A. Proof. Suppose hA , and let v be a valuation. Let a , be a normal sequence in which, for some / S n , A = A(. By definition,» is an otn-valuation, so v(A) 1 from L12, and thus t=A. ■ Theorem T20. (Correctness Theorem) f /D -A then D=A. Proof. Suppose I~h A, and let D i,...J)r be a deduetion of A from r . We prove the theorem by induction on r. A) r = / . Then, either A e T\ and we have nothing to prove, or A is an axiom, so A is valid (by corollary to L13 and F t= A. B) r > 1. If A t T and A is not an axiom, then: (a) for some ] < r, l < r, Di = D j-> A. So T H D/, T H £>, -> A and, by the induction hypothesis, F t= Dj, r i= Di -» A. Thus, for every valuation v, if v t= I~, v(Dj) = v{Dj -> A) 1, and hence v(A) = 1. So T t= A. ' (b) A = Ofl and, for some j < r ,D r = B. In this case, i-D r and HA. By the Corollary to L13, for every valuation v, v(A) = I. So, if v t= T, v(A) = 1. Thus T t= A. ■ 3.4 Completeness Completeness is now easy to prove making use of saturated sets-which are just MCSs. They are defined in lhe very same way as in chapter 1, thus properties like in P5 or P6 also hold: 90 Valuation sem an tics f o r norm al m oda l logics Proposition P21. / / A is saturated, lhen: (a) A e A i f f A iA; (b) -A e A i f f A t A; (c) A -» B e A i f f A t A or B e A. Proposition P 22 .1/T v A, lhen lhere is an A-salurated sei A such that I ' C A. We now consider some properties that we'll need in the completeness proof. Lemma L14. Let A, 8 be any saturated sets c f wffs, T any set ofwffs. Following properties hold: a) L (that is, every normal logic): i) I fT v 04 , then there is an A-saturated se IA such that eOT0) u £ (r^ ° ) C A. i i ) / / F ^ -,0A , then there is an -A-saturated set A such ihat e (r°) u e ( r^ ° ) C A. b) KD. KDB. KD45: i) / / I b' -OA, then lhere is an -A-salurated set A such that eíF 3) u £(í ^ °) C A. ii) I f T * o A, lhen there is an A-saturated set A such that e(r°) u e (r^ ° ) C A. c) KT, KTB, KTS: i) e(A°) Q A; ii) eCA-0) Q A; d) KB, KDB, KTB, KTS: i) e(A°) Ç. 0 i f f £ (0 °) C A; ii ) e (A ^ ° )c :e i f f E(0--0) C A; e) K45, KD45, KT5: i) i/e(A °) v £ (A -°) C 0 then A° = 6 ° and A-10 = B"-0 ; ii) A° C e(A°); iii) A -° C £(A -°). Proof. (a.i) Suppose r v 0 4 . Then r ° o v 0 4 , since boih are subsets of I \ By P19, E (r°) w E (P-°) b* A. Froiti P21, there is an A-saturated set A such that E(r°) u £ (r^ ° ) Q A. (a.ii) If T v -.OA, lhen T v CHA. By (a.i) there is a -v4-saturated set A such that £(1^0 u E(P^°) C A. (b.i) Suppose T ty -UA. Since D fo and D are axioms of these logics, we have lhal D-iA -* -O A as theorem. So T b* d-v4. By (a.i) there is a -^A-saturated set A such that E(r°) u e ( r^ ° ) C A. (b.ii) Suppose r b* OA. Since D is an axiom of lhese logics, r b* 04 . By (a.i) lhere is a A-saturated set A such that E(r°) U £(1^°) C A. (c.i) Let A e e(A°). So 0 4 e A. Since T is an axiom, A e A. (c.ii) Let -w4 e e(Â°). So -.OA e A. Since 7" is an axiom, A -> OA is a theorem, so A t A; -«4 e A. 91 C hapter 3 (d.i) Suppose e(A°) C 8 , and let A e £ (0 °). So oA e 8 ; -0 /1 f 8 and D-.DA t A. But then -iO-iDA 6 A; that is, ooA e A, and, since toqA -> A, A e A. The other direction and (d.ii) are similar. (e.i) Suppose e(A°) u e(Ã'°) C 8 , and let 0 4 e A°. Thus dA e A and, since 4 is an axiom, ddA e A. Thus 0 4 e 8 ; OA e 8 a . Let now oA e 8 a . Then DA 6 8 . If DA É Aa , oA t A; -O A e A. Since 5 Is an axiom we have as theorem 0 4 -> o -o A . So D -oA e A. But lhen -o A e 8 , because e(A°) C 8 , and this is a contradiclion. So OA e A°. Let now -.oA e A-'0 . Thus - .o A e A and, since 4 is an axiom, -.OOA 6 A. Since t(A ^°) C 8 , we have that -.OA f 8 ; ->0A e 8 ^ ° . Let now -.oA e 8 ^ ° . Then -.oA e 8 . If -.OA t A-10; -.oA í A; OA e A. Since J is an axiom, we have as theorem o - ,o A -» - .o A. So 10- .0 A e A; DOA e A; and thus oA e 8 , since e(A°) C 8 . But this cannot be, so If -.oA e A-1®. (e.ii) If 0 4 e A, then, since 4 is an axiom, ODA e A, UA e e(A°). (e.iii) is proved in a similar way. ■ Theorem T21. For every A-saturated set A and every normal sequence a„, the characteristic function f ofA is an â-valuation. Proof. First of ali, it is easy to prove by P2I that ( t) The characteristic function/of A is a semi-valuation. We now prove the theorem by induction on n. If n = 1, the property follows from ( t) above. Let us suppose n > 1. (1) If, for every m < n ,A „ * OAm, A„ * o Am, f is trivially an â-valuation. (2) For some m < n, A„ = OAm. = 0. Then A „ t A, A fc' oAm. From L13, there is an A„-saturated set 8 such that e(Aa) u e(Â°) c 8 . Letf y be the characteristic function of 8 . By lhe induction hypolhesis,/and f y are a * .j- valuations. We also have, since 8 is Am-saturaled, that/e(Am) = 0. We consider now each logic: a ) K, KD and KT: Now, ( a M. / ) a/,i C A, thus e d a , . / ) 0/,)) C e(AD) C 8 ; thus f e n=) E ({a»-/)a/,i) . Let now OA e (" n / ) o/.0Then/e(-iO A ) = 1, so /e(-*4) = l,/e (A ) = 0. T hus/e n 0 E ((a„-/)0/ to). We can thus say that/<n-f >/e; hence/is an a*-valuation. p) K B, KDB, KTB: We prove as in a ) th a t/e M e ((ctn / )a/.l) a n d /e t=o E ((ot„/)°/,o). Now, from L13, e(A°) c 8 iff 6 (8°) C A; and e(A ^°) Ç. 8 iff e (8 ^ ° ) C A. It is easy to conclude th a t/(= | e ((a„ .f )D/e .i) and/t=o £((<V /)0/e.o)We thus can say that f< n-l> fy\ hence/is an Ai,..„An-valuation. Í) KTS: We prove as in a ) th a t /e N i e ( ( c t „ i | a/ ,i) and / e t=o e ( ( a „ . | ) °/,o), and as in P) that />= i £((<*„ ; and/ no £((ot„ / ) °/e.o)Now, from L13, Aa C e(AD) and A ^° c e(Â°). It is then easy to see that (a „ ./)a/,t = ( (V /)n/e ,i, and (a n.;)°/,o = (« * í P/e.OWe thus can say lhat/< n-/> /e; hence/is an a„-vaIuation. S) K45, KD45: 92 Valuation sem an tics f o r norm al m oda l log ics W e p r o v e a s in a ) t h a t / e 1= 1 e ( ( o i n - ; ) a/ , i ) a n d / e t=o c ( ( t t « ; ) ° / , o ) , a n d i t is e a s y to s e e , f ro m (e . i ) o f L13, that = ( a „ . j ) D/e ,i, and (a „ .í ) <>/,0= (a > i l lo/e ,0. We thus can say lh a t/< n - /> /e ; hence / i s an (Xn-valuation. (II)/(A „) = 1. S o aA m 6 A, A HDAm. Let us suppose lhere is som ep, some q ,q < p S n such that Ap = aAq aaAf(Ap) = 0. From L13, lhere is an Af-saturated set 6 such that £(4°) u e(Ã'°) C 6 . L e t/e be the characteristic function of 6 . By lhe induction h y p o thesis ,/and /e are a H./-valuations. With an analogous argument as in case I), we show that f< n-l> fy. Since 6 is A4-5aturated,/e(A4) = 0 and, since Am e e(Aa) ,fe (A m) = 1. Now, in the case of KD, KDB, KD4S it follows from L13, since A i-DA,*, that there is an A m- sahiraled set 6 such that e(A°) u e(A'~'0) C 6 . We prove in a similar way th a t/e is an a„/-valualion , f< n-l >/e and/e(A*) = 1. In the case of KT, KTB, KT5, it follows from L13, since A !-□Am, that Am £ A. So f(A m) = 1. If there is now som ep , some q ,q < p S n such that Ap = OAq and fíA p) = 1, the proof is similar. It follows lhat / is an otn-valuation. (3) For some m < n, A„ = OAm. Proof as in (2). ■ Wilh this resull we come now to lhe following Corollary. v is a valuation iff v is lhe characteristic function o f some saturated set A. Proof. (a) Let us suppose that v is a valuation. Let [v]i = (A : v(A) = 1), and let (v]o = (B : v(B) = 0). Let C e [vjo; so C e [ v] j and we easily see that [v)i C. Let D be a formula such lhat D t M l-Sov(v) = 0, v(-iD) = 1 and -D e [v ] | . Bul, since l- iD-»(D-»C), v(-<D-*(£>-»C)) = 1; thus -D ->(D->C) E [vh. So M l tD->C, and, obviously, (vji u (D) iC. Lei A = M lHence A is a C-saturated set and, by construction, v is its characteristic function. (b) Suppose that, for some saturated set A, v is the characteristic function of A. Let a„ be any normal sequence: then (by T21) v is an oa-valuation. Since On is any normal sequence, v is a valuation. ■ Theorem T22. (Completeness Theorem) //D =A lhen D-A. Proof. Suppose IV A , and IV A . Then there is an A-saturated set A such that r C A. Let v be the characteristic function of A. By lhe corollary lo theorem 3, v is a valuation. Since r C A, v t= T; since A is A-saturated, v(A) = 0. So IV A , against the hypothesis. Hence D-A. ■ 93 Valuation semantics for ciassicai modal logics / have yel 10 see any problem, however complicated, which, when looked at in lhe righl way, did nol become still more complicated. P O U L A N D E R SO N . In (his chapter we are then going to consider valuation semantics for classical modal logics. A system of modal logic is calied classical if it contains D /o (i.e., o A <-> o A ) , and if it is closed under RE: R E : 1 A h B / a B .42 The smallest classical modal logic is calied E. To name other classical systems we write, as usual, ES i ...Sb to mean the extension of E through axiom schemas The axiom schemas which we will be using here are the following: M. D(A a B) -* DA a O B C. OA A DB -» D(A a B) Also at our disposal is the inference rule RN (i-A / 104), already known from the normal modal logics. Thus ECN will mean the logic obtained by adding C as axiom schema and RN as inference rule. Using ali possible combinations of these axiom schemas and rule of inference, we arrive at the following picture (cf. [Ch80], p. 237) of 8 non-equivalent logics (an arrow means that the logic on the arrow's left is a subsystem of the one on the right): 42 M ore abou t classical m odal log ics can be found in C hcllas [1980], chapter 8, w hich 1 am going d o se ly to follow. EM EMC fig. 1» 9 4 Valuation sem antics f o r classica l m odal logics The logic EM is also calied M in [Ch80], because it is the smallest monotonic modal logic, and EMC has the denomination R too, because it is the smallest regular modal logic.43 The system EMCN, by lhe way, is lhe same K which we already knew-that is, the smallest normal modal logic. Now in lhe case of classical logics, things work in a similar way to the normal modal logics: so we have normal sequences and semi-valuations as before. Differences are, of course, lo be expected in the definitions of <4;,...^ -v a lu a tio n s . 5.1 Def inlng A}r..^ n-valuaUons for classical logics As usual, we need some definitions and abbreviations. Let a„ be a normal sequence, and lei us suppose lhal ctn-valuations, and valuations simpliciter, were already defined. We introduce lhe following abbreviations (for 1 S k S n ,and where r.AC (a*)): A «*k B iff for every a*-valuátion v, v(Â) = v(B); A **k B iff for every a*-valualion v, v(A) * v(B); A ®*>k B iff for every a*-valuation v, if v(A) = 1 then v(B) 1; A «>k B iff for every a*-valuation v, if v(A) 1 lhen v(B) = 0; A <*k B iff for every a*-valuation v, if v(A) = 0 lhen v(B) 1; A «k <r. A> iff for every a*-valuation v, v(A) = 1 iff vt= i T and vt=o A; A »<k <r, A> iff for every a*-vaIuation v, v(A) 0 iff v*=o T and vt= t A; <r, A> *»>k A iff for every at-valualion v, if vt= j T and vt=o A then v(A) = 1 <r, a> «>k a iff for every a*-valuaüon v, if vt=o r and v*=i A then v(A) = 0 Now lhe following abbreviations, as in the case of "f<k>g" in normal logics, will be meaning different things for lhe several systems. Lei a„ be a normal sequence, and let us suppose again ihat a „ - valuations were already defined. We introduce the following abbreviations (for l í k £ n , and where I \ A C (a*)): (a) for E, EN: , =df (Be T:B~kA ); Xk[A .n =df ( B e r : B k A ); (b) for EM, EMN: ÇkH,n =df ( B e r:B~>kA ); XkH.H =df (Be r : B <«k A ); ÇkH.H =df (Be r:H.>kB ); 43 A sy stem o í m odal lo g ic ia sa id to be m o n o to n ic i f f il con ta ins D fo and is closed u nder R M , i.e .: A - * B / oA -► ü B . A m odal lo g ic is sa id to b e r e g u la r i f f i t con ta ins D fo and is closed un d R R , tha t i i : A a f l C /o A a q A -» o C . (C f. [C h80], p . 2 34 .) 95 Chtpier4 T l ^ . n =df { B e r : A « > k B }; (c )fo rE C , ECN: Çk[y4, r . A] =df ( < 0 ,1 » : 0 C T, * C A and <4 < 0 , <I»); XkM. r . A] =df ( < 0 ,4 » : 0 C T, <t> C A and A *>k < 0 , <6>); (d) for EM C, EMCN: Çk[/i, r . A] =dí ( < 0 , * > : 0 C r , <t> C A and < 0 .4 » Xk[/4, r, A] =df ( < 0 ,4 » : 0 C r, <t C A and < 0 , * > «>k A ). Before we go into lhe details of what ali these definitions mean, iet us take a look at the definition of an dn-valuation. Perhaps things will be clearer by then. I'll give first the definitions for E, and then we'll see which changes are needed for the other logics. Deflnitlon D l l . v is an a„-valiiation (for E) if a„ is a normal sequence and 1) n = 1 and v is a semi-valuation; 2) n > 1, v is an iv /-valuation and, if for some m < n, A) A n - QAm, I) if V04*) = 0 lhen e((rx„.,)°v,,)] = xn , [*m. e ( ( a „ ; ) ° v.o)] = 0; II) if v(A„) = 1 then for every p, every q ,q < p S n , such that a. Ap = 0 4 , and v(Ap) = 0, e ( (a n ; ) av .l)u Mm)] = Xn l I 'V e((a«i./)0v.o)l = 0; b. Ap = oAq and v(Ap) = 1 ,x n , Mf. e ( (a , í ) Dv .i)u Mm)] = Çn l Mf. e ((a ,,-/)0v,o)] = 0: ®) A» = 1) if v04„) = 1 then xn_1Mm, e((a„.,)°v .i)] = Çn l [-4m. e ((a B. / ) 0v,o)] = 0; ti) if v(Ab) = 0 lhen for every p, every q, q < p < n, such that a. Ap = 0 4 , and v(Ap) = 0, x n , M , . e ( (o t- ;)D,,l)) = Çn l [<V e((a„./)°v,o) u Mm>/ = 0; b. Ap = oA q and v(Ap) = 1, x ^ M , , e(íotn_/)ov,i)J = Çn-'[<4Í , e ( (a „ .; ) ° v,o) u Mm)] = 0This definition, too, looks scaring, but by now the reader has probably got a feeling of how things work with valuation semantics. Everything is like in the noimal logics case, but the way we treat the modal operators. Let us consi(|er the case where A„ = o 4 m. If v gives 0 to it, then the set of formulas belonging to the scope of v's necessities which are equivalent to Am must be empty. This is actually what is required to make RE validity-preserving: we would not want to have some 0 4 , getting value 1, and Aq being equivalent to Am-in which case giving 0 to An would mess things up. So e ((a B.; ) Dv,i)] must be empty. Similarly, lhe requirement thal Çn l [A,j, e((ô, ; ) 0v.o)) should be empty guaranlees the validity of D fo , because we are then sure there is no o -A m, for instance, such that v(o -A m) = 0-in which case v(-iO-u4m) would be 1, and it would be bad to have v(o4m) = 0. The case with v(An) = 1, as in the normal modal logics, requires that v has had a good behavior. For possibilities, the picture is analogous. Maybe it is a surprise for lhe reader that we are not requiring, in the case where v(o4m) = 0, that there is some "accessible" valuation giving 0 to Am. Actually this condition is the one that guarantees that % Valuation sem antics f o r classica l m odal logics RN is validily-preserving. But RN doesn't hold in E, so il doesn't malter if there is or not another valuation with v(Am) = 0. (Things are different in EN, as one can see below.) Having thus defined a„-valuations, the rest is standard: a function v from FOR inlo (0,1} is a valuation iff for every normal sequence 0*. v is an og,-valuaüon. Before going into canonical extensions, normality and so on, I would like to show which kind of modifications are needed in order to get valuations for the other classical modal logics. As in the preceding section, we need only some small changes, so I am only going to repeat each time lhe most important part of the definition. Thus we have: • for EN: A) A . = OAm, I) (as in E ) ... and there is an a„./-valuation v„ such that = 0; II) ... a. (as in E ) ... and there is an a„/-valuaiion vp such lhat vp(Aq) = 0; b. (as in E ) ... and there is an a,,./-valuation vp such that vp(Aq) = 1; B )A n = O A m, I) (as in E) ... and lhere is an a,,./-valuation v„ such lhat v„(Am) 1; II) . .. a. (as in E ) ... and lhere is an a*./-valuation vp such that vp(Aq) = 0; b. (as in E ) ... and there is an a„./-valuation vp such lhat vp(Aq) = 1. What was added here was just a requirement like "and there is an a„./-valuation vH such that Vn(Am) = 0" (or 1), what is, as I said, necessary to guarantee the validity of RN. If a necessity is false, then somewhere its scope must get a 0. But (his is ali: no need lhat this other a„/-valuation be in any kind of relation to v. Let us now see how things look in the case of the other classical logics. » for EM: A) An = OAm, I) (as in E) II) ... ' a. (as in E) h .A p = 0 A q and v(Ap)= l .r i" 1 » , E ((a„-/)aw,l) ^ (^m)) = í n l [A,, e ( (a „ . / j° v,o)] =0; B) An = OAmr I)ifv (A n)= 1 then £((" /.-/)DV,|)] = E ((a „ ,)% ,0)] = 0; II) ... a. (tu in E) b. Ap = <>Aq and v(Ap) = 1, £ ((« „ .;)av.i)l = £((a.-/)°w,o) u (/!„)] = 0. 97 C hapter 4 * for EMN: The only diference in relation to EM is that one should add, as in previous cases, the "and there is an a,./-valuation..." story, which takes care of RN. For flie other logics, now, the differcnces are somewhat greater, so let us write them down whole. « for EC. EMC: A) An = n 4 m, I) if v(A„) = 0 then e ( (a n. / )Qv,i), E ( ( a „ d ° v,o)l = 0; II) if víA«) = 1 then for eveiy p, every q , q < p S n , such that a. Ap = 0 4 , and v(Ap) = 0, e([a« / ) av,l) u (Am), E({a„.í)°»,o)J = 0; b. Ap = OAq and v(Ap) = 1, xn , [* í. £((" »-() °v.o). e ((a„ .,)°„ ,i) u (Am)j = 0; B) A„ = c>Am, I) if v(An) = I then e ( (a „ ./)0»,o). e((a„.;)°v ,i)] = 0; II) if v(AB) = 0 then for every p, every q , q < p S n , such that a. Ap = oAq and v(Ap) = l , x n' , Mí .e ((a « -/)°v .o )u Mm). e ((a„-;)a ,.i)] = 0; b. Ap = 0 4 , and v(Ap) = 0, « d " » / ) Dv.O. «((«/■ í)°v,o) u (Am)] = 0. It is probably not necessary to say that, even if the definition looks the same for EC and EMC, 'Ç* and *x' abbreviate different things) Now for ECN and EMCN ali we need is to add the "and there is an a,<-/-valuation..." story, which takes care of RN (see the case of EN). I hope there is no need to repeat the definition, because I won't. By the way, since EMCN is the same logic K, we have here an altemative definition of an otn-valuation for K. It is a good exercise for the reader to prove the equivalence of this defintion with the one given in the section on normal modal logics! The next definition now considers lhe canonical extensions (first for the logic E). Definition D12. Let a„ be a normal sequence and v an a n.;-valuation. We say that vc is lhe canonical extension of v to a,, if: A) for ali m < n, An * Q4m, An * 0A m and vc = v; or B) for some m < n,A„ = aAm or A„ = OAm and ve is a function from FOR into (0,1) such lhal, for every formula B, 1) if A„ is not a subformula of B, lhen Vc(B) = v(B); 2) if A„ is a subformula of B, then a) for B = An = aA m, vc(fl) = 0 iff £ " '[ * « . £((<*„./l^ .O ) = xn l Mm, e((a„_,)°„.o)) = 0: a') for B = An = o Am, vc(B) = 1 iff xn l Mm. e d a , . ; ) ^ . ! ) ] = £((«»■;)°v,o)] = 0; b) for B = -iC, vc(B) = I iff M O = 0; c )fo rB = C-*D ,V c(B )= 1 iff v^C) = 0 or vc(D) = I; d) for B = DC or B = o C, vc(B) = v(B). 98 Valuat ion sem antics f o r classica l m odal logics This definition now must undergo some changes, in order to be adequate to the other logics. We have: • for EN: a) (as in E ) ... and there is an a„ y-valuaüon v+ such that v+Ôm) 0; a1) (as in E ) ... and there is an a„ ;-valuation v+ such that v+(Am) = 1; • for EM: a) (as in E); a ) for B = A„ = ° A m. Vc(fl) = 1 iff T|n *1^- £((" » / )av.l)l = e((a„-/)°v.o)] = 0; • for EMN: a) (as in E M )... and there is an a„;-valualion v+ such that = 0; a1) (as in E M )... and there is an a„/-valualion v+ such that »+(/!„) = 1 ; • for EC, EMC: a) for B = A„ = aA„, Vc(B) = 0 i f f ^ 'Mm, e ( ( a , . ; )□„,!), e<{<x„.;)%,0)] = 0; a ) for B = A„= O Am,v c(B)= 1 iffXn lMm. e((" 7-/) V o ). E (( tt-z )av,l)l = 0; • for ECN, EMCN: a) (as in EC, E M C )... and lhere is an a,./-valuation v+ such lhat v+(Am) = 0; a1) (as in EC, E M C )... and there is an a„./-valuation v+ such that v*(A„) = 1; In the next slep, as usual, we introduce the notion of normality. Let v be an (V-valuation: for l S k S n , we say that: •fo rE : (a) v is a -a t-n o rm a l iff for every p , every q ,q < p S k , such that Ap = n A q and v(Ap) = 0, ^ [ 4 , , e ((a jt)av.i)] = Xk[^ í. e ( (a * )0w,o)] = 9\ (b) v is o -a t-n e r m a l iff for every p, every q ,q < p S k , such that Ap = 0 A q and v(Ap) = 1, XkM?. E ((« t)a v,l)] = ÇkM ?. é((«*)°v,o)] = 0- • for EN: (a) (as in E ) ... and there is an (V/-valuation vp such that vp(Aq) = 0; (b) (os in E ) ... and there is an On./-valuation vp such that vp(Aq) = 1. 99 C hapter 4 • fo r E M : (a) (as in E) (b) v is O a k n o r m a l iff for every p, every q , q < p S k , such that A p = o A q and v(/4p) = 1, T|k[j4,, e ( (a * l° ,,t) l = Çk[A„, e ( ( a t ) ° v,o)l = 0. • for EMN: (a) (as in E M )... and there is an a„./-valuation vp such that vp(Aq) 0; (b) (as in EM) ... and there is an ctn-f-valuation vp such that vp(Aç) 1. • fo rE C . EMC: (a) v is O O t k n o r m a l iff for every p, every q , q < p S k , such that A p = 0 4 , and v ( A p ) = 0 , Çk[-4,, e((a*)°v ,i). e ( (a t )°„,o)] = 0; (b) v is o -atc-normal iff for every p , every q ,q < p S k , such lhat Ap 0 A q and v(Ap) = 1, XkM?. e({ott)®v,o). E ((« t)Dv,l)] = 0- • for ECN, EMCN: (a) (as in EC, EMC) ... and lhere is an a„./-valuation vp such thal vp(<4,) = 0; (b) (as in EC, EMC) ... and there is an (^./-valuation vp such that vp(Aq) = 1. Now the story repeats itself, just like in normal modal logics. We prove that canonical extensions are semi-valuations, and then that they are â-valuations, if normal. Last but not least, we prove that o»valuations are normal up to n. Lemma L15. I f v is an (Xn-i-valuation and vc is the canonical exíension from v to a„, then vc is a semivaluation . P r o p o s i t i o n P 2 3 . L e t a „ b e a n o r m a l s e q u e n c e , v a n a „ . i v a l u a t i o n a n d v c t h e c a n o n i c a l e x t e n s i o n o f v t o a , , . L e t u í s u p p o s e t h a t v i s O a n d O a ^ . i n o r m a l . I n t h i s c a s e , v c i s a n O n v a lu a t io n . Proof. First of ali, vc is an ô./-valuation, because it is a semi-valuation and, by construction, for l S i < n, vc(Aj) = v(/4j). Now, if, for every m < n ,A „ * a A m, A„ * 0 A m, vc fuifills every condition of Definition Dl 1, so it is an a„-valualion. Suppose, then, that for some m < n,A„ = aAm. We have two main cases, and a lot of subcases: (I) vc(A„, = 0. (A) E, EN, EM, EMN: By D12.B.2.a e ((« „ ./}av ,)] = Z" '[-4m. e ( ( a „ . , ) \ , 0)] = 0 [EN, EMN: and there is an a„ /-valuation v+ such that v+(/4m) = 0 ] . Since v and v c agree for i < n , Ç n l [ A m , E ( ( « „ _ / ) DVf i ) ! = XI'"'['4m, E((an_ ;)°vc,o)| = 0. So vc is an a„-valuation. (B) EC, ECN, EMC. EMCN: 100 Valuation sem antics f o r classica l m oda l logics By D12.B.2.a ^ 'M m , £((" n í )av,l). E((a<i-/)°v,o)] = 0 [ECN, EMCN: and lhere is an a „ / - valuation v+ such Ihat v+(Am) = 0]. Since v and vc agree for i < n, £ "-* [Am, E ( ( a a . / ) Qvc,l). E((tt«i-/)0irc,o)) = 0. So »c is an a„-valualion. (II) vc(A„) = 1. We consider the cases for lhe different logics: (A) E , EN, EM, EMN: By D12.B.2.a, we have: (1) either Ç" e((a„.,)av>l)] * 0, or zn ,[Am, e({a„./)0,,o)] * 0, or (in EN, EMN) for every (Vy-valuation v+, v+(A„) = 1 . Suppose first there is q < p S n such lhal Ap = □A, and vdAp) 0. Then v(Ap) = 0 and, since v is □ - a„./-norm al, ^" '[A ,, e ( (a „ ./)a v,i)] = X""1)^ ,. £((<*»; )°v,o)] = 0 IEN, EMN: and lhere is an a„ j - valualion vp such lhal vp(Aq) = 0], Since v and vc agree for i < n, we have: (2) Ç»'M *. £((«,,-; ) ° ,c.t)] = Zn l [4 ,, e ({a„ ./)V ,o )] = 0. We have now to prove lhal £ ((a„ .; ) V l ) u (A„,)] = 0. Let us suppose this set is not empty: from (2) it follows thal E((a<i/)ave.l) u (Am)l = (Am). a. In E and EN, we thus have thal Aq »„.) Am, and then it is easy to see, from (2) again and from lhe fact that A q and A m have lhe same value, Ihat (i) ^ n l [Am, e ( ( a n . / ) Q»,i)] = 0 and Ihat (ii) x n l [Am, E d a , . ; } °vc,o)] = 0. In EN, since Aq «*„ i Am, and there is an a n./-valuation Vp such that vp(Aq) = 0, we also have (iii) vp(Am) = 0. Now (i) and (ii), and in EN (iii) too, conlradicl (1), so Çn l [Af , e ((a» /} av c ,l)u Mm)l = 0b. In EM and EM N , we thus have that Am =>„-i Aq. Now let us examine the case where Çn l [Am, e ({0t„ - /)av.l)] * 0• So there is some aAj, v(dAj) = 1, such lhal Aj " >n-i A m. It follows, since Am »>„ ! A q, that A j ■»>„.! A q. So Çn l M«, £ ((" « / )Qv.c,l)) * 0. against (2). Well, then x n l W«i. £((" TM-/)0»,o)) must be not empty. So lhere is some OAy, v(oAj) = 0, such that Aj <<"„ i Am. It follows that Aj «•„_] Af , and lhen Xn l [A4, e((ôi-j)°vc,o)] * 0. again against (2). Now suppose, in EMN, lhal for every a„./-valuation v+, v+(Am) = 1. Since Am ■»„-! Aq, we have that for every a n-í-valualion v+, v+(A,) = 1. So vp(Aq) = 1, what cannot be. Hence, in both logics, ^ " '[A ,, e((a„ ; )Dvc.l)>J(A m)] = 0 . Suppose now there is q < p S n such Ihat Ap = OAq and vc(Ap) 1. Then v(Ap) = 1 and, since v is o -a « . /-normal, we'have: Xn''M « . e ((ot„ / )a»,i)] = ^n l [A4, e ( ( a „ / ) ° Vio)] = 0 [EN, EMN: and lhere is an (Xft-y-valuation vp such Ihat vp(Aq) = 1], Since v and vc agree for i < n, we have: a. In E and EN: (3) xn l l<V £((otn / ) uv.i)l = e((a*.;)°v,o)] = 0 [EN: and there is an a„-/-valualion vp such thal vp(Aq) = 1]. Since v and vc agree for i < n, we have: (4) x " 1 , , e( (o t„ /) avc.i)] = t,"H A q, e ( { ( v / ) V o ) l = 0. We have now to prove thal Xn l M«. E((a /i-/)Dvc,l) u Mm)] = 0 . Suppose il is not: it follows lhal Aq " n i Am. Now, if ^" '[A „ , e ((a „ / ) av,i)l were nol empty, there would be an A/ e £ ( (a „ ./)a v>i) 101 C hapter 4 such lhat Al »„-i Am. S o A , «„.) A,, and lhen Xn', l'<9. e ( (a n. ; ) DV(.,0] * 0. against (4). So (i) Çn l [/4m, e((" <i-i}n»,l)] = 0Then e({an. ; ) 0v,o)) must be not empty; so lhere is an A; e e ({ a „ .;)°v,o) such that Ai « „ .| Am. It follows lhat Aq »n_i Ai. Bul lhen, against (4), Çn l [Aí , E ÍfO n/JV .o)! * 0, so (ii) Xn l (^m. e ((« n-/)°v,o)l = 0Now, in EN, since Aq «„_j A m, and since lhere is an a„./-valuation vp such lhat vp(A<,) = 1, we also have (iii) vp(Am) = 0. But (i) and (ii), and in EN (iii) too, conlradict (1), so £ ( (« ,./ ) V l ) ' - ' M „)1 = 0. b. In EM and EMN; (5) t( (n « - /)Dv.l)l = Çn l t>V s((«b-/)°v,o)] = 0 (EMN: and lhere is an a,,.;-valuation vp such lhat vp(Aq) = 1], Since v and vc agree for i < n, we have: (6) nn ,n , . £(("«.-/Iavc,t)l = £((«„., )»,c.0)] = 0. We have lo prove lhat e ( (a „ ./)DVc,i) u (4m)] * 0. Suppose it is no t it follows lhat Aq •*>11-1 Am. Now, if  n , (i4m, E((on. / ] D„,i)l were not empty, lhere would bc an e E ( (a „ /)Dv,i) such lhat Ai ®>n_i Am. It iseasy to see lhat <4, •*>„.] At, and lhen c((ot„ / }avc,l)) * 9 , against (6). So (i) e((a»-/)°v,l)J = 0Then x n l Mm. e((f*n-/)°v,o)l must be not empty. Then there is an Ai e E(l««-l)°v.o) such lhat Aj <••„.) Am. Il follows easily that Aq =>„.| Ai. But then, against (6), e((O„./}°vc,0)] * 0. so (ii) Xn l [<4m. e((ctn-/)0v,o)) = 0Suppose now, in EMN, lhal for every a ni- valuation v+, v+(Am) = 1. Since Aq •» „ .] Am, we have lhat for every a« .7-valuation v+, v+(4 ,) = I. So Vp(Aq) = 1, what cannot be. So (iii) there is an a„./-valuation v+ such lhat v+(Am) = 0. However, (i) and (ii), and in EMN (iii) too, contradict (1), so r|n *(/4 , E((a*./) Dvc,l) ^ Mmí] 0. It follows, in case (A), that vc is an (v-valuation. Let us consider the other logics: (B) EC, ECN, EM C, EMCN: By D12.B.2.a, we have: (1) E ( (d „ /)Dv,i), e((a«i /)% .o)l * 0, or, in ECN and EMCN, for every a„./-valuation v+, v+(/4m) = 1J. Suppose first there is q < p S n such lhat Ap = 0 A q and vc(Ap) = 0. Then v(Ap) = 0 and, since v is □ - an./-notm al, Ç*-'[Aít e ( (a n./)°v,i), E((an./)°v,o)] = 0 IECN, EMCN: and lhere is an ocn./-valuaüon vp such that vp(Aq) = 0]. Since v and vc agree for i < n, we have: (2) V H * * . e ( ( a - / ) Dvc,i). e < ( a i ) 0vc.o)l = «■ We have now to prove lhat E ((a„./)Dvc, i ) u M m), E ((a„./)°vc,o)] = 0. Let us suppose this set is not empty: from (2) it follows that, for some T C E ((a „ ./ |DVc,i), some A C e ( (a „ .|]° v c,o), < ru (A ,„ ) , A > e Çn-, M*. e ( (a » - /)Dvc. l ) '- ' M m). E ( (a „ ./ ) ° w<:,o)]By definition, for every a „ . / - valuation v'- (i*) [EC , ECN] v'(Aq) = 1 iff v 'N ) T >J (4 m) and v't=o A, i.e.: vX/4^) = 1 iff v 'N | r and v't=o A and v'(/4m) = 1. (i<>) (EM C , EM CN ] if v 't= i T u {A„) and v' Nq A then v'(Aq) = 1, i.e.: if v 't= j T and v'i=o A and v'(Am) = 1 then v'(Aq) = I. 102 Valuation sem antics f o r classica l m odal logics Now, suppose, in ECN and EMCN, that e ( ( a „ ; ) DVii), e ( [a „ -í) ° Vio)) * 0. (In EC and EMC it is already so.) It follows lhat, for some 8 C e ( [a fl. / ) DV(;,i>, some 0 C E((a,,./]%<.,()), < 9,0> e E((a/i / ) avc,i). e((0(,-í)0vc,o)l. By definition, we have, for every a„ ;~valuaüon v': (ü*) {EC, ECNJ v'(A«) = 1 iff v' t= i 0 and v' i=0 4>. (iil>) [EM C, EMCN] if v'i= i 0 and then v'(Am) = 1. From (i-ii*) and (i-iib) we get, for every â.j-valualion v': (üi*) [EC , ECN] v ' ^ ) = 1 iff v' N i r and v' Mo A and » 'l= i8 and v' No ®. (iiib) [EM C, EM CN] if v' t=i r and v 'N o A and * V [ 6 and v't-o<D then v'(At ) = 1. But Ihis, for every a*/-valuation v', is lhe same as: (iv>) [EC, ECN] v'(Aq) = 1 if fv 't= , T u 8 and r 'N 0 A u t . (ivb) [EM C, EMCN] if vV= i T u 8 and v' *=o A u 4> then v'(A,) = 1. But then there is a pair < r \ j 0 , A u <fc> e e ( (a „ ./) aVCii), £ ((a„ -/)°„Cio)], what cannot be. Thus: (v) e ( ( a „ . / ) V i ) ^ Mm). £ ((« „ .,) \ c,0)] = 0. In ECN and EMCN we still have to consider the olher half of the disjunction in (1). So suppose that for every on-y-valuation v , v'(Am) = 1. From (i*-1') we have, for every a,,.j-valuation v': (vi*) [ECN] v'(A,) = 1 iff v 't= i r and v'No A. (vib) [EMCN] if v' n=i r and v' t=oA lhen v'(Aí ) = 1. But then there is a pair < I \ A> e ^n l [A?, e ( (a n. ; ) a vc,i), £((" »;)%<:,o)). what cannot be. Thus also here we have that (vii) %n l [A,, t ( ( a „ . i ) a V(.,i) o Mm). e ( la „ .j )« vc.o)] = 0. Suppose now there is q < p S n such that Ap = OAq and vc(Ap) = 1. Then v(Ap) = 1 and, since v is ©a „./-norm al, X"'1!^ ,, e ((an -i)0v,o). e (((V i J0»,»)] = 0 . Since v and vc agree for i < n, we have: (3) x"- >M,, e( (aM.j)0 vc.o)» e( (a„j) o,,.,!)] = 0. We have now to prove that xn l [Aí , e ü a n / l^ c .o ) , £ ([an -/]avc.i) u Mm)] = 0. Suppose it is nol: it followsr that, for some T C E({an-/) ° VCio), for some A C e ( [a „ . ; ) a Vc,i), <r, A u Mm)> e Xn l M,,, e ( (a n ( ] 0vc,o). e((ct„-j l°uc.l)u (Am)]. By definilion, for every ovj-valuation v': (i*) [EC , ECN] v'(A,) = 0 iff v' Nq T and v' N j A u Mm), i.e.: v'(At ) = 0 iff v' t=o T and v' »=i A and v'(Am) = 1. (ib) [EM C , EM CN ] if v 'n o T and » V i A u Mm) lhen v'(A,) = 0, i.e.: if v'>=o r and v 'N | A and v'(Am) = 1 then v'(Aq) = 0. Now suppose, in ECN and EMCN, that ^n l Mm. E ( ( a „ ; ) uv,i), £ ((a„ -;)0u,o)) * 0- (In EC and EMC it is already so.) It follows that, for some 0 C e ( l< V i)uv<;,i), some <t> C e ( (a „ - / ) ° Vc,o), <6,<t>> e e((a „ -/)avc,i), e((an-i)°wc,o)]By definition, we have, for every ot„/-valuation v': 103 Chapter 4 (ii*) [EC, ECN) v \A „ ) = 1 iff V' i= 1 e and v' No 4». (ii1') [EM C, EM CN) if v 't= i 0 and v't=o<D then v'(Am) = 1. From (!* *') and (ii*1'), then, for every otn-i-valuation v': (iii*) [EC, ECN] v'(Aq) = 0 iff v'n>o T and v 't= j A and v 'N |0 and v'n=q <I>. (iiib) [EM C, EM CN] if »' Mo T and v'i= i A and v 't= i0 and v' <6 then v'(A?) = 0. That is, for every otn-i-valuation v', (iV) [EC, ECN] v'(A,) = 0 iff v' no T u ® and v' n i A u 6 . (ivb) [EM C ,EM CN ) if v 't= o T u <6 and v' t=i A U 0 then v'(A?) = 0. But then there is a pair < r u 0 , A u í > e X "1) ^ , £ ( ( a „ ; ) 0Vc,o), E ( ( a „ /)DVC,|)1, what cannot be. Thus (v) Xn l M ,. e((« -/)°vc ,o ). e([o„ / ) ° vc.i) yj M m)l = 0. In ECN a n d EMCN w e s ti ll h a v e to c o n s id e r th e o th e r h a l f of th e d is ju n c tio n in (1). So su p p o s e th a t fo r e v e ry G ta .y -v a lu a lio n V , v\A m) = 1. F ro m (i*-*>) w e h a v e , fo r e v e ry a ,i - ; v a lu a t io n v', (vi*) [ECN] v'(Aq) = 0 iff v' n 0 r and v' N=! A. (víb) [EMCN] if v'l=o T and f 'N | 4 then v \A q) = 0. But then there is a pair <r, A> e Xn 'H , , E ((a „ ./)°V(.,o), e ( [a „ ./)DVc,i)], what cannot be. Thus also here we have that (vii) x ^ U -V e ([a „ -/)V ,o ) . £ ( [« ,- ,) V i ) u M«.)l = «■ tt follows, also in case (B), that vc is an otn-valuation. The proof for A„ = oA m is analogous.a We have thus proved that canonical extensions are (v-valuations on the hypolhesis that the a valuations they are extending are normal. With the next lemma, we can show that (V-valuations are normal without restrictions. Lemma L16. (Normality Lemma) Let vb ea n ox-valuation. Then v is O and o-On-normal. Proof. By induction on n. For n = I it holds tiivially, so let n > / and let us suppose that every u„ / valuation is o and 0-a*.j-nonnal. It follows then from P23 thal ( t) The canonical extensions of ot,./-valuations to n„ are a„-valuaiions. Moreover, it is trivially tnie that, for i < n, and T C (On-í), (1) Ç " l[A „ n = Çn[ /l„ r] ; (2) x n l M i , n = x n[/ii>n; (3) Ç"-, [A i,n = çn[Ai, n ; 104 Valuation sem antics f o r classica l m oda l logics (4) n " ' [ 4 „ r i = ri"[A /,ri; (5) Ç»-l[A |.r.A ] = Ç«lA(i r.A J; (6) Zn-1[A i.r.A ] = x " [^ „ r ,A l; because every a„-valuation is an (Vyvalualion. We have now Ihree main cases: (A) For every m < n , A„ * ClAm, A„ * o Am. So v is trivially □ and o -a ,-n o rm al. (B) Let us suppose thal, for some m < n, AH = Q4n . (I) Let v(A„) = 0. We have: (7) e ((a„ )av,i) = E((aB./) Dv,,); (8) e((a„)0v.o) = e({a„.i)0v,o)From the induction hypothesis, v is D-On j-normal, so we have: (9) for every p, every q ,q < p < n such that Ap = uAq and v(Ap) = 0, (9*) E, EN, EM . EMN: '[A ,, e({a„.j}<=*v.,)l = Xn , l-V E((a« i)° .,o )] = V, (9b) EC. ECN, EM C. EMCN: Ç " '[A q, E ((a„.i)°V-l), £ ((a „ i)% ,0)] = a; moreover, in EN. EMN, ECN and EMCN, lhere is an a*./-valuation vp such that vp(Aq) = 0. From (1), (2), (7) and (8), we get: (10*) E , EN. EM . EMN: ^ [A ,, e ( (a n)" v,i)l = XnH «. e(la„)% ,o)] = 0. (10b) EC. ECN, EM C. EMCN: Çn[Aq, e((a„)DRl), e ( (a „ )°Ro)] = 0. In EN, EMN, ECN and EMCN, for each p , let Vp * be the canonical extension from Vp to a n. Obviously vP'(A<j) = vp(Aq), and, from (t), vP' is an a„-valualion. From this, (9) and (10), lhen: ( II ) for every p, every q ,q < p < n sucht thal Ap = OAq and v(Ap) = 0, (11*) E. EN, EM . EMN: Ç»[A„, e ((a „ )o v,i)] = xHAq, £ ((« ,)% ,o)] = 0; ( l l b) EC, ECN, EM C, EMCN: ^ [A ,, e ( (a „ )DKi)]. E((a„)%,o)] = 0; moreover, in EN, EMN, ECN and EMCN, there is an a*-valuation vp* such that vp*{Aq) = 0. On the other hand, since v is an cv-valuation, we have: (12*) E, EN. EM . EMN:Ç" '[A m .E ((a„-/)° ll-1)] = x"-I[Am.e ( ( a M. / ) <>v,o)] = 0 [E N ,E M N : and lhere is an (^./-valuation v„ such lhat v„(Am) = 0]. (12b) E C , ECN. E M C , EMCN: $" 'lA m, E ((an./} D„,i), E ((a„./)% ,o)[ = 0 [ECN, EM CN : and there is an a„.j-valuation v„ such lhat v„(Am) 0). From (1), (2), (7), (8), and from the fact that Am e (a„ ./), we get (13*) E . EN, EM , EMN: ^ [A m, E (la„ )aVil)] = %"\Am, £(((*„ )%.<>)] = 0. (13b) E C , ECN, EM C, EMCN: ^n[Am, E ((a„)aVii), £((<Xn)0»,o)] = 0105 C hapter 4 In EN, EMN, ECN and EMCN, let be the canonical extension from v„ to 0t„. Obviously = Vn(Am). and, from (t), v„* is an cgr-valuation. Thus we have (14) for p = n, q = m, Ap 0 4 , and v(Ap) = 0, (14*) E . EN, EM , EMN: £"[<4,. e ((a„)°„,i)l = x nM ,. e((a„F „,o )] = 0; (14b) EC , ECN. EM C. EMCN: E ((a„)av,i). e({a„)o»,0)] = 0: moreover, in EN, EMN, ECN and EMCN, there Is an oî-valualion vp' such that vp'(,4,) = OJ. From this, together with (11), then ,» is an CHV-normal. Now, from the induction hypothesis, v is o -a „ /-n o rm a l, so we have, together with the fact that A„ ClAm: (15) for every p, every q ,q < p < n such that Ap = OAq and v(Ap) = 1, (15*) E, EN: x" >[*«. E ((a„ ./)°v,|)] = Ç"-'[-4,.e(fan./)\,o)] = 0; (15") EM . EMN: e ( (a ,. ;) ° , , i ) ] = Ç " '!* ,. E((a„.,)% .o)] = «: (15°) EC, ECN, EM C. EMCN: x n l M ,, E ( ( a „ / ) \ , 0), e ( (a „ .i)° ,,i) ] = 0; moreover, in EN, EMN, ECN and EMCN, there is an a„./-valuation vp such that vp(Aq) = 1. From (1), (2), (7) and (8), we get: (16*) E, EN: n "M ,. e ( (a „ )° v.l)] = t" [ -V E((a„)% ,0)l = 0; (I6>>) EM , EMN: x n[-4 ,.E Ü a„)°v.i)] = e ( (a „ ] \ ,o ) ] = 0; (16') E C , ECN. EM C . EMCN: E((a„)°v,o), e(ta» )°» .l)l = 0In EN, EMN, ECN and EMCN, for each p. let vp• be the canonical extension from vp to a*. Obviously vp*(Aq) = v^Aç), and, from (t), vp* is an a*-valuation. From this and (I6*c), then, v is o-a„-norm al. (II) Let i<A„) = 1. We then have: (7) E ( ( a ,) a»,i) = E ((aB. / ) D*,i) >J {Am); (8) e((a„)%.o) = E((an./)ov,o). Since v(A„) = 1, we have from Dl 1 that: (9) for every p, every q ,q < p < n such that: i. Ap = OAq and v(Ap) = 0, (■) E, EN, EM , E M N :^ l |^ ,E ( |a , . / ] o v, i ) u (Am)] = x" 'M ,. E((a„./)% ,o)] = 0; (•>) EC , ECN, EM C, EMCN: E ((aB. / ) av,i) u E ((a „ ./) °v,o)) = 0; moreover, in EN, EMN, ECN and EMCN, there is an (^./-valuation vp such that vp(Aq) = 0; ii. Ap = OAq and v(Ap) = 1, (») E,EN:x"-|lAí .E((a,./)n,li)c» (-4m)] = Ç"'[<4,.E((a„.,)%,<>)] = 0; (•>) EM, EMN: £ ((« „ .,)° ,,t) u (Am)l = e((a„ i)%,o)l = 0; 106 Valuation sem an tics f o r classica l m odal logics (<=) EC. ECN, EM C. EMCN: E((a„.;)<>v.o). e ( ( a . ; ) ° v.l) u (Am)] = 0; moreover, in EN, EMN, ECN and EMCN, lhere is an a„./-valuation vp such Ihat vp(Aq) = I. In EN, EMN, ECN and EMCN, for eachp, lei vp* be the canonical extension from vp lo a„. Obviously vp'(Aq) = vp(Aq), and, from (t), vp" is an a„-valuation. From (1), (2), (7), (8) and (9), and from the fact lhal Am e (a „ ./) , we gel: (10) for every p, every q, q < p S n such that: i. Ap 0 4 , and v(/4p) = 0, (*) E, EN, EM. EMN: e((a„)°„,l) u M » .) ] = Xn [ A , . E ( ( a „ } « v,0)] = 0 ; (•>) EC, ECN. EM C, EMCN: Ç"[A,, e((a„)a ,,.,) u [A„). e({aB)% .0)J = f>; moreover, in EN, EMN, ECN and EMCN, there is an a„-valuation vp such lhal vp(Aq) = 0; ii. Ap o Aq and v(Ap) ~ 1, (*) E, EN: xn[/4,,e((an)av,i)u (/»„,)] = E ( ( a „ ) \ , o ) ] = 0; (b) EM , EMN: ri"l<4,, e ( ( a , )"„.,) o (y4m)] = e ( | a „ ) \ , 0)] = 0 ; (<=) EC. ECN, EM C, EMCN: x n[A,. e ( (a „ )ov,0), E ((a„)°„,i) u M m}] = 0; moreover, in EN, EMN, ECN and EMCN. there is an (X* valuation vp such that vp(Aq) I. That is, v is O and o -a„-nonnal. (C) Let us suppose that, for some m < n ,A n = oA m. Proof as in (!() ■ As a direcl result of combining this Lemma wilh P23, we have lhe following Corollary. Let a„be a normal sequence, v an a„-i~valuation and vc the canonical extension o /v to a n. Then vc is an On-valuation and, for I S i S n -I, vc(Ai) = v(Aj). The next theorem makes use of ali we got unlil now: Theorem T23. y is an cc„ vulualion iff: I ) (Xn is a normal sequence: 2) v is a semi-valualion; 3) v is Dand o-on-normat. 5.2 Correctness Having then proved lhese properties of ( ( v )valuations, we are now ready to consider correclness. The stralegy is analogous to lhe case of normal logics. 107 C hapler 4 Lemma L17. Lei vb ea n â-valualion; then, for 1 < / <n, i f Ai is an axiom o f some classical modal logic L then v(Aj) = I. Proof. The axioms of said logics are either those from PL, and it follows from the fact that v is a semivaluation , that v(Ài) = l , or they aie one of the modal axiom schemes. We consider each case. (a) A,- o A «-> ~U- Suppose v(°A «-> -õv4) = 0. Then we have, say, v(oA) = 1 and v(-.o-u4) = 0, so v ( í j vA) = 1. From the normality lemma it follows that, for every p, every q, q < p < n, such that Ap = oAq and v{Ap) = 1: (i) E, EN: XnM ÍP E((a„)D, t)] = Ç"[<4Í , E((a„) ° ,o )] = 0. But, for every semi-valuation, and, consequently, for every a„-valualion, A «„ A , s o A e X "^». E((ot„)D„,i)|, what cannot be. (ii) EM, EMN: T|n[j4, E ((aB)D,.i)] = Çn[/4, e({a„)°v,o)I = 0But, for every semi-valuation, and, consequently, for every a„-vaIuation, A <">„ -v4; hence, since -v4 e E((a„)D»,i), ~A e E ((a„)Dv,i)], what cannot be. (iii) EC, ECN: X"[A, E((a„}°»,o), E ((a „ )u„,i)] = 0. Let us take the pair <0 , (-«4J>. Obviously 0 C £({<*,,) °„,o). (-iAJ C e ( (a „ )a Wii) and for every a„-valuaúon v', v'(A) = 0 iff v'No 0 and v V j {-«4}; that is, A «„ <0, {-u4]>. Hence <0, (-</4)> e x"!^ , e((«n)°v.o). e ( (a „ )D»,i)|, what cannot be. (iv) EM C, EMCN: X"[-4, £ ( (a „ )% -o), E ((a „ )av,i)] = 0. Let us take the pair <0, (-v4}>. Obviously 0 C E ((a „ )° v,o), (-■A) Ç E ((aB)Dv,l) and for every a„-vaIuation v \ if v't=o0 and v't= 1 (-«4) lhen v'(A) = 0; lhat is, <0, (-w4}> • » n /4. Hence <0, (-v4 )> e X"M. E ((a„)°„o). e ( (a „ )a w,i)), what cannot be. If now v(<M) 0 and v(-iÕ*4) = 1, the proof goes for every logic in a similar way. Hence v(oÁ «-» - .□ ^ 4 ) = 1. We must now consider the special axioms of each system. (b) Ai = 0 4 A DB -> 004 A B) (EC , EMC, ECN, EM CN). Suppose v(A j ) = 0. So v(0 (A a B)) = 0 and v(aA) = v(OB) = 1. Lei us take the pair <(A, B ), 0>. Obviously (<4,fl) C ê(a„)a „ i), 0 C £((<xn)°»,o), and for every a„-valuation v', (EC, ECN] v'(A a B) = 1 iff v ^ i ( A í ) and v't=o 0; lhat is, A <1^43), 0>. [EMC, EMCN] if v*N= j [A fi] and vt=o 0 lhen v'(/4 a B) = I; that is, <[AJB), 0> «>„ A. Hence <(i4,fl), 0> e ^"[A, E ((an)Dv,i), £((<*«) °w.o)]. what cannot be. So v(<4,) = 1. (c) A i D(v4 a f l ) -» 0 4 a C]B (EM , EMC, EMN, EMCN). Suppose v(/4,) = 0 . So v(U(A a B)) = 1 and v(uA) = 0 , or v(DB) = 0 . We consider lhe logics scparately: [EM , EM N] Obviously A a fl " > „ A, and A / \B •»>„ fl. Since A a f l e £((a<i)Dv,i), we have that ^n[/4, e ( (a „ )Dw>i) | * 0, and Çn[fl, £ ( (a „ )nv |) | / 0, so v(o4) = v(üfl) = 1, what cannot be. So v(/4,) = I . 108 Valuation sem an tics f o r classica l m odal logics (E M C , E M C N ] Let us take lhe pair <(A a B ), 0>. Obviously (A a fl) C e ({ a „ )a v,i), 0 C e({an) ° Vto), and for every a n-valuaiion v\ if [A a 5} and v'nq 0 then v'(<4) = v'(B) = 1; that is, <(A,B), 0> » > ,A ; <(/4,B), 0 > < »„B . Hence <{A a í ], 0> 6 £n[A, e ((a * )av,i), e ( (a „ ) 0 Vio)], <(A a B ), 0> 6 Çn[B, E ((a„)a„,i), e((ot„)°»,o)], what cannot be. So v(A,) = l .a Theorem T24. IfA is an axiom o fL and v is a valuation, then v(A) = I. Proof. Let A be an axiom of one of the said logics, and v a valuation. Let a„ be a normal sequence such that, for some i S n , A = A/. By definition, v is an a„-valualion and, from L12, v(A) = 1 .■ L e m m a L 1 8 . For ali n, ali i, I S i S n , and L a classical modal logic, if v is an (X„ valuation and > i. Aj, lhen v(Ai) = 1. Proof. By induction on lhe number r of lines of a proof of A, in L. A )r = /.T hen A,-is an axiom, and theproperty follows fromL12. B) r > 1. If Ai is an axiom, the property follows from L12; else: ( a ) Ai was obtained by MP from B and B - » Ai. Proof as in the case of normal logics, since valualions here also are semi-valuations. (b) Ai = aB <-» □C and was obtained by RE from B «-> C. We have lhat t-B C. Obviously, for every normal sequence to which OB <-> UC belongs, B, C, aB and DC belong too, so they belong to a„. If B <-> C also occurs in a„, let a = a „ and vc = v. Else let a = A / .....A„,A*+i, where <4„+; = B <-> C. o is obviously a normal sequence, so let vc be the canonical extension of v to O. vc is thus a o-valuation. Since t-B <-» C, we have by the induction hypothesis that for ali n, if v is an a„-valualion and i-B <-> C, lhen v(B ♦-> C) = 1. So vc(B C) = 1; besides, B » n+l C. If now vc(dB < > dC ) were lo be 0, we would have, say, v(Ofl) = 1 and v(aC) = 0. From the normalily lemma, ^n+,[C ,£ ((a„+/ ) DVCii)] should now be empty, but it isn't, because B C and thus B belongs to it. Thus vc(dB <-» DC) = 1, and hence v(Dfl <-» DC) = 1. (c) Ai = Ufl and was obtained by RN from B (for EN, ECN, EM N, EM CN). Well, in every normal sequeflce a in which Ai occurs, B occurs too; so, by the induction hypothesis, for every a-valuation v, v(B) = 1. If now v(Ai) were to be 0, lhere should be an (1 ,,-valualion v„ such lhat v„(B) = 0, what cannot be. So v(Ai) = 1. ■ C o ro llary . / / i-A lhen »=A. P roof Suppose i-A, and let v be a valuation. Let On be a normal sequence in which, for some i S n , A = A;. By definition, v is an a„-valuation, so v(A) = 1 from L18, and thus t=A. ■ Theorem T2S. (Correctness Theorem) / / H A then I > A. J ro o f. Suppose T H A, and let D i,...J)r be a deduclion of A from P. We prove the theorem by induction on r. A) r = I . Then, either A ( I , and we have nothing lo prove, or A is an axiom, so A is valid (by the corollary to L18) and í i-A. 109 C hapter 4 B) r > l . l f A t Tand A is not an axiom, then: (a) for some j < r, i < r, Di = Dj -> A. So T H Di, T lDj -» A and, by the induction hypolhesis, T N= Dj, T m Dj -* <4. Thus, for every valuation v, if v t= T, v(Dj) v{Dj -* A) = I, and hence v(A) = 1. So I" »= A. (b) A = □B <-» dC and, for some j < r, Dr B C. In this case, t-D r and i-A, so, for every valuation v, v(A) = 1. Hence F >A . (c) A = aB and, for some j < r ,D r = B (for EN, ECN, EMN, EMCN). In this case, t-D r and i-A. By the Corollary lo L18, for every valuation v, v(A) = I. So, if v t= r , v(A) = 1. Thus T t= A. ■ 5.3 Completeness Compleleness will now be easily proved in the same way of normal logics-Ihat is, making use of saturated sets. In lhe following, let L be a classical modal logic, and let us understand as referring to L. We are now able to prove some results about saturated sets, showing some of their properties which will be of use in compleleness proofs. Proposition P24. I f A is saturated, then: (a) A e A i f f A H A ; (b) -u4 e A i f f A c A; (c) A -> B e A i f f A t A o rB e A; (d) A h B e A i f f A e A and B e A ; o r A t A and B t A. Proposition P25. ! f T M A, then there is an A-saturated set A such that T c A. Definition D13. Let T, P be any sets, A and B wffs, and 6 , 0 finile sets of wffs: r c 0 r =df for every A e T, A t T; r « o =df (OA í T); e ( r ° 0) =dí (A : O A t P); A = s « ,iff for every saturated set A, A e A iff B e A; A * s S irr for every saturated set A, A e A iff B t A; A « > s B iff for every saturated set A, if if A € A then B e A; A * > s B iff for every saturated set A, if if A e A then B t A; A <**s B iff for every saturated set A, if if A t A then / i r A; A « s <8 . *1» iff for every saturated set A, A e A iff 8 C A and <I> Co A; A -»s < 9 ,4 » iff for every saturated set A, A t A iff 8 Co A and ® C A ; <8 , <I» «> s A iff for every saturated set A, if 8 C A and <I> Co A then A e A; <8 , *>g A iff for every saturated set A, if 8 Co A and O C A then A t A; 11 0 Valuation sem antics f o r classica l m oda l logics Definition D14. Let r , A be any set of wffs. We define: (a) for E, EN: =df { B e r : B ~ s A ) ; XlA. n =df ( B e r :fl«* sA )• (b) for EC, ECN: Ç[A, T, A] =iif ( < 0 , <I» : 0 C T, <l> C A and A «<s < 8 , <t>>); X(A, T, A] =df ( < 8 ,0 > : 0 C T, <I> Q A and A * s <8 . *1»); (c) for EM C, EMCN: Ç[A, T, A] =df ( < 0 , <I» : 0 C F, ® C A and < 0 , ®> * » s A ); X(A, T, A] =<jf ( < 0 , «I» : 0 Q T, <P C A and < 0 ,0 > <»>s A ); (d) for EM, EMN: Ç H .r j =df ( B e r : B » > s A ); X H .n =dr ( B 6 r : B < s A ) ; Ç(A,r) =jf [ J e T M w s J ) : h [j4 , r j =df [ í e r M < > s S | . Lemma L19. Let A b e a saturaled set. I f Am 0 4 , then: (i) E, EN, EM , EMN: (ii) EC , ECN, EMC, EMCN: (iii) EN, ECN, EMN, EMCN: //A lo A, then: (iv) E, EN: (v) EM, EMN: (vi) EC. ECN, EMC, EMCN: (vii) EN , ECN, EMN, EMCN: Ç[A.e(A°)] = x(A ,e(AoO)] = 0. Ç[A,E(Aa), £(A°0)] = 0. there is an A-saturated set A'. %IA, e(A°0)J = x|A , e(A°)] = 0. ÇM. e(A°0)J = nM , e(A°)J = 0. X[A, e(A°0), e(A u ) | = 0. there is a -A-saturated set A'. Proof. (i) Suppose A 0 4 , and let B e E,(A, e(A°)]. So ufl e A. We then have: (i*) (E, EN| Since B « s A, B «-> A is a member of every saturated set, so B t-» A e A, and Qfl < * LIA e A. (ib) [EM, EMN] Since B ~>s A, B -> A is a member of every saturated set, so fl -» A e A. ofl -> 04 e A; and lhen 0 4 e A. But then, in both cases, it cannot be that A ^ 04 . So Ç[A, E(AD)] = 0. Now let C e x(A, e(A°0)]. Thus: (ic> [E, EN] C « s A, and OC e A. Hence C >» -.A. So i-C *-» -<A; hO C í-> o-*4; OC « 0-^4 e A. Since OC t A, o-,A e A, so - .o -v l e A. Since HQ4 <-> -> 0-tA ,q4 e A, (id) (EM, EMN] C <*s A .o C t A. Hence C «=>>s A. So - £ - » A is a member of every saturated set, so G-iC -> 0 4 e A. Since OC e A, -O -iC £ A, thus Q-.C e A. It follows lhal 0 4 e A. 111 C hapter 4 H e n c e , in b o th c a s e s , A t u 4 , w h a t c a n n o t b e . I t fo llo w s th a t x ( A , e íA ^O )] = 0 . ( i i ) S u p p o s e A M d A , a n d le t 8 = ( 0 / .......6* ) , <í> = (<p/....... f m ] s u c h th a t < 8 , 4 » G Ç [A , e ( A ° ) , e ( A ofl) ) . T h e n , s in c e 8 <-* e (& ° ) , { O 0 / , . . . ,O 0 i ) C A . S in c e <6 C e ( A ° 0 ) , ( o <pt ...... O <pm ) C o A . I .e ., ( - . o <pm ) C A . I t fo l lo w s th e n th a t 0 0 / a ... a O 0 * e A , a n d th a t 0 ( 0 / a ... a 0 * ) g A. S im i la r ly , - . 0 ^ / a ... a -*oq>m g A , a n d th a t - . o - , ( - , p / a ... a ~<q>m ) g A ; D(-iÇ>/ a ... a -><pm) g A. F ro m a x io m C , th e n , 0 ( 0 / a ... a 0* a -,<pi a ... a <pm) G A . T h u s w e h a v e : ( ii" ) [ E C , E C N ] S in c e A « s < 8 . < I » , f o r e v e ry s a tu r a te d s e t A ', A e A ' i f f 8 C A ' a n d C o A '. T h a t is , f o r e v e ry s a tu r a te d s e t A ', A e A ' i f f ( 0 / .......0 * ) C A ' a n d ( <pi....... <pm ) C o A '; i .e . , ( - .$ > / ....... C A '. I t fo l lo w s th a t f o r e v e ry s a tu r a te d s e t A ', A e A ' i f f 0 / a ... a 0* a - ,q j ; a ... a -^<pm g A '. T h u s , f o r e v e iy s a tu r a te d s e t A ', A <-» 0 / a ... a 0 j a -<ç>/ a ... a e A '. B y / ? £ , OA 4-» 0 ( 0 / a ... a 0* a - . 71/ a ... A -iÇ>rn) G A '; LM < > 0 ( 0 / A ... A 0* A -»Ç>/ A ... A -.Ç V l) G A. (ii*>) [ E M C , E M C N ] S in c e < 8 , í> > « » s A , fo r e v e ry s a tu ra te d s e t A ', i f 8 C A ' a n d <I> C o A ' th e n A e A '. T h a t i s , f o r e v e ry s a tu r a te d s e t A ', i f ( 0 / , . . . , 0 * ) C A ' a n d C o A ' th e n A g A '; i .e . , i f ( 0 ; .......0 /t] C A ' a n d { - " ? > / , . C A ' th e n A g A '. I t fo llo w s th a t fo r e v e ry s a tu ra te d s e t A ', i f 0 / a ... a 0* a -.<p/ a ... a g A ' lh e n A g A '. T h u s , f o r e v e ry s a tu ra te d s e t A ', 0 ; a ... a 0* a -.ç»/ a ... a -iç»m - » A e A '. B y R M , 0 ( 0 / a ... a 0* a ->ipi a ... a -1 ç>m) -» OA g A '; 0 ( 0 / a ... a 0* a -iÇ>/ a ... a -iç>m) OA G A. In b o lh c a s e s , s in c e 0 ( 0 / a ... a 0* a -,ç> / a ... a <pm) € A , OA g A , A h o A , w h a t c a n n o t b e . T h u s Ç (A , c (A D) , £(A O O )] = 0 . ( i i i) S u p p o s e th e r e is n o A s a tu r a te d s e t A '. S o , f o r e v e ry A ', A g A . L e t B b e a th e o re m : s o , fo r e v e ry s a tu r a te d s e t A ', A ' 1B , B g A '. T h u s B « s A . F u r lh e r , s in c e l B , w e h a v e t O B a n d , f o r e v e ry s a tu r a te d s e t A ', A ' t 0 8 , O B g A '. B u t lh e n O B g A , B g e (A u ) , a n d , in th is c a s e , Ç [A , e íA 0 )] * 0 , w h a t c a n n o t b e . S o th e re is a n A -s a tu ra te d s e t A . T h e p r o o f o f c a se s ( iv ) - (v ii) is s im ila r to c a se s ( i) - ( ii i) . ■ T h e o r e m T 2 6 . F o r e v e r y A s a tu r a t e d s e t A a n d e v e r y n o r m a l s e q u e n c e a „ , th e c h a r a c te r is t ic fu n c t io n f o f á is a n a " v a lu a t io n . P r o o f. F i r s t o f a li , i t is e a s y to p ro v e b y P 2 4 lh a t ( t ) T h e c h a ra c te r is t ic fu n c tio n / o f A is a s e m i-v a lu a tio n . W e n o w p r o v e . th e th e o re m b y in d u c t io n o n n . I f n = / , th e p ro p e r ty fo l lo w s f ro m ( f ) a b o v e . L e t u s s u p p o s e n > 1 . (1 ) I f , fo r e v e ry m < n , A „ * Q 4 m , A „ * O A m , / i s I r iv ia l ly a n a „ v a lu a t io n . (2 ) F o r s o m e m < n . A * = o A m . O / M u ) = 0 . T h e n A n t A , A Ir' o A m . F ro m L 1 9 w e h a v e n o w th e fo llo w in g : 112 Valuation sem an tics f o r classica l m odal logics (i*) E, EN, EM , EMN: Ç[Am, e(A°)] = X[Am, é(A»0)] = 0. (ib) EC , ECN, EM C , EMCN: £(AD), e(A°0)] = 0. We have lhen that (0 ,./)° / ,! C A° and {aB./)°/,o C A<>Oi thus: (ii*) E, EN, EM , EMN: E((a,,.,)o/t1)] = Xn l [A„, edâ/JO /.o)] = 0. (ii**) EC, ECN. EM C, EMCN: ^ E d a , , ) ^ . , ) . £ ((« „ .;)0/,o)] = 0. Now, in EN, EM N, ECN and EM CN , from L19 there is an Am-saluraled set A'. By the induclion hypolhesis, lhe characteristic function f of A' is an a„./-valuation. So lhere is an a n./-valuation / such ihat f (A m) = 0. H ence/is an og,-valualion. II)AAn) = 1. So DAm e A. Let us suppose there is some p, some q ,q < p S n such that Ap = aA q and ÂAp) = 0. From L19, we have (i*) E, EN, EM , EMN: e(A°)J = x[Aq, e(A°0)] = 0. (ib) EC, ECN, EM C , EMCN: £[/»,. e(A°), e(A°0)] = 0. By lhe induclion hypolhesis,/is an íín l valuation, so it is U (Xn l normal. Thus for every p, every q, q < p S n such that Ap = aAq and fiAp) = 0, (ii*) E. EN. EM, EMN: £>-'[* ,. E((a„.,)ty,,)) = x n l H ,. £ ((« „ ./)»/,0)] = 0 . (üb) EC. ECN, EMC, EMCN: Çn l [/t,, £({«„-/)"/.,) , E d a , . , ) » ^ ) ] = 0Now, since f iA n) = 1. E((a„ / ) D/,t) u (Am) = E((a,,}a/,i); and since (a „ )a/,i C A°. (iii*) E. EN. EM. EMN: £ ((« „ .,J tf r ) u M „)J = Xn 'H , , E d ^ ./lV .o ) ] = 0. (iiib) EC, ECN. EM C. EMCN: Çn l [<4í , £ ((« „ .,)°/,,) u M m), £((«„_,)»/.o)] = 0. Now, in EN, EMN, ECN and EM CN, from L19 lhere is an ^-saturated set A' and, by lhe induclion hypolhesis, lhe characteristic functio n /o f A' is an on-j-valuation. So lhere is an ocn-z-valualion/ such lhat/*(/4^) = 0. If lhere is some p , some q ,q < p £ n such lhal Ap = <>Aq and f{Ap) = 1, the proof is similar. H ence,/is an a„-valuation. (3) For some m < n, An = <Mm. Proof as in (2). ■ C o ro lla ry . v is a valuation ijfv is lhe characteristic function o f some saturated set A. Proof As in normal modal logics ■ T h e o re m T 2 7 . (Completeness Theorem) / / H = A then Í V / l . Proof As in normal modal logics. ■ 113 GTTs for K B r a d y 's F ir s t L a w o f P r o b le m S o lv in g : W h e n c o n fr o n te d b y a d i ff ic u l t p r o b le m , y o u c a n so lv e it m o r e e a s ity b y r e d u c in g i t to th e q u e s tio n , "H o w w o u ld th e L o n e R a n g e r h a v e h a n d le d th is? " . Now lhat we have seen how a valuations semantics looks like, let us look for a way of obtaining GTTs out of it. I'm going here to take K as an example; changes for other logics aie more or less straightforward. If we have a wff A, it is easy to construct a finite normal sequence A /,...A n where A is the last lerm-one just takes A and its proper subformulas. Now let V(<4/...../4„) be the class of ali A i,...A nvaluations. Let us define an equivalence relation over this class as follows; v = v' iff, for I S i S n , v(A,') v'(A|). Since A j .....A„ is a finite sequence, V (A i..... An)/= is obviously finite too. Thus a decision procedure for K consista in a procedure which allow us lo reduce, for every Ivt e V{Ai,...An)l&, the restriction v* of some v 'e Ivl to the set (A/ .....A„). Such construction, which we will designate by JJA/„..An1. and call the GTT for A/,...An. will decide on the validity of any formula belonging to the sequence (and, consequently, of the property of being a theorem). In particular, of the formula A. Let us first examine, by means of an example, how things are supposed to work. Let A be the formula 0->-.p-»qp, where p is a propositional variable. We'll construct a normal sequence A i,...A n where A„ = A by listing ali subformulas of A. As a result we get the sequence p, p , -r~p, Op, n->-p, □ -i-p -» q p , which has six elements. The procedure I am going to show consists in constructing the table for A i-i.e., T[A/] and then extending it successively to lhe rest of the sequence; T [ A iA 2 ]..... T[Ai ,...A6]• At the end, the table looks like lhe following: 1 2 3 4 5 6 p p T a P o - ^ p -> n p 1) 1 0 1 i 1 1 2) 0 1 0 i 0 0 3 ) 1 0 I 0 1 í 4 ) 0 1 0 0 0 0 ng. 19 1 1 4 G TT iforK T [ A i ) , T I A j . A í ] and T [A /, A 2 ^ 3 Í are construcled in lhe usual way, i.e., like in classical propositional logic: you assign values to the variable(s), and then proceed by calculating the value of more complex formulas. In the picture, this correspond to lines 1) and 2), rows 1 to 3. But in T [ A i , . . . ^ 4*] two new lines were added: 3) and 4). I II try to explain why. T [ A i A i A } ] l = has obviously just two elements: Ivjl and Ivjl. The elements of Iv/l and Iv l, resuicted to A 1A 2A 3 are represenled in lhe table by lines 1) and 2). Now, since (A;,A2,A.j)a = = 0, we have, for i 6 (7,2): I) v/(p) = 0, v /N i e((A/,A2,A.})Dvji|) . v/i= 0 e((A/,A2,Aj) %;,(>). i.e., v ,< J> v /. In this way are fulfilled the necessary conditions which juslify the existence of v' and v " in T I A i , . .A 4 ] such lhat »' e lv/1, v" e IV2I and v '(A 4) = v"(A<) = 0. 10 vacuously, for every p, every q ,q < p £ 4 such that A p = oA , and v;(A p ) = 0 [Ap = o A, and v $ A p ) = 1 ], there is a j e ( / , 2 ) such that v / A q) = 0 [= 1], v j i A ^ = 1 and v ,< 3 > v j. So are fulfilled the necessary conditions for the existence of t>*, v** in T \A i,...A 4 \ such lhat v* e Ivjl, v** e Ivjl and v*(A < ) = v**(A.<) = 1. By I) and II), it is plausible lhat T[A i,...A 4 \I& has thus four elements lw/1...1 v^l, such that the reslriclion of each one to A i,..A 4 is represenled by lines 1) to 4) of the GTT. This is how and why we got these two extra lines. Such an unfolding, now, doesn't happen in lhe construction of 7'{A/,...,Aj]- which was lo be expected, since we p and 1p are equivalent, and in consequence qp and u 1-.p should get the same value. Let us see, for instance, how \ ) * P cannot possibly lake the value 0 in line 1). Let V £ Ivjl, whose restriction to A ] , . . . A 4 is represenled by line 1). We have thal { A i , . . . A 4) ° v ,0 = 0 and ( A / , . . .^ ) a »,i = (Dp); i.e., e ( (A ;,.. .^ ^ )aVl|) = (p). If we had v(D-i-ip) = 0, we should have a v;, 1 e (7,.. . .4 ) such lhat Vj(-1->p) = 0 and v<4>vJ-i.e., vj n=i £({Aj,. A a ) s o v,(p) = 1. Bul v,( -.-./?) = 0 and v,(p) = 1 is in every valuation (and so in every line) an impossibility; hence o-r-y> cannot get a 0 in line 1). On the other hand we can see lhat v(U-i-p) = 1 is possible, for, vacuously, for every p, every q, q < p < 5 , such that A p = 0 4 , and v (A p ) = 0, lhere is a j £ ( / ......4 ) such lhat v j(A q) = 0, v / A j ) = 1 and v < 4 > v j. This situalion also happens in line 2), where once again o-i->p cannot take lhe value 0, only 1. In lines 3) and 4) il's the other way round: D -i-^ can only take a 0. So lhese are the reasons why we don't get a splitting of lines in lhe construction of T[Ai,...As]And by 1\A i ,...A ó] we are again in the realm of things classical. I hope that this example has helped to make things a liltle bit more clear, because now we will have to define everything rigorously, and this is far from being easy-lhe definition of a GTT is a very big one. I'U stale the definition first and give some explanalions later on. Definition DI5. Lei a „ be a normal sequence. A g e n e r a l i z e d tr u th ta b le (GTT) for a„ is a function T ia , ] : (a„) x J(a„) -> {0,1), where: 1) for n = l , J ( a / ) = (7 ,2 ) ,71" /1(Aí, 1) = 1 and 71a/l(A ;,2) = 0; 2) for n > 1, and J(an-i) = {1.... q) : (a) if A„ is a propositional variable, lhen J(a„) = (1.... 2q) and: i) for i < /!,> £ J(Ofl-;), l \ a „ \ ( A i , j ) = '/]a„./|(A „j); ii) for i < n ,j ' e J(a„./) and; = q + j \ 71an|(A„;') = / |a „ .; |( / i„ / ') ; !Í5 Chapter 5 iii) for i = n ,j e J(a„_;), 7l0nl(Ai.i) =1; iv) for i = n , j 'e J (a ,.í) and j = q + / , T[a„\(Ai,i) =0; (b) if A„ = A k, t < n , I(a„) = J(tt„./) and: i) for i < n, nanU A i.j) = T[a„ i](At,jy, ii) for i = n, TlanHAj.y) * l\a „ i] (A k,j); (c) íf A„ = A*-»A,,, k< n ,e < n, J(a„) = J(ot„í) and: i) for i < n, T\a„](Aí,j) = T[a„i](Aj,j); ii) for / = n , n a nJ('4' , » = * iff H a* /](-4»,y) = 0 or T[an i)(Ae,j) = 1; (d) if A„ = 04*, k < h , then for every j e J(a„ /) : I) let a(j, n l) y e J(a„ ./): 7'[<*/i-/J('4*,y') = 0 and, for every r, 1 S r S n , if A r = OA, and n a „ i)(Ar.j) = 1 [Ar = oA s and T\a„.i\(Ar, j) = 0], then T {a„.i)(A„n = 1 [= 0]); II) for every p , every q ,q < p < n such that Ap = a A q and T [aK-i\(A p, j ) = 0 [Ap = o A q and T[aH. i W p,j ) = 1). let P (p .;, n -l) = ( / 'e J ( a „ /) : n a ^ / ] ( A , ,y ) = 0 [= 1], 71a„.; ](A *,/) = 1 and, for every r , l S r S n , if Ar = 0 4 , and T \a„ i](A r,J) t [Ar = o A, and 71a»-jKAr, j) = 0), then TtOfi-;](Aj,/) = 1 [=0]); III) let C í ( a „ /) such thal: 1 )jm-<jmif m' < m"; 2) jm-6 if n-J) Jt 0 and, for every p < n, n -l) * 0. Then J(aB) = (1.....q .....<?+m) and: i) for i < n ,j S q , 1\a„](At,f) = 71a„./](Ai,y); ii) for i < n.y = q + m \ 7Ta„](Aj,y) = r [a „ ;](A/,ym-): iii) for f « n,y such that a(y, n-l) = 0, 71ot,](A,,y) =1; iv) for í = n j such that a ( / , n-l) * 0 and, for some p < n, P(p,y, n-/) = 0, 71ot„j(Aj,/) =0; v) for i = n,y such that a(y , n-7) * 0 and, for every p < n, fHp.j, n l) * 0, in which case, for some m 'e (1.....m ) , j= jm o i j = q + m \ 1) ify=ym Üien 71a»t](A|.y) =i; 2) if j = q + m' then T[<x,,KA|,y) =0. (e) if A„ = o Ak, k < n , then for every j e J(a„./): I) let y{j, n l ) = 0 ' e J (a„ ./): T lO n/K A ^.y-) = 1 and, for every r , 1 S r S n , i í A r = o A , and 7Ta„.;](Ar,y) = 0 [Ar = 0 4 , and 71a,/](A,.,y) = 1], then n a n /K A ,,/ ) = 0 [= 1]); II) for every p , every q ,q < p < n such that Ap = O A , and 7 '[a„.;](A í,,y) = I [Ap = ü A q and T[an.,)(Ap, j ) = OJ, let 8 (p ,j, n l) = ( / e J (a „ /) : 71a„.,](Aí ,y') = 1 [= 0], 71 « „ ,] ( / ! * , / ) = 0 and, for every r, 1 S r S n, if Ar = o A, and 7[a„./](A r,y) = 0 [Ar = 0 4 , and 77a„/)(A r, j) = 1], then n < V /]( /4 „ /) = 0 [ = l] ) ; III) let U lj m ) c J(a„./) such that: 1)jm'<jmif 2)jme l ii . Jm) if yijm 1. n -l) * 0 and, for every p < n, S (p ,jm-, n -l) * 0. Then J (a n) = (1.....q .....í+ m ) and: i) for i < n , j S q , T[aJ(.Ai,j) = 7T(V/](Aj,y); ii) for f < it,y = q + m',71a„l(Ai,y) = T[an-iKAi,jmf , iii) for i = n . j such that >0, «•/) = 0, 71««)(y4i,y) = 0; iv) for i = n ,j such that 7O, ñl) * 0 and. fw some p < n, h(p,j, n-l) 0 ,71(*„](Ai,y) = 1; 116 GTTs fo r K v) for i ~ n ,j such that yij, n-I) * 0 and, for every p < n, 6(p ,j, n-I) * 0, in which case, for some m 'e [ l,...,m ),j = j m-o r j = q + m', D if í = jm■ «hen HOnHA;,./) = 0; 2) if j = q + m' lhen 71a,,](/4/,y) =1. Now to lhe explanations of this ali. Intuilively, a GTT is a function with two arguments, the first being a formula (in a normal sequence), what would correspond to a row in a "normal" table, and the second pointing to a line: thus for instance we can express lhe fact that in our example the formula Cp gets the value 0 on line 3) by stating that l \A i,.. .A 4 ](Op, 3) = 0. Of course, as the table expands, lhe number of rows and lines increases, but formulas in lhe expanded table preserve the value lhey already have. Thus T [A ,....A 6\(ap . 3) = n A , , . . .A s ](ap, 3) = 0. Back to the explanations. J(a„) denotes the set of lines of the GTT. When n = I we have a normal sequence with just one element, which must in this case be a propositional variable (or the sequence wouldn't be normal). So J(ot/) = ( / ,2): that is, we have two lines in our table, and lhe variable gets value 1 in line 1) and 0 in line 2). Clause a) just states the fact that when we found another proposilional variable, we must double lhe number of lines (from ( l,...,v) to ( l,...,2q)). The new variable gets value I in lines 1 up to q, and 0 from q+1 up to 2q. Clauses b) and c) offer no problem: the new function T[A i,...An] gel lhe same values as T[Ai,...Ah-11 for wffs whose index is smaller than n; for A„ the classical condilions must be preserved: so for instance in b), for i = n, An = A t gets lhe opposite value of At. Similarly for implication. Let us lhen consider clause d), where we handle the case where An = 04*. for some k < n. First we define for each line j a certain sets of lines called a(j, n-I)- this is just lhe set of lhose lines/ who give 0 to A t and which salisfy ali Aj such lhat 04,have value 1 in line j , and reject ali <4, such that o A/ have value 0 in line j . In a similar way we define, for each p such lhat for some q , Ap = Q4f [= oA?] and which gets value 0 [1] on line j , the set |Mp.j, n-I). Now (o III): i) marks a subset of the set J (a„ ./) of lines, namely lhose lines which potentially lead to spliuings: lhat is, lhere are two possible ways of extending them, one in which uA* gets 1, the other in which it gets 0. These splitting character of some lines comes from lhe fact thal their a-sels are nol empty (i.e., some line / gives 0 to A,and satisfies/rejects lhe scope of j), lhe same holding of iheir p-sels: for every p which is a false necessity there is a line ensuring lhal and giving 1 to A,-. Now m is lhe number of lines*which split, so the table gels m extra lines: J(a„) = { /.... q .... q+m\. The formulas whose index in the sequence is less than i conserve their values in lhe extension: clauses i) and ii) (for i < n). Clause iii) states lhal, if no line satisfying lhe scope of line j gives 0 lo At, lhen n A t gels 1 in j . In clause iv) there is such a line, so 04* gets 0, and, since the f}-sel is empty, for some p, 0 is the only possibility. Compare this lo clause v): lhere both seis are non-emply, so lhe line has lo split. The "old" line <Jm') gives 1 lo 04*, and lhe new one 0?+m). Üie value 0. Ali clear? Then let us try to prove some results about this ali. Lemma L20. Lei a„be a normal sequence. For every j e J(a*), I <k S n , lhere is j* e J(a„) such lhat, for every i, I Z iH k . 71a*](A„y) = I\a„\(Ah ]•). 117 Chapter S Proof. This lemma can be easily proved by induction on n-k, based on conditions i) and ii) of DIS.I.a through D15.1.d. Lemma L21. Let a„ be a normal sequence, and v a valuation. Then there is j e J(a*) such that, for 1 S i S n , v(Ai) =T[an](Ai,J). Proof. By induction on n. Let v be a valuation. (1) n = 1. By D15.1, J (a ;)= ( / ,2 ) . There are two possibilities: I) v(A i) = 0. From D15.1.1 we have lhat T [a i\(A i, 2) = 0. Hence there is j = 2 such that v(Aj) = T \a ,\(A ,,i) . II) v(A i) I. From D15.1.1 we have that 7 '[a/](A /, I ) 0. Hence there is j = I such that v(A i) = T [a i)(A ,,i) . (2) Let n > I , and let J(a„./) = <f). Induction hypothesis: for every valuation v there is j e J(a„./) such that, for I S i < n, v(Ai) = TUVlKAfaj). That is, for every valuation v there is a line j of the table until n-l, such that j and v agree for i < n. We need now to prove there is a line j such that v and j agree until n, what we'll do examining the construction of the GTT. By D I5.I, for i < n, l\a*\(A i, j ) = I\O ni](Ai,j). It follows from L21 that: (t) For every valuation v there is j e J(a,,) such lhat, for I S i < n, v{Ai) = T[a„KA,■,./). (a) Let An be a propositional variable. By D15.2.a, -1(0*) = ,2 <y) . I) v(An) = 0. From ( t) there is j e J(a«) such thal, for I S i < n,v(A i)-T\a„](A[,j). Let us suppose lhat j e J(a„ ./)- i.e., j e ( / .....i?). By D15.2.a.ii, there is j* e J(<x„) such lhat, for / < n, r[«n)(A(,y*) = 7'[a„](/4;,/); hence, for i < n, v(A,) = 7Tot„](/t>*). By D15.2.a.iv, for i = n, 71otn|(Ai,y*) = 0. Hence there is j* e J((X„) such that, for I S i S n , T lan](At,j*) = v(/4,). Let us now suppose that j = q + j'. By D15.2.a.iv, n a »](An,7) = 0-thus there i s / e J(otn) such that, for / S i S n , v(A,) = T[an\(Ai,f). II) v(/4„) = 1. Proof as in case I). (b) Let An ~ -«4*. for some k< n. By D15.2.b, J(a„) = J(a„./). I) v(AB) = 0. Then v(A*) = 1, since v is also a semi-valuation. By (t) there is j 6 3(a„) such lhat Tln„l(At ,j ) = v(Ak) = 1, By D15.2.b.i, n«„K-4*,y) = r{a„.i)(Ak,j) , and, by D15.2.b.ii, T{ar\(A„,j) * r [ a „ |] ( A t , / ) . That is, n ^ n U A » ,/) = 0. Hence there is j e J(a„) such that, for I S i S n , v(A|) = noU M i.y ). II) v(A„) 1. Proof as in case I). (c) Let = A t -» Ae, for some e ,k < n. By D15.2.b, J(aB) = J(tV /). I) K^b) = 0. Then v(Ak) = I and v{Ae) = 0, since v is also a semi-valuaüon. By ( t) there is j e J(<Xn) such that r[a„](A *,/) = 1 and n a„ ](A t ,j ) = 0. By D15.2.b.i, T[an)(Ak, f ) = T[an i\(A k,j) , T[anKAt ,j ) = T\a„ i)(At ,j) . By D15.2.b.ii, 71a„](/1n,7') = 0. Hence there is j e J(ra„) such that, for I S i S n , v(/t;) = n«„](A „y). II) v(An) = 1. Proof as in case I). 118 GTTs for K (d) A„ = uAm, for some m < n. (I) v(An) = 0. By definition, v is an a„-valuation, and thus there is an a„./-valuation v„ such that = 0 and v<n-I>v„. By ( t) lhere is j n e J(a*) such that, for i < n, 71an](A,,yn) = v„(Aj). Then 7'[an](Am, in) = 0 and, for every r, I i r < n such lhat A , = 0 4 , and v(Ar) =1 [Ar = o A , and v(Ar) = 0], 7[a„](i4r, j) = 1 [= OJ and n a n K A i.Ã ) = 1 [= 0], By definition, a(j, n-1) * 0. If we now check D15.2.d.iv and D15.2.d.v we see lhat, whatever the case, there is always a j* such lhat, for i < n, T[(X„](Aj,y) = riOn]^,-, j*), and, besides, r i a n](<4„,y*) = 0. Hence there is j* e J(a„) such thal, for 1 i i i n , v(A,) = /[a„)(/t,, /*)• (II) v(A„) = 1. By definition, v is an oq-valuation, and thus for every p, every q ,q < p i n such that Ap = 0 4 , and iiA p) = 0 [Ap = 0 A q and v(Ap) = 1], lhere is an a„./-valualion vp such that vp(Aq) = 0 [= 1], vp(Am) = 1 and v<n-l>vp. By ( t) there is j e J(a„) such lhal, for / < n, T[a„)(A(,j) = v(A,). That is, T[a„\(Ap, j) = 0 [= 1J. Besides, for every r , I i r < n such lhat Ar = 0 4 , and v(/4r) = i[ A , = OA, and v(/4r) = 0], 7 '[an|(/4,,y) = I (= 0]. Also by (t) lhere is j„ e J(a„) such that, for i < n, 7'[a„)(/t,,;„) = vn(Aj). That is, 71 a*](A?.;,,) = 0 (= 1), T\a„\(Am,j„) = 1 and, for every r , l i r < n such that Ar = 0 4 , and v(Ar) = I \A, = o A , and v(Ar) = 0], 71aM](A,,y„) = 1 [= 0]. Thus it follows lhat, for every p < n, that P(p .j, n-1) * 0. By DlS.2.d.iv, lhere j 6 J(otn) (/ = jm-) such lhal, for 1 i i < n, v(A,) = T[a„l(A,,y') and n a * K A n,j) = 1. Thus, for l i i i n , v(A,) = r[a .](A „ ;). (e) A„ = 0 A m, for some m < n. Proof as in (d). ■ Lemma L22. For every normal sequence a „ , if, for some i i n, t-g A, lhen fo r every j e J (an), T [a n l(A ,/)= l. Proof By induction on lhe number r of lines of a proof of A,in K. (1) r = / . In this case, Aj is an axiom. (a) Let A/ = A -» (fl -> A). Let us suppose lhal, for some j e J(a„), /[a„](A |,y) = 0. By DI5.2.c.ii, r[a„](A,y) = 1 and 71a„](B -> A ,j) = 0. Again by D15.2.c.ii, TlOnKB.v) = 1 and T{an\(A,j) = 0, what is not possible. Thus for every j e J(a„), l]a„\(A ,j) = 1. (b) If now A,is an instance of A2 or A3, the proof is analogous as in (a). (c) Let Ai = d(A -> B) -> (04 -» OB). Lei us suppose lhat, for some j e J(a„), 7 '[a„|(A ;,/) = 0. By D15.2.C.Ü, r[anl(D(A -> B ),j) = 7'(a„](Q4,y) = 1 and r[a„](uB ,y) = 0. By D15.2.d we see lhat, since 7'[a„](Dfl,7) = 0, thal a(j, n-I) * 0 (if not, by D15.2.d.iii, we would have 7 '|a„ |(afl,y) = 1). From this fact il follows lhat there is j„ e J(a„) such that 7'[ol„1(B,;„) = 0 and, for every r, I i r < n such that Ar = 0 4 , and T[a.„)(Ar,j) = I [Ar = o ^ , and n a n}(Ar,J) = OJ. 7'[a„](A„;n) = 1 [= 0], Well, 7I<i„](u(A fl)./) = 7(0^1(04 ,» = 1, thus 7'(a„](A - ) B ,jn) = 7'[a„l(A,;n) = 1, and it cannot be thal 7'(a„](fl,y„) = 0. Thus for every j 6 J(a„), 7'ia„](A,y) = 1. (2) r > 1. In this case, either A, is an axiom, and lhe property is already proved in 1), or: (a) A,was obtained by MP from fl and B > Aj. We thus have lhat t /i and Hfl -» Ai. Let us take a normal sequence On+*. k> 0 , where A„+* = fl -* A,. Of course fl occurs in this sequence. Since i-fl and Hfl -> Aj, we have by lhe induction hypothesis that, for every j* € J(a„+*), 7 [a „ +*](fl Ai,7*) = 1 19 ChapíêrS n « n+*l(B. j* ) = 1. It follows that, for every j* e J(a„ t *), T la „ +tKAj, j* ) = 1. Now, by L20, if, for some j e J(a„), T \a„\(A i,j) = 0, then, for some j* e J(a„+t), notn+tKA;,./*) = 0, and Ihis cannot be. Thus for every j e J(a,,), T[d„](A,j) =1 . (b) Aj = dA* and was obtained by R N from A*. We thus have that l-A* and, by the induction hypothesis, for every j e J(a„), r[a«i](A*.y) = 1. But then, by D15.2.d.I, for every / e J(a«), a ij, n-I) = 0 . Thus; by D15.2.d.iii, for every j e J(ot„), T\a„\(Aj,f) = 1. ■ Theorem T28. i-A ifffo r every normal sequence a„, where A = A,-, / £ i £ n, and fo r every j e J(a„), H a ,] (A/,y)= 1. Proof. A) If i-A the proof follows immediately from L22. B) Let us suppose that for every normal sequence a„ , where A = A,-, 1 S i i n , and for every j e J(otn), nOoKA/.j) = 1. Suppose lhat b■M,-. Then kA,-; thus, there is some valuation v such lhat v(A/) = 0. By L21 there is a y e J(a„) such that TlouKAi, j) = 0, against the hypothesis. Thus >=A;. ■ As we see, K is decidable by GTTs. And, as the reader certainly noticed, the definition of the GTT was exactly "copied" from the definition of an a„-valuaüon. Defining GTTs for lhe olher logics, thus, is not that difficult. I am not going to do it here for reasons of space and because it is really straightforward. (It is a good exeicise, though.)44 44 W e are, how ever, go ing lo take a look a t G T T s for S4 and fo r the E D L Z 5 , from w hich defin itions one can have an idea of, for instance, how reflex iv ity o r euclideanity are handlcd. The S4 problem 1 0 ) Nobody knows the troubles Vve seen... 6.1 The problem... In this chapter we are going to consider, albeit briefly, lhe problems that valuation semantics have with some normal systems of modal logic. Aclually S4-i.e., KT4- is nol alone lhe problem, as lhe tille of this chapter could suggest, but, since il is the most known logic among lhe problemalic cases, it takes the blame. To tell the truth, lhe problem concems axiom schema 4; so many systems containing this axiom are bound to make trouble. (This also includes, by the way, inluitionistic logic, which is not surprising at ali when one Ihinks that there is a translation function relating it and KT4.) As I said in a previous chapter, there is a "natural" definition of an a„-valualion for KT4.1 will show how it looks like, but, for simplicily reasons, I'U let possibilites out of lhe picture. It goes like this, if we consider the case where An = OAm: I) if v(/4„) = 0, lhere is an a„/-valualion v„ such that = 0 and » ,N | II) if v(j4„) = 1, then v(Am) = 1 and for every p, every q, such thal Ap = OAq and v(Ap) = 0, lhere is an (Xn.j-valuation vp such that vp(Aq) = 0, vp(Am) = 1 and vp (a „ ./)Dv>i. Now why is this definition nalurai? Well, for the one part, it takes care of the characleríslic axiom of KT4; thal is, axiom schema 4. Suppose v(uA t QQ4) = 0. Then v(LH) = 1 and v(DQ4) = 0. By the definition, there is an a„./-valuaiion v„ such lhal v„(Q4) = 0 and v„ (a„ .j ]uv,i. Bul, since 0 4 e (A i.....A„_i)av,i, we also should have v„(dA) = 1, and this is a contradiction. So v(04 -> d q 4) = 1. It is easy to see thal this definition renders also the olher modal axioms Irue. So it seems lhal this definition would lake care of KT4. But it is nol lhe case, as far as I can tell. The problem is, one cannot prove-to be hones, I couldn't prove, wilh this definition, the normality lemma, that is, Ihat a„-valuatións are Do-On-normal. The proof comes lo a hall because in KT4 we have, as a derived inference rule, the following one: 0 4 -> B □A -> a B. 121 ihnplfr 6 Suppose now we have the normal sequence A, B, OB, 0 4 ; and let us suppose that, for every Onvaluation v, if v(o4) 1 then v(B) = 1. Let us suppose further that there is some v, such that v(o4) = 1 and v(LlB) 0. In the example, our Ap = Ufl, and A„ = oA. By clause II) of the definition, we should have an a„ /-valualion vp such thal vp(B) = 0, vp(A) = 1 and vp >= i |re„_/)avj . But this is not sufficient to derive a contradiction I In fact, we would need the following: (•) there is an a„-valuation vp such that vp(B) = 0, vp(A) = 1 and vp 1=1 (a» )av,i. If that were the case, we would have vp as an (vvaluation, and we would have vp(o4) = 1. Since the hypothesis was that for every ô-valtiation v, if v(o<) = 1 then v(fl) 1, then we would have vp(B) 1, a contradiction. The rule would be validity-preserving. However, writing the definition this way is obviously something we are not allowed to do. since it would amount to trying to define cvvaluations by means of themselves. A similar problem occurs with axiom schema 5, and also because we then have, as a derived inference rule, the following one: o A - * B o A -» Dfl. The reasons why the natural definition (for 5 alone, like in K5) doesn't work are pretty much the same. Why, then, do we have valuations semantics for logics such as KT5, K45, KD457 Well, in these logics the requirements are that, for instance, if f< k> g, then (a * )Dy;i = (ot*)aj ,r , and (a * ) °/.o = (a*)°g,oThis is enough to guarantee that things work. (See proof of the Normality Lemma L U , particularly in the case of the mentioned systems.) But maybe we can find a way out of the KT4 predicament. Let us think a bit about KT4-models, and let us consider some world in it: call it 0. Because the accessibility relation is transitive, we notice that every world x which occurs "under" 0-that is, which is accessible to 0-has either the same number of tnie necessities as 0, or more. It doesn't happen that in x has less necessities are tnie than in 0. Witness the following example picture (black circles denote true necessities): fl*. 20 As one can see, the set of true necessities increases. One can also see that there is a world, namely t , which gives Imth to every necessary wff (at least, to every one of the Ihree here represented). That is, if t 122 T he S 4 p ro b lem we consider only a finite set of formulas, there is a point where every new accessible world has the same set oftrue necessities as the one to which it is accessible.This would be a world with a "maximal" number of true necessities. 6^...andasolution It is now clear how (his characleristic could help us: with valuation semantics, or GTTs, for that matter, we are also working wilh a finite set of formulas. So we would also have some ''maximal" a „ - valuation, or a "maximal" line in a table. Now to solve the KT4 problem we could make an induclion within our already induclive definition: in the case of modalized formulas, we begin by defining the maximal ones-the basic case-and then proceed "upwards" to the olher ones, uniil we reach lhe levei zero. In this chapter, we will try to pul this idea into practice. Since our main inierest is lhe construction of GTTs, D l skip lhe definitions of valuation semantics and go direclly to lhe GTT definition ilself, after what we'll prove thal a wff is a theorem of KT4 iff it gets 1 in every line of the table. A next remark is: for reasons of simplicily, we'll let the possibilities out of the picture. Bul lhe method here described can be extended to cover them loo. I will introduce two conventions. Let a„ be a normal sequence. For 1 < i < n, we say lhal T; denotes lhe cardinalily of (o tj-/)°, if Ai = OAj, for j < i. If í = 1, or if A,- * QAj, lhen T,- = 0. The second convention-aclually an abbreviation-has lo do wilh lhe GTTs we are going lo define below. Let j be some line of of a GTT for KT4: we will use x,(/) to denote lhe cardinalily of lhe subset of (a,-./)a consisting of those wffs which are given the value 1 in line j . In other words, T,(/) denotes the number of necessary wffs, for k < i, which are true in line j. Now we can go lo our definition. Bul, inslead of a GTT being a function T[a„], it will now be r [ a „J], where lhe I parameter takes values from T„ (maximum) down to zero. Detinition D16. Let ct„ be a normal sequence. A generalized truth-table (GTT) for a n is a function T ía „ ,l] : (a„) x j'(a„) -> (I,* ,0), where: 1) for n = l , / = 0 ,J '( a ; )= (1,2), 7Ta( ,/](/»;, 1)= 1 and T[a,,ll(A/, 2) = 0; 2) for n > 1, a n d J° (a „ ;) = (1.... (j): (a) if An is a propositional variable, then x* = / = 0, J^ a ,) = (1 ,...,2^), and: i) for i < n .j e J°(a„ ,). 71a„. l\(Ai, j) = H a ,. / . 0](/lf,/); ii) for i < n .j ' e J°(an í ) and; = q + j \ T[an. IKAi.j) = / (an ( , 0](/l„j'); iii) for i n. j e )°(a„.;), ÍTOn, l\(Ai.j) =1; iv) for i = n .j 'E J°(a„ /) andy = q + / , T{a„, l\(Ai,j) =0; (b) if An = A k,k < n , i„ = l = 0, J'(a„) = J°(a„ ;) , and: i) for i < n, 7'[a„, IKAi.j) 0 )(Ai,jy, ii) for i = n, 71 a„, l\{At.j) * T lan l , 0](A*,;); (c) if An = A t-iA e , k < n ,e < n , J'(a„) = J°(a„ /), t„ = I = 0, and: i) for i < n, T |a„, ll(A,.j) = 71a„ ; , 0)(A„j); ii) for i = n, ?1a„. l\(Ai,j) = 1 iff 71a„./, 0)(At ,j) = 0 or 7[a„./, 0](At .j) = 1; 523 Chapler 6 (d) if A» = 04» , k < n, lhen, for T» 2 12 0: For every y e J°(a» ./), I) let a ( / , n-l) - (y 'e J°(a» ./): 71a» ./, 0](A*,y") = 0 and, for every r, 1 S r < n, such lhat Ar 0 4 , , 71a»-/, 0](Ar ,y) = 71a»./, 0 ](A r ,/) ); II) where / < r„(f), let '((j. n-I) = (/' E J,+,(a„): 71a», /+ /) (A * ,/) = 0 and, for every r, 1 £ r < n, if Ar ~ 0 4 , and 7 Ja» ./, OJ(Ar ,y) - 1, then / + / ](Ar, j') - 1); III) for every p, every q ,q < p < n such that Ar = oAq and H a » .; , 0](Ap, f) 0, let p(p,y, n-I) = [/" e J° (a» .;): 7 1 a ,. ; , 0)(AÍ , / ') = 0 , 7 1 a » / , OKA*,/-) = 1 and, for every r , 1 S r < n, such lhat A , - Q 4„ T la n.i,0 ](A r . / ) = T[a„-i, 0 ](A„ j") ); IV) for / < T„(/), for every p, every q ,q < p < n such that Ap = 0 4 , and T\a„i](Ap,j , 0) = 0, let &(p, y, n-l) = c J,+ /(«n): 71a», Í+^K A ,,;") = 0 , 71a», /+/|(A *,y") = 1 and, for every r, 1 i n , if Ar = 0 4 , and 71a* i . 0 ](Ar ,y) = 1, then 71a*, /+ /](A „y ') = 1); (a ) Let us take lhe case where / = T„. For every / e J°(a» ./) such that T„(/) = <■ V) let U l Jm) C J ° (a „ ./) such lhat t„(f) = / and: 1 )jm-<jmif 2) jme i i l . -jm) if a ( Jm'. n-l) * 0 . for every p < n, p (p , /m\ n - /) / 0 , and 7 1 « „ i , 0 ](At,y) = 1. Then J ^ a , ) = (1 .....q ..... q+m ) and: 1) for i < n J S q , 71a». J](A/,/) = 71a»./, 0)(/4,.y); ii) for i < n , j = q + m \ 71a», íl(A„y) = 71a»./, 0)(A;, ;„,■); iii) for i = n , j such that a (y , n i) = 0 , 71a», IRA,,/) =1; iv) for i = n , j such lhat a( j , n l ) * 0 and, for some p < n, P(p,y, n-l) = 0 , 71a», í](A/,y) =0; v) for I = n, 71a» ./, 0](A*,y) = 0 , 71a», l\(Auf ) = 0; vi) for i = n,y such lhat a (y , n-l) * 0 , for every p < n, P(p,y, n - /) * 0, and 71a» ./, 0](A*,y) = 1, in which case, for some m 'e {l , . . . j n ) , j= jm'O t j = q + m \ D i f ; = jmthen 71a», fl(A„y) =1; 2) if / = q + m ' then 71" », IWAi,j) =0; Now, for lhose j e Jc,( a » / ) such that T„(/) < I. vii) for i < n, riO ». /](Aj,y) = 71a» ./, 0](A/,y); viii) for i = n , j = q + m', 71a», /](A|,y) = *; (P) Let us now take lhe case where I < T». For every j e J°(a» ./) such lhat T»(/) = I, VI) let Ui,...jm) C J°(a» -/) such that: 1 ) jm'<jmif m',< m"\ 2 ) jm -e Ul, .Jm) if a ( j m', n l ) * 001 a ' ( )„■, n l ) * 0 ; for every p < n, P (p ,yms n i ) * 0 or P!(p./i»-. n-l) * 0; and 71a» ./, 0)(At,y) = 1. Then J(a„ ) = (1 .....q,...,q+m) and: i) for i < n , j S q , 71a», f](A /,» = 71a»./, /+ /)(A i,y); ii) for i < n,y = q + m \ 71a», /](A,-,/> = 71a»./, /+ /](A j,ym-); iii) for i = n,y such that a (y , n - /) = 0 and •/(/, n-7) = 0 , 71a», /](A,,y) =1; iv) for i = n , j such that a (y , n-l) * 0 or y ( j. n ' ' ) * 0 . an(l, for some p < n, p (p ,j, n l ) = 0 an<f* 8'( p ,; , n - /) = 0 . 71a„,/](A „/-) = O; v) for i = n ,7 1 a» i , 0](At, j) = 0 , 71a», I](A i,j) = 0; 12 4 The S 4 problem vi) for i = n ,j such lhat a ( j , n-J) * 0 or •/(/, n-J) * 0; for every p < n, fi(p,j, n-1) * 0 or 8l(p ,j, n7) * 0; and 7T<Xn-/. 0](/4*,7) = 1, in which case, for some m 'e ( / .....m ) ,j = j m'O tj = q + m\ >) if j = lhen lia » , /](^«./) =1; 2) if y = q + m' lhen 71a,,, H(Aj,j) =0; Now, for thosey e J°(Ob-/) such lhat T„(/) < U vii) for í < n, 71a». /KA/.;) = 71a ,./, 0)(A,,;): viii) for j = n j = q + m', T[a„, l\(Ai,j) = *; And for j e J°(a„./) such thal f„(/) > /, >*) for i S n . 71a„, fl(Aitj) = T io ,, f+/](Ai,j). So lhe main idea of lhe definilion is lo first extend lo A„ the lines which have a maximum number of true necessities, and lhen proceed slepwise until we reach lhe "lop" lines-lhe ones which have a minimum number of troe necessities. It is also to notice lhat we introduced a third, dummy truth-value. This is so just in order to have the GTT function giving a value lo every line on every slep. The next steps are now quite straightforward. We first prove lhat lines in a GTT which extends anolher preserve lhe values lhe wffs get: Lemma L23. Lei a„ be a normal sequence. For every j e J°(a*). 1 í * í n, lhere is j* e J°(a„) such that, for every i, 1 £ i í k, 7]a*. 0](<4/,y) = 71a,, 0)(A/,y*). Proof. This lemma can be easily proved by induction on n-k, based on conditions i) and ii) of D16.1.a through D ló.l.d. The fírsl important result here is lo show lhat saturated seis "coincide" with lines of a GTT, giving some finile set of wffs. Lemma L24. Let be a normal sequence. For every saturated set A lhere is a j € J°(a„) such that, for 1 S i £ n. Ai e A iffl\a .„ , 0J(A/,y) = 1. Proof. Let A be a saturated set, and let A i,...A m (i.e., a m) be the longest initial segment of a n in which no wff occurs whose main operator is a necessity. We make first an induction on m. (1) m = 1. By D16, J° (a /) = (1,2). If A, t A, from D16.1 lhere i s j 2 such lhat 7 [a m, 0)(A j,f) = 0; and, if A,e A, from D16.1 there isy = 1 such lhat 7'((xm, 0](Aj,j) 1. (2) Lei m > 1, and = (1 .....q). Induction hypothesis: for every saturated set A there is a j e J°(am-/) such lhat, for 1 S i < j , Ai e A iff T[ami , 0](Ai, j) = 1. Now we have in this segment only propositional formulas, and lhe cases are where Aj is a variable, a negation or an implication, which are treated in a similar way as in the proof of Lemma L21. Thus we conclude ( t) for every saturated set A there is a j e J°(am) such lhat, for 1 < i< m, A; e A iffllcim , 0\(Aj,j) = 1. We now proceed with lhe main proof of this lemma, which goes by induction on n. For n = l,...,m, the lemma is proved, so let us consider n > m. (A) n = m+7. Here we have lhe first occurrcnce of a modalized wff, lhat is, for some k< n ,A „ = 04*. Obviously enough, i„ = 0. 125 C hapter 6 (I) Suppose An e A. By P22 (which also holds for KT4) we have an Ai-saturaled set 8 such that e(A°) C 8 . From ( t) there is a line j e J°(a« ./) such that, for '1 S í < n, Ai e A 0](Ai,j) = 1; and also that there is a line j ' e J°(o„./) such lhat, for 1 S / < n, Ai e 8 0](A/,/*) = 1. That is, 7Ta„./, 0](A/t,y) = 0. Thus a (/, n -/) * 0. By D16.2.d.a.iv (or vi), T\a„, 0](An.j) 0. (II) Suppose An e A. From ( t) lhere is a line j e J°(aB./) such lhat, for 1 i i < n, Aj e A iffT [a n-i, 0](Ai,J) = 1. Ali we have to show (since lhere are no olher modalities) is that 71a*, 0](An,j) = 1. Since * < n, and since A t e A, 77a n 0)(A t,j) 1; lhen it follows from D16.2.d.a.vi.l, that r [a „ , 0](A„,j) = 1. (B) n > m + /. Having thus proven lhe lemma for the base case of the first occurrence of a modal formula,* we arríve lo the following first inductive hypothesis: (IH1) for every saturated set A there is a j e J°(a*./) such that, for 1 <. i < n. Ai e A iffT lc in i, 0](At,j) = 1. The cases in which A* is a propositional variable, a negalion or an implication are proved as usual. Let us again consider the case in which, for some k < n ,A n = 04*. Suppose also that x„ * 0. Now let I denote the cardinality of (a«./ )a n A. We proceed by making a new induction on I. (a ) I = t*. (I) Suppose A„ t A. By P22 (which also holds for KT4) we have an A*-saturated set 8 such that efA13) C 8 . From L14.e.ii, A° C e(A°); thus A° C 8 . From ( t) there is a line j e such that, for 1 < j < n. A/ e A iffT[o«.i, 0](Ai,j) = 1; and also that lhere is a line / e J°(ot„ /) such lhat, for 1 S í < n, Ai e 8 iffT[a„.i, 0](Ai,j') = 1. Then we have that, for every Ar 0 4 ,, such that Ar e A, 71an ; , 0](Ar, j ') = 1. Now, since I = T„- i.e., I is maximum- (a „ ./) a n A = (a „ ./)° r> 8 . Thus 1Jan i,0](Ar,j) = r ia „ . ; , 0](Ar, j '). Moreover, T[a„ i, 0 ) (A t,j ') = 0, since 8 is A*-saturated. This means that a (/, n-1) * 0, hence, by D16.2.d.a.iv (or vi), 7Ta„, f](A„,/) = 0. (II) Suppose An e A. From ( t) there is a line j e J°(a» ./) such that, for 1 S l < n, Ai 6 A i f fT \a „ i, 0](A i,j) ~ 1. Since k < n, and since A t e A, 7'[an 0 ]{A t,j) = 1. Since 1 is maximum, there are no necessities in (a„_/)Q which doesn't belong to A, so T (an, t](.A„,j') = 1 and we are done with the base case. (P) I < i„. We have the second induclive hypothesis: (IH2) for every saturated set A, if the cardinality of ( a , ; ) 0 o A is greater then 1, there is a j e J,+ ,(a„) such that, for 1 £ i S n, Ai e A iffT[(tn, í+/](A/,y) = 1. (I) Suppose An t A. By P22 (which also holds for KT4) we have an At-saturated set 8 such that e(A°) C 8 . From L14.e.il, ÒP C e(A°); thus A° C 8 . From ( t) there is a line j e J ° (a „ ;) such that, for 1 £ i < n, Aj e A iffT [an-i, 0](Ai,j) = 1. Now, if the cardinality of (a „ ./)n n 8 is equal to /, the proof goes as in (a.1). So suppose il is bigger than I. From (IH2) there is a line/ e J,+ ,(a*) such that, for I S i < n, Ai e 8 i f f1 \a n, l+ l](A i,j") = 1. Then we have that, for every A , = oA ,, such thal Ar e A, if 71" »-/. 0)(Ar,j) = 1, then l \a „ J + l\(A rJ ') = 1. Moreover, 71a„, f+ /|(A * ,/) = 0, since 8 is A*-saturaled. This means that y o , n-1) * 0, hence, by D16.2.d.p.iv (or vi), T[a„, I](A„,j) = 0. (II) Suppose A„ e A. From ( t) there is a line j e J°(a„ ./) such that, for 1 5 i < n, A, e A 0](.Ai,j) = 1. Since k< n, and since A t e A, 7 [a„ 0 \(A t,j) = 1. Suppose now there is some Ap 0 4 , such that Ap t. A. Similar to case (I), there is an A,-saturated set 8 such thal AD C 8 . It is easy to see that An € 8 . A similar reasoning as in case (I)-now considering P and 5* will prove lhe lemma. ■ 126 T he S 4 p ro b lem This lemma is im portai because, as we can easily prove (like in chapter 2 for EDLs, or chapter 3 for normal modal logics) Ihat a wff A is a theorem of KT4 if and only if it belongs to every saturated set. We are not going to prove it here, but will make use of this. Lemma L25. / / h A then, fo r every normal sequence where, for some i, A Ai, and for every j e J0(a„). T[a„, 0)(A„j) = 1. Proof. By induction on the number r of lines of a proof of Aj in KT4. (1) r = 1. The cases in which Aj is an axiom of PL are straightforward. Let us consider the other cases. (a) Let Ai = o(A -» fl) -» (04 -* Ofl). Let us suppose that, for some j e J°(â), 7 '|a„ , 0](A„ j) = 0. By D16.2x.ii, 71 a „ , 0](d(A -> B ),j) = 71a„, 0 ](o 4 ,/) = 1 and 7'[a„, 0](Dfl,y) = 0. By D16.2.d we see that, since 7'lrt*, 0](l Ifl./) = 0, that a(J, n-l) / 0 or y (/, n-l) * 0 (if not, by D16.2.d.f}.iii, we would have 71a«, 0](t3fl,y) = 1). From this fact it follows that there is jn e J°(a„) such that 71 a„, 0 |(fl,;„) = 0 and, for every r , l S r < n such thal Ar = 0 4 , and 71a„, 0](A„f) = 1, 7'(a„](Arl l ; n) = 1. Well, /[a „ , 0](D(A -» B ) ,]) = 7 |a „ , 0](oA , j) = 1. thus H a . , 0](/t -> B .j„ ) = 7 (a „ , 0)(A .;„) = 1 (else, by D16.2.d.(i.v, 7'[a», 0)(O(A -> f l) ,/) = r (a „ , 01(04,;-) = 0). It ihus cannot be that 7 [a„ , 0](fl,;„) = 0. Thus for every ; e J^a* ), 71a„, 0](/4,7) = 1. (b) Let Ai = 0 4 -» A. Let us suppose lhal, for some ; e J°(an), 7 (a„, 0 J ( A f) = 0. But then T[a„, 0](o4,y) = 1 and 7[a„, OKA,;} = 0-against D16.2.d.p.v. (c) Let Ai = 0 4 -» 0 0 4 . Let us suppose Ihat, for some ; e J°(a„), 7'[an, 0 ](<4 ,,/ ) = 0. By D16.2.c.ii, 7TO(i. 0](Q4,y) = 1 and 71a„, 0J(EH4,;') = 0. By D16.2.d we see that, since 7[a„, 0](DU4, j) = 0, that a(j, n -l) * 0 or 'f tj , n l) * 0 (if nol, by D16.2.d.p.iii, we would have 71a,,, 0](DQA,;) = 1). However, in both cases we'U supposing lhere is a line;" of the table in which OA has value 0 (since 7'[an, 0J(CO4, j) = 0), but which also has to give value 1 to this formula (since 71a„, 0 |(Q 4,;) = 1 and, by definition of a (/, n -l) and y o , n-7), lhey salisfy the necessities of j). Since this is impossible, it cannot be thal 71ot„, 0](Aj.Ã) = 0. Thus for every; e J°(a„), H a , . 01(A.» = 1 . (2) r > 1. In this case, either A,is an axiom, and the property is already proved in 1), or it was obtained by one of the inference rules MP or RN, in which case the proof is analogous to the case of K in the preceding chapter. ■ Theorem T29.V A iff for every normal sequence a„ where, fo r some i, A = Aj, and for every j e Í°(otn). T[an, 0](Aj, j) = 1. Proof One direction is the preceding lemma, so suppose that, for every normal sequence a„ where, for some i,A = A lt and for every j e J°(an), T[a„, 0](Aí,j) = 1. Suppose furher that ^ A. It is easy to prove that, for some saturated set A, A tf A. By L25 there is a j e J°(a*) such that T[an, 01(4/.y) = 0-against our hypolhesis. Thus h-A. ■ As we just saw, we have lhen used sucessfully GTTs for S4. A similar technique can be used lhen wilh lhe other normal modal logics like K5. 127 1 Valuations, possible worlds, and tableau systems Non ovum tam simili ovo, quam hic Uli est. I guess il would be interesting, now lhat we just finished our joumey through lhe (alelhic modal) Valuation Semantics Jungle, to say a few wocds comparing this semantics to other ones. Thus, in this small chapler, we'll give some thought to two questions, namely: what is the relation (if any) between valuation semantics and the ordinary Kripke or possible-wortd semantics, on lhe one side, and, on the other, what is the relation between GTTs and tableau systems. 7.1 Vahiation and Possible World Semantics 1 already mentioned in chapter 4, while inlroducing the subject, that there is some kind of relation between valuation and possible world semantics. If we want, we can certainly see a valuation as a world, or as a function describing a world. Remember, we have proven lhat valuations are characteristic functions of maximal consistent seis, and what is an MCS, one could ask, if not a world? From this point of view, lhe only model we have to consider is lhe class of ali MCSs-we don't need anymore to introduce acessibility relations as primitive elements of the model. To be precise, in the first chapter of this work, where we were discussing semantics for EDLs, we came exacüy across this fact: lhat lhe class of ali saturaled seis (which are maximal consistent), and the class of seis of wffs true in some world of some (EDL) model, were the same. For the sake of completeness, let us write everything down here, taking a normal modal logic as an example, say KDB. The first step is to define a possible-world model for KDB. As usual, it is a structure < W. R>, where W is a non-empty set (of worlds) and R a binary relation o ver W which has lhe properties of seriality and symmetry. To this structure we add an interprelation function l which assign a truth-value I or 0 to the atomic formulas. In lhe usual way we exlend I to a function from IV x FORkdb inlo (0 ,1 ), such that, for some w e W: 128 V aluations, possib le worlds, a n d tab leau system s l(-A , w) = 1 iff I(A ,w ) = 0; /(A-+fl,w) = 1 iff I(A, w) = 0 or l(B, w ) = 1; I (o A , w) = I iff for every v e W such ihat w/fv,I(A, v) = 1. Now this tuple <W, R, /> is a possible-world model for KDB. I am not going to prove that it is: the interested reader can consult, say, [Ch80], or any good textbook on modal logic. Now, in a similar fashion as we did in Chapter 1 wilh respect to EDLs, we define an equivalence relation among KDB-models. Let K be the set of ali KDB~models. If fMand 9{jz K, we say that <M<* 9^ in case A iff A, for every wff A. If íM*is a model, tíM) will denole equivalence class of 9 4 (thus \94\ € K/«). Let now $4= <W ,R,I> be a model. For each w e W, let [Mtw\ = (A: l(A,w) ~ 1J. Let W = (T C FOR : r = [ M r f, for some íW, some h»}. And fínally, let S be lhe class of ali KDB-saturated seis. We can now prove the following results: Lemma L26. If[94,w] tA then I(A,w) = 1. Proof. Analogous lo the proof of L2. Lemma L27. W = S. Proof Similar the proof of L3, or, by that matter, of L7. Theorem T30. There is a bijective function h from the set V o f ali valuations into the set K/„ such that if v e V, A(v) = líMl, and if% {^ <WJÍJ> then fo r every A e FORkdb> v(A) = I(A,w). Proof. Follows from the Corollary to T21, and from L3.4 ■ So this is the way in which valuation and possible-world semantics are relaied. But I would like to stress again that, in spile of the similarities, they are nol lhe same ihing, specially because it is the induetive definilion of an /4;J...^4rt-valuation, for some normal sequence Ai„ ..A n, which allows us lo easily obtain a decision procedure via GTTs. 7.2 GTTs and ta b le a u System s It is very likely that the first thing that comes to your mind, if you are familiar with lableau-style theorem provers for nonclassical logics, is lhe queslion whether GTTs are just the same as these. What lhey aren't, as litlle as, in classical propositional logic, Irulh-lables are lhe same Ihing as tableau systems. But of course lhey are relaied. To give a short answer-I'm going to explain it later on-lhey tackle the decision problem from opposile ends. ^ Cf. |l^o77|, p. 152, where this Iheorem is pioven in the case o f K. 129 Chapter 7 The history of tableau systems (cf. [Fi83|, pp. 3-10) could be said to begin in 1935, with the introduclion by Gerhard Gentzen of the proof systems nowadays known as "Gentzen's sequent calculus". The nice feature of these proof systems (once you've proven that a certain inference rule named cut can be disposed o f without losses) is that they obeyed the so-called subformula properiy. lhat is, in a proof of some formula <4 we only need to consider subformulas of A. That this is an awfully nice property should be clear to everyone who spent some time trying to find proof of theorems in a "get it from the axioms" way. Gentzen's work was developed first by Beth, and later by Smullyan (for classical logic). The result were "upside down" Gentzen type systems, which consists in what we now call tableau systems (fFi83), p. 5). In the modal logic case, we have the contributions of Hinlikka, Kripke, Hughes and Cresswell, and, of course, Fitting himself.4* Now the main feature of tableau proof procedures is that they are refutation systems. That is, one tries to generate a countermodel for the formula in question: the formula is assumed to be false, and one proceeds by computing which value its subformulas would have under this assumption. Since aleach step we reduce a formula to its subformulas (which are smaller), and since formulas have a finite length, lhe method is sure lo tcrminate: somewhcn there are no more subformulas to be processed, and atoms, of course, cannot be further reduced. The whole conslruclion is made in an inverted-tree manncr we write at the top (of a sheet of paper, for instance) the formula A to be (dis)provcn, precedcd by a sign: T or F, which informs us whether the formula is true or false. Since lhe first wff is lhe one we are trying lo refute, it gets an F. Now we proceed way down on lhe paper by adding new nodes to this seed of a tree: at each node there is going to be a signed subformula of A. In other words, we enlarge the tree using a set of extension rutes ([Fi83], p. 29). In this adding of subformulas to the tree we can distinguish two cases. First, sometimes there is only one possible way of assigning values to subformulas, like when we have a true conjunction A a fl: both conjuncts must be true, if their conjunction is; so we extend lhe tree by adding both TA and TB. Sometimes, however, we are confronted with two possibilities: with a true implication A->fl, for instance, one has that either the antecedem is false, or the consequent is true. In order to account for these two possibilities, the branch we are working on must be split into two new branches, each of them represenling a way of going on (a possible assigranent). Branches can of course split further into sub-branches, and sub-sub-branches, and this is why the tableau ends up being a tree. For the classical propositional logic, the tableau extension rutes are the following (cf. [FÍ83J, pp. 29-30): T a : T (A aB ) F a : F(Aa B ) TA Tfl FA | FB Tv: T(AvB) Fv: F(AvB) TA | TB FA FB More historical delaili can be fbund in [Fi83|. 130 Valualions, possib le w orlds, a n d tab leau system s T->: T (A-*B) F->: F(A-*B) FA | TB TA FB T-,: T -A F-i: F-iA F A TA After having applied lhe extension ruies, we find lhat, in lhe end, two things can occur: (1) We find lhat every branch leads to a conlradiction, i.e., for some alom p, Tp and Fp belong to lhe branch. In Ihis case, lhe branch is said to be closed. AU branches being closed, the supposition that the original formula could be false is absurd, hence it must be valid. (2) Some branch remains open, i.e., there is no more complex formulas in lhe branch which we have still not processed, and no conlradiction arose. In Ihis case, what we have done amounls lo actually crealing a model which falsifies our formula-hence it is not valid. Since a picture is betler than ten lhousand words, let us look at a tableau for A = 0 * F (a->6) -> 1 * T a-»6 2 * F -V>->-vi 3 * T-V> 4 * F v 5 F b 6 T a 7 J L f . I J l X X fig. 21 We began by writing down 'FA'-this is what line 0 means. It is a false implication, so its antecedent must be true and its consequent false; hence we add both of them, wilh the corresponding signs, and cross FA out (we put a star in fronl of it lo show it was already reduced). Now we have two new formulas to which we can apply lhe ruies: Ta->6 and F-A-»-«J. Since lhe firsl of these would enlail a branching in the tableau, we reduce first lhe second one, thal is how we obtain T -i> and F -a on lines 3 and 4. This is the First Very Importam Tableau Rule: if you can avoid branching, then do il, else you'll be complicating things unnecessarily. After further reducing we get the atoms Fí> and Ta (lines S and 6). Now we come to lhe point where there are no more non-branching formulas, so we work on Ta-yb . An implication is true eilher if its antecedent is false, or its consequent is true: these two possibililies are represenled by branching lhe tableau in line 7. Each node begins a possible continuation. Now in each 131 C hapter 7 branch we find a contradiction (the underlined atoms in the tree), so both branches are closed (denoted by lhe 'X ' at the boUom). Since every possible way of assigning values lo lhe propositional variables of A led lo an absurd, it must be a tautology. And it is. Summing it ali up, we say that a tableau is closed iff every one of its branches is closed. And a closed tableau for a wff <4 is lhen said lo be a proof of A. (Cf. [Fi83], p. 30) Now, how are we going to extend this constmction to modal logics? An answer is to be found in, for instance, [HC72], or, more complete, in JFÍ83]. I'm going to take here K as an example, since we are wanting lo make a small comparison to GTTs, and we did that for K in a previous chapler. What we need are, obviously enough, extension rules for the cases in which we want to reduce some modalized formula. For this we could use some inluitions from the semantics. Drawing again on metaphors, we can say that a lableau (for PL) is (part of) a world. In a modal logic, to evaluate a formula we have sometimes to consider formulas in other worlds as well. So the answer should be something along the following line: when you find a modalized formula, say FoA, create another, altemate tableau (Cf. [Fi83], p. 34). For the general case, as Fiting points out, the diffículty with this idea is lhat "in practice it gets rather messy keeping track of the altematives in a tableau proof of even a moderalely complex formula" ([Fi83], p. 34). But for some systems, K included, that will do nicely. Now one just has to define what to carry on to the altemate tableau, when we create one. Again, the semantics gives the answer. Having found a F o4 , of course we are going to create an altemate tableau ("an accessible world") in which we have FA. And since, if OB is true in a world, lhen B is tnie in ali accessible worlds, we have to add TB to the new tableau for every Tdfl we had in the older one, the case of (im)possibilities being handled in a similar, mirroring way. Thus, if S is the set of formulas in a branch, we define S# = {TA : To4 e S ] u (FA : FoA e S). Then we have lhe following two rules (in K we don'i have rules for Fo and Ta): TO: S, TOA Fd : S, Fü A S#, TA S#. TA An example: 0 * F ü ( a ->fr) aD a -O b V 5 F b f * T 0 (a -* 6 ) a D a 6 * T a-»6 2 * F D b 7 T a 3 T D (a->b) 4 T Ba 8 F a T b n«.22 X X In the picture we see lhat we could work on tableau w until we were only left with modalized formulas (lines 2-4). Since K has no rules for Fo and TU, our only possibility was to process line 3. We thus opened a new tableau v, into which we carried F6 and the scopes of lhe true necessities in w: Ta->6 132 Valuations, possib le w orlds, a n d tab leau system s and Ta. With these three lines we were then able to find a contradiction in v, closing both branches. Thus ^ ie original wff is valid in K. One can now show that the tableau proof procedure just described for K in fact works. That is, a wff A is valid in K iff there is a closed tableau for A. (I'm not going to do this here, because it is a standard result; the reader can consult e.g. [Fi83], chapter 2.) What is of concem to us here is the fact that a branch of a tableau is satisfiable iff there is a world w in a Kripke model M such thal ali wffs on the branch get are true in w. Since a world in a Kripke model coiresponds to a saturated set (cf. L28 above), and thus to a valuation (Corollary to T21), and thus (by L21) to a line of a generalized truth-table, we conclude that a branch of a tableau is satisfiable iff there is a line on some GTT which gives l to every formula on the branch. What is then the difference? As I put it before, these methods tackle the decision problem from opposite ends. Whereas tableau systems are refutation procedures-one tries to build a countermodel, reaching a contradiction if none exists-GTTs try to examine ali relevant models, and see whelher the formula is true in ali of them. This is the same situation which obtains between truth-tables and tableau systems for classical propositional logic. As I said in the introduction, they are the two sides of a coin.47 47 T here is still an o th e r w ay o f do ing tab leau >yslenu , w hich is som ew hat d iffe ren l from F ittin g 's fo im ulation : I m ean the w ay G .E . H u g h es a n d M .l . C rcssw e ll p rc sen t sem an tica l lab lea u s in th e ii (11C721. T h e ii fon riu la tio n is d iffe ren t from F ittin g 's in tha t they use a m ore graph ica l approach : w orlds are re p resen ted through boxes con ta in in g form ulas, and the accessib ility re ia tions ho ld ing am ong them are represented through arrow s. M oreover, in system s like K T S o r K T B , o n e goes back and forth betw een w orlds, a feature that F itting tries to elim inate because it increases lhe com plexity o f the com putatíon. H C 's fo rm ula tion is w hat F itting has in m ind when he says (quo tation ab o v e) that it is ra th e r m essy keeping track o f a lo t al lem a te tableaus. In the ca se o f va luation sem an tics , they have som etim es m ore sim ilitudes to I I C 'i w ay o f do ing th ings than to F ittin g 's , because, i f w e try to generate a tableau procedure ou t o f a G T T 0 ík e we do in a next chapter on im plem entations) w e 'U see that w e wiU a lso be go ing back and forth betw een "w o rid s" , o r betw een tab leaus, using resu lts from a " new er" one to deriv e a contrad iciton in an "o ider" one. 133 Valuations & GTTs for Z5 Kinkíer's Second Law: Ali lhe easy problems have been solved. After having developed in the preceding chapters the method of valuations for several systems of modal logic, now it has come lhe time to tum our auention again to epistemic-logical matters. In this chapter we are going to apply to EDLs what we have leamed so far. I will choose here just one EDLogic as an example, and give for it the valuation definitions, GTT construction, and then proving that they are correct. After lhat we'll be ready to consider some implementation queslions. The logic I have chosen to take as an example is ZS, for the very simple reason that it is the system HM already mentioned. The strategy of this chapter is pretty much lhe same as in the modal logics case. Some things are of course going to stay lhe same-for instance, semi-valuations, and valuations simpliciter. Hence the main point is again finding a nice definition of an A/,.../n-valuaUon for Z5. 8.1 Defining Ai,...^n-valuatlonsforZ5 Before we get thipgs rolling, il is worth mcnlioning that lhe situalion here, on the one hand, is going lo be more complicated than in the modal case, since we have two strong primitive operators ('B' and 'K'), in comparison to On the olher hand, we won'l be considering weak operators (like o ), so things get in this aspect simpler. As a first consequence of this we don't need anymore lhe distinguishing subscripts in '1= 1 ' and 'No', because we'll be considering just the satisfaction case, and not rejection. That is, plain '*=' will be meaning our old V j \ Just lo remember, we'll continue to use 'a ' as a typographical substitution for '<4/,...^4', so 'a*-/' means actually 'A i,...A k-i', and so forth. But let us now begin by introducing the Z5 analog of/<*>*: we'll have here actually two analogs, since we have two strong operators (which has already given us, in lhe possible world models, lhe two 134 V aluations <1 G T T s fo r Z 5 accessibütiy relations S and R). So let a n be a normal sequence and / , g two functions from FOR into (0,1). We say that, for / £ k £ n , (a )(belief) f< p ,k> g iff g t= e ((a * )B/ i ) , l « t ) B/.l = (a*)8*,!, and (b) (knowledge) f<K,k> g iff (a* )K/.l = (" *)**,1A few words on this. The belief case is, firsüy, similar lo KD45 one (wilh 'B ' inslead of '□ ')- lhal's whal lhe firsl Iwo clauses say (cf. Abb2 on chapler 4). Now the additional requirement lhat the true knowledge formulas o f/and g must be lhe same is there in order lo capture the idea that knowledge implies belief (or, in possible world semantics lalking, that the belief accessibility relation is included in lhe knowledge one). The knowledge case, as one can see, is just plain KT5 (wilh 'K ' inslead of '□ '). We can now go immedialely lo lhe definition of an a*-valuaüon. Definition D17. v is a an-valualion (for Z5) if On is a normal sequence and: 1 ) n = 1 and v is a semi-valuation; 2) n > 7, v is an a*./-valuation and, if for some m < n, A) A„ = BAm, I) if v(A„) 0 then lhere is an on j-valuation v„ such thal vn(Am) 0 and v<p, n-l >v„; II) if v(A„) = 1 lhen lhere is an a„/-valuation v„ such that v„(Am) = I and v<p, n-l >v„; moreover, for every p, every q ,q < p S n , such thal Ap = BA, and v(Ap) = 0 lhere is an a* / -valuation vp such that vp(Aç) = 0, vp(Am) = I and v<P, n-l>vp. B)A„ = KAm, I) if v(A„) = 0 lhen lhere is an on-z-valuation v„ such that v„(Am) = 0 and v< K, n-l >v„; II) if x(A*) = 1 then v(Am) = 1 and for every p, every q, q < p í n , such lhal Ap = KA, [Ap = BA,] and v(Ap) 0 lhere is an a„./-valualion vp such lhat vp(Aq) = 0, vp(Am) = 1 and vcic, n-I>vp lv<P, n-l>vp ]. As one can see, precious lillle has changed from the cortesponding deflnitions for the normal modal logics-we just needed some minor adjuslments. It is however worth noticing thal, in the knowledge case, g lse belief formulas (lhe requirement " ... [Ap = BA,] and v(Ap) = 0 ...") must also be taken into account (they entail that the corresponding knowledge wff will also be false). Having now defined on-valuations, the rest follows as usual. Firsl we make the necessary changes in lhe dcfinilion of a canonical extension: Definition D18. Let On be a normal sequence and v an a„./-valuation. We say lhat vc is the canonical extension of v lo a„ if: A) for ali m < n ,A n * BAm, A„ * KAm and vc = v; or B) for some m < n, An = BAm [A„ = KAm] and vc is a function from FOR into (0,1) such lhat, for every formula B, >35 Chapter 8 1) if A„ is not a subformula of B, then v^B ) v(0); 2) if An is a subformula of B, lhen a) for B = A*. vc(B) = 0 iff lhere is an a„.;-valuation v+ such that v+(Am) = 0 and v<p, n-l >v* (v<ir, n-l>v*]\ b) for B = -C , vc(B) = 1 iff vc(C) = 0; c) for B = C -> D, M B ) = 1 iff V(< 0 = 0 or vc(D) = 1; d) for B BC or B = KC, Vc(B) = v(fl). We only need now the definitions of normality, and then things can get going. Let v be an a * - valuation: for / i k i n, we say that v is Bo-at-normal [K^-â-normaí) if for every p, every q ,q < p í k, such that Ap = BA, [= KA,] and v(Ap) = 0, there is an a*-valuation vp such that vp(Aq) = 0 and v</J, k > V p [v<«r, k > V p ] . We say that a valuation vis Bi-at-norm al if for every p, every q ,q < p i k , such that Ap = BA, and v(Ap) = 1, there is an a*-valuation vp such lhat vp(Aq) = 1 and v</3, k>vp; vis Ki~a* normal if for every p, every q .q < p i k , such that Ap = KA, and v(Ap) = 1, v(A,) = 1. (B iand K i- normality, remember, are the conditions which lake care of axiom schemas Ifi and k.) We are now ready to get our results. That canonical extensions are semi-valuations should be by now obvious-it is proved exactly in the same way as in the normal modal logics case, so nothing new here. Proposition P26. Let a * b e a normal sequence, v an (I*-/ valuation and vc the canonical extension o f v to a n Let us suppose lhat v is Bo-, B i-, Kõ and normal. In this case, vc is an an-valuation. Proof. First of ali, vc is an (^./-valuation, because it is a semi-valuation and, by construction, for 1 i i < n, vc(Ai) = v(A,). Now, if for every m < n, A„ * BAm, A„ * KA„, vc fulfllls every condition of D17, so it is an dn-valualion. So suppose that, for some m < n ,A n BAm or A„ = KAm. We have two cases: (I) vc(An) = 0. By D18.B.2.a there is an a,./-valuation v+ such that v+(Am) = 0 and v<p, n-l >v*\, or v<«r, n-l >v*. Since v and vc agree for i< n , vc<p, n-l>v*, or vc< k, n-l >v*. So vc is an dn-valualion. (II) Vj(A„) = 1. We get first lhat: ( t) By D18.B.2.a, for every a„./-valuation v+ such that v<p, n-l>v* [or v<r, n-/>v*], v+ÍA,*) = 1. We consider now separately lhe belief and knowledge cases: a ) Belief: Suppose there is q < p i n such thal Ap = BA, and vc(Ap) = 0. Then v(Ap) = 0 and, since v is B o-a*-/normal , there is an <v;-valuation vp such lhat v<p, n-l>vp and vp(Aq) 0. Since v and vc agree for i < n, we have that vc<P, n-l>vp, and, from (t), lhat vp(Am) = 1 (else we would have vc(A„) = 0). We have now to prove that there is an u„ /-valuation v„ such lhat vn(Am) 1 and v<p, n-l >vn. If there is some q < p < n such lhat Ap BA, and vc(Ap) = 0, then we have already got this a„-/-valualion vp such that v<p, n-l>vp and vp{Am) = 1. Suppose lhen there is no q < p i n such that Ap = BA, and vc(Ap) 0. We have two possibilities: 136 V aluations <& G T Ts fo r Z S (i) lhere is some q < p S n such lhal Ap = BA, and Vc(Ap) = 1. Then v(Ap) = 1 and, since v is noimal, lhere is an tvy-valuation vp such that v<p, n-I>vp and vp(Aq) = 1. Since v and vc agree for i < n, we have Ihat v</3, n-I>vp; il follows from (t) lhal vp(Am) = 1. (ii) there is no q < p S n such that Ap = BA, and vc(Ap) = 1. Well, in this case, (a*)8 = 0, in which case vc n £ ( (a * )B„c,i), and obviously enough (a * )Bvc,i = (a * )Bvc,i; (a * )K„c,i = (a * )Kvc,r, so vc<f), n-l>vc and, from (t), Vc(Am) = 1. It follows, in both cases, Ihat vc is an ot„-valuation. [}) Knowledge: If there is q < p S n such that Ap = KAq and vc(Ap) = 0, lhe proof goes as in a). If now lhere is q < p S n such Ihat Ap = BA, and v^Ap) = 0, lhen v(Ap) = 0 and, since v is B o-an.j-normal, there is an a „ . / - valuation vp such thal v</3, n-l>vp and vp(Aq) = 0. Since v and vc agree for i < n, we have that vc</3, n-l>vp. Now this means, among other things, lhal (ô)K„c i = (a*)Kwp, 1 ■ That is, vc<if, n-l>vp. From (t), we lhen get vp(Am) = 1 (else we would have vc(A„) = 0). We have now to prove lhal Vc(A„) = 1. Since, for every a n./-valuation v+ such that v< K , n-l >v*, v*(Am) = 1, we only need to prove that vc< r, n-l>vc. But this is immediate, because (a* )Kvc,i = (a* )Kvc,lHence vc is an tx„-valualion.B Now to lhe next lemma, where we can show that cu-valualions are normal wilhout reslriclions, and thus that they can be exiended as long we we wanl them lo be. Lemma L26. Let v be an â-valualion. Then v is Bo-, B |- , Koand Kj-a„-normal. Proof. By induclion on n. For n = I it holds Irivially, so let n > 1 and let us suppose Ihat every IX„ / valuation is Bo-, B |- , Koand Ki-<x„ /-nonnal. Il follows lhen from P26 that (t) The canonical extensions of a„ /-valuaüons to a„ are og,-valualions. We have now our usual Ihree cases: (1) For every m < n. A* * BAm, A„ * KA m. So v is Irivially Bo-, B |- , Ko-and K]-a„-normal. (2) Lei us suppose Ihat, for some m < n. A* = BAm. (I) Let y(A„) = 0. We have: 1) ( a „ )Bv,i * {On-/)Bv,i; 2) (a„ )Kw+,l = (o*-/)Kv+,l, for every dn-j-valuation v+; 3) e ((a « )Bv,i) = E ((a„ ./)Bv,i). It follows that, for every a„./-valuation v4, 4) if v<p, n-I>v* lhen v<P, n>v+. From lhe induclion hypolhesis, v is Bo-a„ /-normal, so we have: 5) for every p, every q ,q < p < n suchl that Ap = BA, and v(Ap) = 0, lhere is an a n./-valuation vp such that vp(Aq) = 0 and v<p, n-l>vp. 137 C hapler 8 Now, for each p, let vp ' be lhe canonical extension of vp to a„. Obviously vp'(A q) = vp(Aq), and, from (t), Vp* is an (1,,-valuation. From this, 4) and 5): 6) for every p, every q ,q < p < n sucht that Ap BA, and v(Ap) = 0, there is an «,-valuation vp* such that »>*(A,) = 0 and v</3, n>vp*. On lhe olher hand, since v is an aa-valuation, we have: 7) there is an (V/-valualion v„ such lhat va(A ^ = 0 and vcfl, n-l >vn. Now let vn* be the canonical extension of v„ to a*. Obviously v„*(Am) = v„(Am), and, from (t), v„* is an a,r-valuation. Thus we have from this fact, together with 4) and 7): 8) for p = n, q = m, Ap BA, and v(Ap) = 0, there is an an-valuation vp* such lhat Vp*(A,) 0 and v<p, «>vp*. From 6) and 8), lhen, v is an Bo-tv-normal. Now, since A* = BA„ and v(A„) = 0. v is trivially B i-, Ko-, and K i-a„-norm al. (II) Let v(A„) = 1. We then have: 1) = (ttn-/)®,,! u (An); 2) (a fl)K»+ i = (a„ / ) K»+,i, for every a»./-valuation v+; 3) e ((o „ )B,,i)= e ( (a „ ./)B, , i ) u (Am). Since v(An) = 1, we have from definition I that: 4) for every p, every q ,q < p S n , such lhat Ap = BA, and v(Ap) = 0 there is an (V/-valuation such lhat VP(A,) = 0, vp(Am) = 1 and v</3, n-l>vp. For each p, let vp' be the canonical extension of vp to a*. Obviously vp*(A,) = vp(A,)t and, from (f), vp* is an (ifl-valuation. It follows that: 5) for every p, every q ,q < p S n , such that Ap = BA, and v{Ap) 0 there is an dn-valuation vp* such that V (A ,) = 0, vp*{Am) 1 and v<P, n-l >vp'. We only need to prove now lhat v</}, n>vp»; the Bo-oin-normality follows. First, since vp*(Am) = 1, vp' >=) e ( (a B. / ) B»,i) <J (Am); thus, from 3), 6) V i = ie ( ( a „ ) Bv,i). We have now two cases lo consider: (A) If, now, Vp*(A„) = I, (a„}BVj,.,i = (a „ ./]BUp.,i u (A„); (a „ )B„,i = (a n)BVp<,i and since (from 2) (tt<i)KVp*,l = lttn-/)Kij>*,l. we have, togelher wilh 6), that v</3, n>vp*. Hence v is Bo- (v-normal. (B) Suppose now Vp*(A„) = 0. We define, for every p , a new function vp* from FOR into (0,1} in the following way: for every formula fl, 1) if A„ is not a subformula of B, lhen vp*(B) = V p * ( f l ) ; 2) if A„ is a subformula of B, then a) for B = A„, vp*(fl) = 1; b) for B = - £ \ vp*(B) = 1 iff vp*(C) = 0; 138 V aluations A. G I T s fo r Z 5 c) for B = C -» D, k /(B ) = 1 iff v /(C ) = 0 or v / ( 0 ) = 1; d) for B = BC or fl = KC. v /(B ) = vp '(B ). It is now obvious thal vp* is a semi-valualion. Besides, for I S i < n. Vp*(Aj) = vp*(A,). Since vp' is an a„./-valuation, Vp* >s an oa-j-valuaiion. We prove lhal vp* is an a*-va!uation. First, since we have v</), n>vp*, and since vp* and vp* agree for i < n, it follows that v<p, n>vp»-from what we get that (<x„_/)BVii = (a„ / ) Bvp*,l and (a „ ./)Kv i = (a „ ./)KVp#,|. Let us suppose now that there is r, s, s < r Sn , such thal Ar = BA, and vp*(Ar) = 0. Then v(Ar) = 0. Since v is an a„-valualion, from D17 il follows Ihat there is an a*./-valuation vr such that v,(As) = 0, Vr(Am) = I and v<p, n-l >vr. Since lhe set of Irue belief and knowledge formulas of v and vp* is lhe same, il follows lhal vp*<P, n-l>v,. If now lhere is no r, s, s < r S n , such lhal Ar = BAs and Vp, (Ar) = 0, we gel, since v is an a„-valuation, Ihat there is an U * . / valuation v„ such that v„(Am) = I and v<p, n-l>vn. Thus vp*<p, n-l>v„. It follows Ihat vp* is an a „ - valuation. Now, since vp*(Am) = 1, ( a „ ) By,i = and ( a rtJKv.i = ( a n) KVi,íf,i, and, since vp*(Am) = vp*(Am) = 1, Vp# n e({a*)Bv,i). Thus v<p, n>vp* and il follows that for every p ,every qt q < p < ,n , such that Ap = KAq and v(Ap) = 0 there is an (Xn-valualion vp* such that vp*(Aq) = 0 and vcp, n>vp*. That is, v is B o-an-normal. We prove now that v is B |and Ki-on-normal. That v is K |-a„-norm al follows trivially from the fact that it is K]-<Xfl-/-normal, because An * K/4m. By induclion hypolhesis, v is B |-a„./-norm al, and, from definition 1, we have that for p = n, q = m, there is an a n./-valuation vp such lhal vp(Am) = 1 and v<p, n-l>vp. We take the canonical extension vp* from vp to a n. It is of course an a„-valuaiion, and, since vp*(Am) = 1, it follows from 3) and 4) that v<p, n>vp*. So v is Bj-on-normal. (3) Let us suppose that, for some m < n ,A n = KAm. We prove as in KT5 (with *K' for '□ ') that v is Koand Ki-a„-norm al. What we should show is that v is Boand Bi-a„-normaI as well (remember, true knowledge formulas are also involved, and here we can have one more). We have again two cases, but, if v(j4„) = 0, lhe proof goes as usual. So let us consider the case where v(/4„) = 1. We lhen have: 1) (o. ) k»,i = (o , . | ) k , ,i u M ,] ; 2) (<Xfl)®v+,i = (a* ./)B»+,i, for every a„ /-valuation v+; 3) e((a»)B*+,i) = e((a*-í)Bv+,l), for every a„.;-valuation v+. Since v(i4n) = 1, we have from DI7 lhal: 4) for every p , every q ,q < p S n , such that Ap = B Aq and v(Ap) = 0 lhere is an «„ / valuation vp such Ihat vp(Aq) = 0, vp(Am) = 1 and v<p, n-l>vp. For each p, let vp* be the canonical extension of vp to a„. Obviously vp*(Aq) = vp(Aq), and, from (t), vp* is an dn-valualion. It follows lhal: 5) for every p, every q .q < p S n , such lhal Ap = BAq and v(Ap) = 0 lhere is an a„-valuation vp' such that vp'(A q) = 0, vp'(Am) = 1 and v<p, n-l>vp'. We only need to prove now that v<p, n>vpr; lhe Bo-a„nonnalily follows. First, since Vp^fA") = 1, vp* N j e d a , . / ) 8,,,!) u [Am\-, ihus, from 3), 139 C hapter ê 6) vp* N i £ ({a„)B„,i). We have now two cases lo consíder: (A) If, now, Vp*(A„) = t , (a * )K»p.,i = ( a „ .j )K,p»ti u (A„); ( « , ) Kv,i = (On)Kvp*,i and since (from 2) {«(i)Bvp*,l = (a» / ) BvpM, and since (from 3) e ( (a B)BVp«,i) = e((a„_ ;)BVp«ii), we have, together with 6), lhat v<P, n>Vp'. Hence v is Bqan-noimal. (B) Suppose now vp'(A n) = 0. We define, for every p , a new function vp* from FOR into (0,1) in the foltowing way: for every formula B, 1) if A„ is not a subformula of B, lhen vp*(B) = vp*(fi)\ 2) if A„ is a subfonnula of B, lhen a) for B = A„. v /(B ) = 1; b) for B = ->C, V p * (B ) = 1 ilf vp*(C) = 0; c) for B = C -> D, v /(fl) = 1 iff Vp*(Q = 0 or vp*(D) = 1; d) for B = BC or B = KC, v /(B ) = vp'(B). It is now obvious thal Vp* is a semi-valuation. Besides, for 1 S i < n, vp*(Aj) = vp, (Ai). Since vp* is an <V/-valuation, vp* is an ct„/-valuation. We prove that vp" is an a„-valuation. First, since we have v<p, n>vp ' , and since vp» and vp* agree for i < n, it follows thal v</J, n>vp»-from what we get that (<*n-/)Bv,l = (<*»-/)Bvp#,l and (a*./)*»,! = {ot*.|)KVpir,lLet us suppose now lhat there is r ,s ,s < r S n , such thal Ar BA, and vp*(Ar) = 0. Then v(Af) 0. Since v is an on-valuation, from D17 it follows lhat lhere is an (^./-valuation v, such thal v,(A,) = 0, vJA m) = 1 and v</3, n-l>vr. Since the set of true belief and knowledge formulas of v and vp" is the same (for i < n), il follows lhat Vp*</J, n-I>vr. Now suppose lhere is a r , s , s < r S n , such lhat Ar = KA, and vp*(Ar) = 0. Then v(Ar) = 0, and from DI7 we get that there is an a„-|-valuation vf such that vr(A,) = 0, vr(Am) = 1 and v<»r, n-I>vr. Since the set of true knowledge formulas of v and Vp* is lhe same (for i < n), it follows lhat vp* < K, n-l >vr. And finally, we have thal Vp»(Am) = 1, so vp#(Am) = 1. It follows that vp* is an on-valuation. Now, since Vp*(A„) = 1, ( a „ ) B»,i = (a„}BVp#.i; and |a „ ) Kv,l = ( a „ ) KVp#,|, and, since Vp*(Am) = VptiAm) = 1. t= E ((a„)B„ |). Thus v<p, n>vp* and il follows that for every p, every q ,q < p S n , such lhai Ap = BÂ and v(Ap) = 0 lhere is an on-valuation vp# such that Vp^Aj) = 0 and v<p, n>vp*. That is, v is Bq normal. In a similar way we can prove thal v is B |n n normal, and this completes the proof of the lemma. ■ Now we have, as consequences of Ihis and the other lemmas the following two proposilions: Corollary. Let a R be a normal sequence, v an â i-valuation and vc the canonical extension o fv toOn. Thenvc is anat-valuationand.for I S i S n l , v</AJ = v(Aj). Theorem T31. v is an On-valuation iff: l ) a „ is a normal sequence; 2) v is a semi-valuation; 3) v is a* normal . Proofs are exactly the same as in the normal modal logics case, so there is no need to repeat them here. 140 V aluations A. G T T s fo r Z 5 8.2 Correctness Having lhen proved these properties of (a„-)valuations for Z5, we can now move lo considering correctness. The notions of satisfiability, validity and semantical consequence are defined in the standard modal logical way (cf. chapter 4). Actually lhe only big difference now is lo prove thal lhe axioms of ZS are valid under this semantics; the rest follows in the same old way. So let us get to lhe following Lemma L27. Lei v be an a„-valualion; then, for 1 S i S n , ifA j is an axiom o fZ S lhen v(A,) = 1. Proof. If Aj is an axiom, then it is either an axiom from PL, and it follows from the fact lhat v is a semivaluation that v(Aj) = I, or it is one of the modal axiom schemas. Now, if Aj is one of the pure belief axioms, the proof is the same as in KD45 (with 'B' inslead of and, if Aj is one of lhe pure knowledge axioms, the proof goes as in KT5 (with 'K' inslead of We consider lhen only the cases of lhe mixed axioms. In ZS there is jusl one of them, m, so suppose Aj = KA -* BA. Suppose v(Aj) = 0. Then we have v(KA ) = 1 and v(BA) = 0. From lhe normality lemma it follows that for every p, every q, q < p S n , such that Ap = BAf and v(Ap) = 0, there is an a„-valuation vp such that Vp(A4) = 0 and v<p, n>vp. Thus Vp(A) = 0. Now v<P, n>vp means, among other things, that ( a * lKw,l = (ot„)KVp,l. So Vp(KA) 1. Since, now, vp is K i-a„-noim al, we have vp(A) = 1. a contradiction. So, for every valuation v, v(Aj) = 1. ■ Now we have no trouble to show that, if A is an axiom of Z5 and v is a valuation, lhat v(A) = 1. For the next lemma and its corollary, too, lhe proof is the same as in the normal logics case (with lhe care of subslituting 'B' for '□')■ The same holds, flnally, for the Correctness Theorem, so we can jump without delay to lhe next seclion. Lemma L28. For ali n, ali i, I S i S n , i f v is an -valuation and t /. Aj, then v(A,) l. C o ro l la ry .//h A then)=A. Theorem T32. (Correctness Theorem) / /D -A lhen IV-A. 8.3 Completeness Completeness is again easy lo prove making use of saturated seis-which wc already have defined for knowledge and belief on pari I. The main task here is to proof thal characteristic functions of saturated seis are a„-valuations; the rest follows smoothly in the good old way. So let us consider our Theorem T33. For every A-saturated sei A and every normal sequence a n. lhe characteristic function f ofA is an a n-valualion. 141 C hapter 8 Proof. First of ali, it is easy to prove by P5 lhat ( t) The characteristic function/of A is a semi-valuation. We now prove lhe theorem by induction on n. If n = 1, the property follows from ( t) above. Let us suppose n > 1. (1) If, for every m < n ,A n * BAm, A„ * KAm, / i s trivially an a„-valualion. (2) For some m < n, A* = BAm. (I) /(An) = 0. Then A„ t A, A b* BA„. From P7, lhere is an Am-saturaied set 9 such that e(AB) C 8 . L e t/e be the characteristic function of 9 . By the induction hypothesis,/and f g are (^./-valuations. We also have, since 9 is Am-saturated, lh a t /e (A J = 0. Now, (a,i-/}B/,i C A, thus E (((V í)B/,l) C e(AB) C 9 ; th u s /e t= e( (a „ / ) B/,i) . Now, from P8, AB C e(AB). It is lhen easy to see lhat ( a „ . ; ) = l«fi-l)B/e ,1 Now, from PR, AK C e(AB) (because t-KA -* BKA). It is then easy to see lhat = We thus can say (hat/<n-/>/e; hence/isan dn-valuation. (II) f[An) 1. So BAm e A, A i-B Am. Let us suppose there is some p, some q ,q < p S n such that Ap = BA , and f(A p) = 0. From P7, there is an A,-saturated set 9 such that e(AB) C 9 . L e t /e be the characteristic function of 0 . By the induction hypo(hesis,/and/e are (^./-valuations. Wilh an analogous argument as in case I), we show that f< n-l> fy. Since 9 is A7-salurated, /e (A 4) 0 and, since Am e s(AB),/e (A m) = 1. Now it follows from lemma S, since A t-BAm, that there is an -iAm-satmated set 9 such that e(AB) Ç. 9 . We prove in a similar way thatf y is an a*/-valuation,/<n-l> /e and fo(Am) = 1. (3) For some m < n, A„ = KAm. Proof similar to (2) and lo (he KT5 case. ■ Finally, lhe proof of this lemma's corollary, and of the Completeness Theorem, suffer no change from the modal logic case: Corollary. v is a valuation iff v is the characteristic function o f some saturated set A. Theorem T34. (Completeness Theorem) / /IV A then D-A. 8.4 GTTs for Z5 The last thing needing consideration in this chapler, before we go on, is the definition, based on lhe semantics just seen, o f GTTs for Z.5. There are of course differences in comparison to what we have done lo K, but they show up only in the case of lhe modal operators, lhat is, cases (d) and (e) of the old definition will have to be changed. 111 give only the importanl part of lhe defmjtion: 142 V alua tions A G T T s fo r Z 5 Definition D19. Let a . be a normal sequence. A generalized trulh-table (GTT) for a . is a function T[o«) : (a„) x J(a„) -> (0,1), where: 1) for n = I , J(ot;) = (7,2), 71a/](Aj, 7) = 1 and 71a/](A /,2) = 0; 2) for n > 7, and J(aB./) = (a) propositional variables: as in K; (b) negation: as in K; (c) implication: as in K; (d) if A , = BAt, k < n , lhen for every j e J(a*.j): I) for h e (1,0), let a (u .j. n l) = (/' e J(a„.;): 7T" «-j](A*,y') = u and, for every r, l S r S n , if Ar = BA, and JJtXg-il(Ar.j) = 1, then 7'[a„.;)(AJ,y') = 1; if Ar = BA, or A, = KAs,T[a„i](Ar,f) = 71a„. lV A r.D ); II) for every p, every q ,q < p < n such that Ap = BA, and 71a„ ;](Ap,/) = 0, let p(p ,j, n -l) = (/' e J (a n-i): Tla-n-lVAq.j') = 0, T[a„-i](Ak,j') = 1 and, for every r, 7 S r S n , if Ar = BA, and 7[u„_ l](Ar.j) = 1. Uien 71afl.;](A „;') = I; if Ar = BA, or Ar = KA,,71a„.;](Ar,y) = 71a„./](Ar,;') ): III) let U lJm) C J(a„ .|) such that: 1)7m* <7m* if rnt < m"; 2)jm ' 6 if " Om'. " 0 * 0 and, for every p < n, p (p ,jm-, n-l) * 0. Then J(a„) = {1 ,...,q,...,q+m) and: 1) for i < n .j S q , 1 \a n}(Aj,J) = ii) for i < n .j = q + m', 71oU(A;,y) = T[a„.i](Aj,j„); iii) for i » n .j such that a ( 0 ,j, n-l) = 0 ,7'[a„](A;,/) = 1; iv) for i = n .j such that a ( l . j , n -l) = 0 ,71a„](A,,y) = 0; v) for i = n . j such lhal a(0, j , n l) * 0 and, for some p < n, P(p,y, n-7) = 0, or a(7, j , n -l) = 0, 7Ta„){A„y) =0; vi) for i = n .j such that a(0, j . n -l) * 0, a(7,y, n-l) * 0 and, for every p < n, P(p,y, n-7) * 0, in which case, for some m' e (7,...,m),y = jm' or j = q + m', O if7 = jm lhen7'[a„](A,,/) =1; 2) if j = q + m' then T\an](Ai,j) =0. (e) if A„ = KA*, t < n, then for every j € J(On-;): I) let 7Ü, n-l) = W e J (a» í): H O n-llM *./) = l and, for every r, 1 S r S n , if A , = K A ^ a ^ /K A ,, j) = T[*n.i\(Ar.j')Y. II) for every p, every q ,q < p < n such that Ap = BA, and T\a.nl\(Ap,j) = 0, let 8(p ,j, n-l) = [ / 'e J(" n-í): 7'[<x#i-/](A,,y') = 0, 7 '[aB i)(A t,y ') = 1 and, for every r, 1 S r S n, if A r = BA, and n a n.l)(Ar,j ) = 1, then 71a» /] (A „ j l = I; if Ar = BA, or Ar = K A „r[a„.,](A r .y) = r ( a B./](Ar, 7")); III) let U i,...jm) C J(tt„ /) such lhal: \)jm -< jm ii>n' < m "\ 2) jm' e U/ J m l if a ( jm', n -l) * 0, for every p < n, p(p ,jm\ n -l) * 0, and 7(a„ y](A*,j) = 1. Then J(a„) = 11,....q.....<7+m) and: i) for / < n .j S q. 7Ia„](A;,y) = 7'(a„.í ](A„y); ii) for I < n .j q + m , 71a„](A„y) = 7Tan /](A„jm); iii) for i = n .j such thal y(j. n -l) = 0, 7'[a„](A„y) = 1; 143 C hapter 8 iv) for ( = n . j such that a(J, n -l) * 0 and, for some p < n, h(p,j, n l) = 0 ,7I®n)(A,-,y) = 0; v) for i = n, and r[a*-/](A*,./) = 0, U a«KA,,j) = 0; vi) for / = n ,/su c h that a (/, n -l) * 0, for every p < n, &(p,j, n -l) * 0, and T[a„i](A i,,f) = 1, in which case, for some m' e | J .....m), j = j m' o t j = q + m', 1) if j=Jmthen 1 \« n M i,j) =1; 2) if y = q + m' then TlOaKAi,,/) =0. Now there is probably not very much to explain: this definition just mitrors what we've already done in the valuation definitions for ZS in the preceding sections. We only have to prove that things work, and the only result we have to get is the following Lemma L29. Let a„ be a normal sequence, and v a valuation. Then there i s j ç J(rt*) such that, for 1 S i S n , v(A,) =T[a„](Ai,j). Proof. (cf. the normal modal logics case.) ■ Lemma L30. For every normal sequence a n, if, for some I S n, \-g Ai then for every j e J(a„), na .104 ,;) = 1. Proof. (cf. lhe normal modal logics case.) ■ Theorem T35. i-A iff for every normal sequence a*, where A = Ai, 1 S i S n, and for every j 6 J(a„), n « „ J 0 4 „ /)= 1. Proof. (cf. the normal modal logics case.) ■ As we then see, ZS is also decidable by GTTs. 144 Intermezzo 2 So Part Ii is nnished, and now, wilh our thus acquired knowledge about valuation semantics and generalized truih lables, we can move to the thírd Part of this dissertatkm, where we are going to try putting into practice a little bit of what we leamed. The next three chapters will thus discuss some implementations. Chapter 9 presents a GTT-builder for an example EDL (ZS). Building a whole table, however, is something costly in time and memory, so in Chapter 10 we take another EDL (ZP5) and present for it a tableau-like theorem prover. The last Chapler, 11, shows an implementation of the algorithm for characterizing minimal belief states from Part I. m i u Implementations Implementing a GTT Builder for Z5 O h,Iam a Cprogrammer a n d ím okay / muck with indices and structs ali day And when it works, / shout hoo-ray Oh, la m a C programmer a n d ím okay. In this chapter we are going to examine a simple C program which implemenis the construclion of GTTs for Z5 formulas. This is going to be a very straightforward implementation of the GTT definition which we have just given in chapter 8. It is not intended to be a real, fast theorem prover for Z5, even if its performance it's not that bad-it is more a 1 to 1 implementation of the semantics, mainly pretending to show how things can be done. It has been chosen to be close to lhe definition, not to be fast. (Surprisingly, however, it can be very fast in certain cases, when com pared to other implementations.) The program, which is called GTT. 25, has three main parts, which are split over several files. The main loop of the program does the following things: • reads a formula (string) from the standard input; • parses the formula, that is, the read string is transformed into a trce-like internai representation; • calls a function which constnicts the table; • and, finally, printes lhe output on the screen. I am not going to discuss every liule thing on the program (for example, parsing and printing routines are just going to be mentioned), but ralher the building of the table. So let us begin. 9.1 Data Structures and main () We examine first the macros and globais, which are lo be found in the file "macros. h". 9.1.1 Macros Id efin e STB_LEN 256 This is the maximum allowed length of the input slrings. 149 C hapter 9 fdefin e MAXLINES 64 fdefin e MAXROWS 30 Ditto for the maximum number of lines and rows in the GTT. We are going to have the tables statically defined as a bidimensional array, where the rows will held places for the wffs. Another allemative, of course, would be using dynamic memory allocation, but I think that with a statically defined array things are easier to grasp. The next lines define, first, a "pointer" to null: since arrays in C begin with 0, and since, as we'll see, conlenls of fields will be pointing to other places o f the arrays, we have to know when we are meaning, say, location 0, or meaning nothing. In the other lines connectives are given an internai representation code by means of integers. For atomic proposilions ( 'a ' till V ) we'll use its ASCII-code. fdefin e CNULL (-1) /* a "p o in te r" to NIL */ fdefine UND 5 /* conjunction */ fdefin e ODR 6 /* d is ju n c tio n */ fdefine IMP 7 /* im plica tion */ fd efin e NEG 8 /* negation */ fdefine EQU 9 /* b ico n d itio n a l */ fdefine KNW 10 /* knowledge */ fdefine BEL 11 /* b e l ie f */ In the next lines we define two kinds of "atomic propositions": on input, when lhe wff is being read and parsed, atoms are small letters. But intemally, after thal, they'11 be everything bul connectives-we have to make just one check, instead of two. Thus the program will run a liule bit faster. fdefine P_ATOMIC( c ) ( c >« 'a ' ( t c <- * t ' ) /* atoms on input */ Idefine ATOMIC( c ) ( c > BEL ) /* atoms in te rn a lly */ The last macro line is just a definition for the parsing mechanism. 1 *11 explain that later. fdefine INIT_NODE(a, b, c) HFF[wf_ptr](0) a; \ WFF[wf_ptr](1) b; \ WFF[wf_ptr)[2) c 9.1.2 G otels We'll also be using some global variables, which we get on lhe next lines: sho rt WFF(STRLEN)[3], w f p t r , TBLfMAXROWSj{MAX_LINES); char theWff(STR_LEN); long k l, k2, k3, k4; ' w f f ( s t r _ le n ] [3] is our bidimensional arrray where lhe internai tree-like representation of formulas is stored; wf p t r points to the next free place on it. The parsing mechanism takes as input a string, like 150 Im plem enting a G T T B u ild er f o r Z 5 a4b->-»av-«b and converts it inlo a tree, which is stored in h ff like 0 1 2 3 4 5 6 a b & V -> 0 0 1 3 2 1 4 5 fig . 24 The rightmost used place is, of course, the "top" node of lhe tree. Numbers under each operator denote lhe row in the array where lhe arguments are to be found. So, for example, lhe disjunclion in 5 has as lhe left disjunct lhe wff of row 3, and as lhe right disjuncl, lhe one in row 4.** The table (t b l) is also a bidimensional array. I have chosen a small number of rows and lines, but Ihis can be easily changed lo suil one's needs. Finally, lhe long integers are used for performance measurements. Now the main (i function has the following code: main () ( short i, j, k, u; printf <M\n* GTT BUILDER FOR Z5 *"); printf ("\n* Cesar A. Mortari *"); printf {"\n* VI.0, May 1990 *"J; printf("Xn***************************************************"); printf ("\n\nSyntax:\n a..t (v^riables), K, B, i, v, ->, <->\n"); printf {"\nTo exit type ';•<CR>\n")/ printf ("\nPlease type in a formula:\nM); This was just printing slartup information. Now we enter lhe main program loop (with for ( ; ;) ).First we have two inilializations: »f p t r is set to o (next free row in h f f ) , and h f f I o h o ] is set lo c n u l l (lo ensure lhat lhe parser won't think that this place, automatically fílled wilh a o on slartup45, is pointing to row zero...) wf_ptr 0; WÇF[0| (0) CNULL; The next step is to read lhe formula of which we want lo have a table. If the input is *;'» that means "end (he program",30 else a small routine remove ali blanks from t hew f f: gets { theWff ); if < theWff(0) - *;') return 1 ; i 0; 48 In rea lity w e d o n 'l have , fo r in stance in row 6 , lhe characlers and *>' ilo red lhere , bu l ra lh e r lhe value 6 , w hich , by m acro defin ition , rep resen ti the arrow . Sim ilarly for the o ther ly m b o li. ^ 1 'm im plem en ting Ihis w ilh lhe THINK C Compiler , vereion 4.0 , w hich has ihis characteristic. W ilh the THINK C Compi l e r this is ac tually unnecessary, because a standard console w indow is provided . together with a "Q u it" m enu op lion . W ilh o lh e r com pilers w hich d o n 't p rovide ih is op tion , o n e has lo in troduce a w ay o f in lerrup tíng the m ain loop. \5 l Chapter 9 j 0; w hile ( th eW ff[i1 1- ' \ 0 S ) ( i f ( theW ff[1J !- ' • ) theW ff[ j*+| " theW ff(i); *+i; » theW ff(j) - ' \ 0 ' ; Next we initialize lhe lime counting, and call lhe parser (with formula o ) to convert lhe string into a tree, at the same time checking if the synlax was correct. If everything is OK, we get lhe time used in parsing and go build the table ( ma k e t a b l e ()). After that lhe contents of the wff array are displayed (di.spLayWFF o ) and the table isprinted. The iast lines print lhen lhe time used for the different routines on lhe screen. kl TickCountO ; /* i n i t i a l i z e time counting */ i f ( form ula( 0, íu , &1 ) theWff(u) - - ' \ 0 ' ) < k2 TickCountO; k m ake_table<); k3 ■ TickCount (); dlsplayWFFO ; p rin t_ ta b le { w f_ptr, k ) ; k4 - << TickCount() k3 ) * 100) / 6; k3 - <(k3-k2)*100)/6; k2 - < <k2-kl)*100)/6 ; p r ln tf {"Parsing tim e: %ld ms", k2); p r i n t f ("\tMake ta b le : %ld ms",k 3 ); p r ln tf (" \n P rin t ta b le : %ld«ms**, k4); p r i n t f (" \tT o ta l tim e: %ld ms\n\n",k2+k3+k4); ) If we had a syntax error, of course, nothing applies, so we print an error message and start again. e ls e p r i n t f ("\nSyntax e r r o r . . . \n \n " ) / We are now ready to consider more detaíls of the program. The other routines are in difTerent Files, which are included (with lln c iu d e < fiie>) just before main o . They are lin c lu d e "macros. h" lin c lu d e "p ro to ty p es . h" lin c lu d e "p a rse r.h " lin c lu d e "ou tpu ts.h" lin c lu d e " ta b le .h " , p ro to types.h isjust a small file containing lhe prototypes of ali GTT.Z5 functions-similar to the "forward" declarations in Pascal procedures and functions. In C it is normally not needed, but see listing in Appendix B. 9.3 parser.h This is the file containing the parsing functions. I won't discuss it here in detail, just give general informalion. It is an adaptation from a parser once wrilten in Prolog by Franz Guenthner. I've changed it 1 5 2 Im plem enting a G T T B u ild er f o r Z S here lo use my data struclures, as well as introducing a mechanism lo check if a node is already lhere. For instance, if we had a formula like a ib -> a«b, we could end up having a tree wilh twice lhe same node, as following: fig . 25 In order not to get repeated wffs when building lhe tables, which is totally unnecessary, we need a checking routine that looks, before crealing a node, to see if it is maybe already there. We would then have the following representation: fig. 26 And that's ali about parsing. See listin'g in Appendix B, if you are interested. Il needs some improving, too. 9.3 outputs.h Liule lo say about Ihis: just Iwo functions which, first, display lhe conlens of h f f and, second, print the table (that is, the t b l array). See lisling in Appendix B. 9.4 table.h And finally, lhe routines lo build the table. We First have the code of lhe make ta b le o function, which we are going lo discuss now: short make_table() /* -> b % b -> b */ int i, row, line, ad lines; Ü53 Chapler 9 1 is a loop variable; row and l in e denote, respectlvely, the current row and line which we are computing. «d i i n e s is the number of added lines in the case of knowledge and belief formulas (correspond to the m parameter in the definition of GTT (as in D19.d, for example). Now the first step is lo initalize the first row and first two lines of the table (the flrst row is always a propositional variable...). The current (bottom) line number is then set to l, and ad lines is initialized to o. Just note that the actual number of lines in lhe table is equal to line + í , because lines are numbeied from o to line. TBL10110) 1; TBL[0]11] 0; lin e 1; ad_ lines 0; Now we enter the Main Table Loop; we do a looj) creating each new row, slarting wilh 1 (lhe row o already contains a propositional variable) and going until w f p t r l (which points to the top node of our formula). If the number of rows or lines gets greater than hax rohs or m a x u n es , the loop is stopped and we exit with o. Else We make a switch on the main (current wfl) connective, and act accordingly. for ( row « 1; row < wf_ptr; ++row ) ( tf ( row >MAXROWS I | U n e >MAX_LINES ) printf<M\n\n***** ERROR j TABLE TOO LARGE! ******"); return 0; ) switcM WFF[row](0] ) C The cases in which the operators are boolean ones, we just do a boolean computing of its argument(s), slarting with line o and going until line. case NEG : for ( i 0; i <line/ ++1 ) TBL[row]{i1 - !TBL(WFF[row][1]](i ]; break; case UND % for ( i " 0; i <• line; ++i ) TBL(row][i] TBL[HFF(row][1)][i] && TBL(WFF(row][2 ]][1 ] ; break; case ODR : for i i * 0; i <line; *+i ) TBL[row]íi] TBL[WFF[row][ 1 ] ] tiI I | TBL[WFF[row][ 2 ] ] [ i ] ; break; case IMP : * for ( i 0; i <line; ++i ) TBL[row}[i] - !TBL(WFF[row][1]][i] • I| TBL[WFF[row][2]](I]; break; * case EQU : for ( i ■ 0; 1 < « 'line; +<-i ) T B L l r o w H H nBHWFFlrow] [1] M U -TBL[WFF(row][ 2 ] ]( i } > ; break; Now we have to consider the epistemic operators, where things are slightly more complicated. Let us begin with knowledge (and lefs begin on lhe left margin, for clarity): 154 Im pU m eftíing a G T T B u iU U r /o rZ S case KNW : < for ( 1 0; i <line; ++i ) if < TBL(WFF(row|(1)|[i] - 0 ) TBL(row)(i) 0; else if ( gammaNE( row, line, WFF(row)(l), 0, 1 ) ) < if ( deltasNE( row, line, WFF(row)(l), 0, 1 ) ) < split_lines( row, line+l+ad_lines, 1 ); ++ad_lines; } else TBL(row)(1] 0; I else TBL[row)[1] 1; ) line line + ad_lines; ad_lines 0; break; ) Let us examine ali that. Let us suppose we have some KA as our wff in the current row. First, of course, we do a fo r loop to compute the value of KA in each line i . And this goes as follows: first, if TBL(WFF [rowj [ i )) [ i ) - o-that is, if A has a value o in line i , obviously KA will have to get o, because we are in S5 for knowledge, and reflexivity holds. This corresponds to clause (e.iii) in lhe GTT definition D18. Else, if A has a value 1, lhere is more lhat we have to check. The function gammaNE o checks if 7( 1, row) is not empty (cf. D18.e.iii-v). That is, wheter there is a line j giving 0 to A and being "accessible" to lhe current line i . If there is not, then KA gets the value 1 (cf. Dl 8.e.iii). Else we check the normality conditions (for every p, every q ,q < p < n, which is a modal wff and gets zero, etc). This is what the function deitasNE o (5 is non empty) does. If it fails, lhen KA gets 0 (cf. D18.e.iv). Else everything checks, and we then arrive to the case where we have to split lhe line, what we do wilh s p ü t iines o : the current line is copied at lhe bottom of the table, and JG4 gets lhen 1 in lhe old line, and zero in lhe new one. Just like lhe definitions. We then increment ad_ünes to signa! that a line was added. Exiting lhe for loop, we set the global line number to its old value, plus the added lines, and reinitialize ad iines to zero. I am not going to discuss here the code of subroulines like gammaNE (), and so on. But see listing in Appendix B. The case of belief is now similar. We have of course the belief correspondents of gammaNE <) and deitasNE (), lhat is, aiphaNE o and betasNE o . The first check does not more cares for reflexivity, because here we have seriality. The resi is pretty much the same. case BEL : { for ( i 0; i <line; ++i ) if < !aiphaNE < row, line, WFF(row)(l), 0, i, 1 ) ) TBL(row)(il 0; else ( if ( aiphaNE< row, line, WFF|row)(l), 0, i, 0 ) ) ( ^ if ( betasNE( row, line, WFFlrow)(lJ, 0, 1 ) ) I 155 C hapter 9 spllt_llnes( row, llne+l+ad_llnes, i ); ++ad_llnesí ) else * TBLÍrow){1) 0; ) else TBLtrow]{1) 1; > U n e line + ad_lines; ad_llnes 0? break; ) And, last but nol least, we have to consider the case where new propositional variables appear! default j / * Atomic proposltlons */ { copy__llnes( row, line + 1 ); for 1 1 0 ; 1 <line; ++1 ) TBLlrowJ[11 1; line " 2 * (line ♦ 1) 1; whlle ( 1 o line I TBL[row)[1++J 0; ) ) When we get a variable, we have of course to double the number of lines. This is what the routines do: c o p y _ i i n e s o make of course a copy of theold lines; then the values 1 and 0 are set, and thenumbef* of lines is updated. After exiting the main table loop, we retum the number of lines, which will be needed by prlnt_table (). ) return ((short) llne+1); > And that's it. 9.5 A worting sesslon with GTT.Z5 Now some examplcs from what happens when the program rans: let us type some formulas and see what happens. I ran it on my Macintosh Plus, with I MB RAM: GTT BUILOER FOR Z5 Cesar A. Mortarl VI.0, May 1990 Syntax: a..t (variables), - K, B, &, v, ->, <-> To exlt type '/*<CR> Please type in a formula: Ka->Ba 156 Im plem enling a G 1 T B u ild er f o r Z S [a][K)[Bi(>) ( 1(0 ) (0 ) [1] ( I ( ) ( 1(2) *** TABLE *** [0) (1] [2)(3) I I I I I I III III 101 101 III III l l l i o i l l l l l l 101|0| | 0| III l l i I 0 | 101|1| Parsing time: 0 ms Make table: 0 ms Print table: 1550 ms Total time: 1550 ms -iKa->K-»Ka ía|(K)[-)[K){>) ( U0 U1 H 2 H 2 I ( ) ( ) ( ] [ 1[31 *** TABLE *** ( 0 ) ( 1 ) ( 2 ) í 3 1 { 4 ] I1II1M0II0IUI 101101l l j I I I | 1 | l l l i o i | 1 | I I I | 1 | Parsing time: 0 ms Make table: 1 6 ms Print table; 1 0 6 6 ms Total time: 1 0 8 2 ms Ba->a (a)[B][>} í 1C0][11 ( )[ 1[01 *** TABLE *** (0)[1][2] m u m i 101 | 1 | 101 11 I I 0 1111 10110| III Parsing time: 0 ms Make table: 16 ms Print table: 1133 ms Total time: 1149 ms ( ] (0) [1) (1) ( II )l 1(2) *** TABLE *** [0][ 1 ] [2)(3) I I I I I I I I I I I I 101| II | 1| m I I I I 0 | | 0 | | I I I0 | | 0| | 0 | I 1 | m m toi ioi ioi i n ioi | 0| Parsing time: 0 ms Make table: 0 ms Print table: 1316 ms Total time: 1316 ms Ka->BKa [a|[K|[BI(>} [ 1(01[11(11 157 C hnpíerÇ [ )( 11 112) * * * TABLE *** 10J(1)[2J(3) i i i m m i i f i o i ; oiio> t i i II! i o i i o i i i i Parsing time: 0 ms Make table: 0 ms Print table: 1050 ms Total time: 1050 ms Ba->BKa [aj [BI [K] [Bj [>] l ) 10) (0) [2] [1] ( H 1 [ I [ I ( 3 ] *** TABLE *** [ 0 ! [1 1 [ 2 ! [ 3 | (4 1 1111I I I I I I I I III 101 | 1 | |0|101 101 I I I 101|1| |1| |1| 101 I 0 M0 I | 0 | | 1 | Parsing time: 0 ms Make table: 16 ms Print table: 1116 ms Total time: 1182 ms As we see, the right results are coming out. Some formulas are lheorems of Z5, and others not, as we can see b. I suggest the reader try himor herself the program. 158 n ® A tableau-like theorem prover for ZP5 Third Law o f Computer Programming: Any given improvement costs more and takes longer. In this chapter we'll also be discussing a C program which implements a theorem prover for ZP5, but in a more efficienl way than what was done in the last chapter for Z5. The idea is to adopt a refutation proof procedure, instead of building a whole trulh lable. Basically we'll take lhe tableau extension ruies for the classical case, and enlarge them adding ruies lo cope with lhe knowledge and belief operators. The intuition for these ruies comes to the conditions defined in valuation semantics.51 This program is called T T P . Z P 5 , and it is very similar, on lhe overall structure, to lhe one considered in the previous chapter ( G T T . Z 5 ) . The parsing mechanism is not exacüy lhe same-two importanl changes were made because the proof procedure makes olher assumptions. The big change, ofcourse, is that we no longer build and display a GTT, but use this tableau procedure to gel a simple "yes" or "no" as whether some wff is valid in ZP5 or not. The main loop of the program does the following things: • reads a formula (string) from lhe standard input; • parses the formula, transforming lhe read string into a tree-like internai representation; • runs the tableau proof procedure; and • prints the answer on the screen. 10.1 DataStructuresandmaino We examine first some macros and the globais. 31 TTP . ZPS i i based on an o ldcr version o f FT L , a lableau theo rem prover fo r lhe classical p ropositional log ic w ritien by 1. Iludelm aie r and m yself, ((M dM 89)), but il underw ent ex tensive icw riting lo cope w ith m odalities. In particu lar, I w ould like to m ention lhat som e tricks to cu t branches used in later versions o f FTL are not being im plem ented here. 159 Chapter 10 10.1.1 Macros Some things are going to remain the same as in the previous program (like length of strings). We don't need m a x l x h e s and m a x _r o w s anymore, but as new stuff have the following: Idefine EMPTY <-2> Idefine LMK (-3) / * left marker * / Idefine RMK ( 4 ) / * right marker */ Idefine L SUB 1 Idefine R SUB 2 Idefine UND 6 / * negation */ Idefine KNW 7 / * knowledge */ Idefine BEL 8 / * belief */ Idefine U KNW 9 / * knowledge, used */ Idefine U BEL 10 / * belief, used */ Idefine USD 11 As the reader may have noticed, also the "codes" for the operators has changed: first, from the classical functors, only conjunction remained. The reason is that formulas will be being rewritten on parsing, so we eliminate negations, disjunctions and implications. As a side effect of this policy, I'll be leaving equivalences out fdlr simplicity (cf. latcr on parsing). And second, for knowledge and belief we also have an u k nh and a u b e l : this shall show the search mechanism that the corresponding wffs were already processed in the branch. u sd has the same function, but for other wfTs. Now, if we take a look at the listing of macros .h (in Appendix C) we'll find that there is much more. I'll discuss some of them macros when opportunity arises; I can't do this here without first explaining how the program is supposed to work. 10.1.2 Globais The global variables which we'll be using are: short WFFISTRLEN)(3), PRF|STR_LEN][3|, ALTISTR LEN), BLF1STRLEN], BCKT1STR LEH) H ) , MODISTRLEN][2], wf_ptr, prfptr, mod_pt r, alt_ptr, bckt_ptr; char theWff(STR_LEN); /* input formula * / long kl, k2; /* for time measurement */ w f f and w f_p t r are know from the last program. But, as one can see, there are some arrays more, in p la c e o f t b l (w h ic h s to re d th e ta b le ) . p r f is w h e re w e a re g o in g to s to re th e c u r r e n t b ra n c h . a l t (w ilh h e lp o f b l f ) h a s to d o w ith th e p o s s ib le a l te m a te w o r ld s w e s ti ll h a v e to c o n s id e r . b c k t h a s to d o w ith b a c k tra k in g : w h e n w e g e t s o m e b ra n c h in g , w e s to re th e o th e r p o s s ib i li ty ( to g e lh e r w i th th e a c lu a l s ta te o f 160 t A lableou-like theorem p ro ve r fo r ZP S affairs) on b c k t and, after getling a contradiciton, reium (o it and try the other continualion. mod slores the modalized formulas of the branch. The olher short inlegers are place-pointers for each of these arrays, and the long integers are used for performance measurements. The code of the main o function is very similar to the one in the previous program, so I won't bother to repeat it here whole. Basically, lhere is some startup information being outputed on the screen, after what one reads a wff (or V lo end the program), and proceeds to par se it. The only change worih mentioning appears in lhe following piece of code: if ( formula< 0, tu, (i ) ti theWff[uj - * I if ( K B ( WFF[i][0] ) ) ( ++mod_ptr; MOD[0)[0] - -WFF(i1(0); MOD(0) 11) 1; lse • \0 ' ) ♦+prf_ptr; PRF(0)10) - -WFF(i)(0); PRF[0)[1] i; PRF[01(2) 0; if ( tableauO) /* «11 branches were closed... */ k2 - (< TickCount() kl ) * 100) / 6; putchar( 'y ' )/ putchar ( 'e' ); putchar ( * s ' ); e /* some open branch tableau() returned zero */ k2 - ( ( TickCount() kl ) putchar ( 'n* ); putchar( 'o* ); 100) / 6; » printf ("\nTime: lld ms\n\n\n", k2); After having succesfully parsed the wff, there is a call lo the proof procedure with tableau o , bul, before this, we have to add lhe formula lo be (dis)proven to lhe branch. As I cxplain laier on, the program keeps modalized and unmodalized formulas of the branch in different places. That explains lhe line if ( kb ( wff [11101' ) )_ in which 1 is lhe address of lhe wff. The macro kb only checks whether lhe w ffs main operator is a modal one. If yes, lhe branch begins in mod, else in prf. Now tableau o is calied, and rans as long as lhere is something to do, only stopping and reluming 1 if ali branches are closed, or 0 if there is an open branch which cannot be fuither processed. If we had a synlax error, of course, nothing applies, so we print an error message and start again. We are now ready lo consider more details of the program. The olher routines are in different files, which are included (wilh linclude <íl fe>) jusl before main o . They are linclude "macros.h" iinclude "prototypes.h" linclude "parser.h" linclude "tableau. h'* Chapter 10 10.2 parser.h Here we find (he parsing functions. I won't discuss it again in detail, just remark that there is two important changes in comparison to the parser on GTT.Z5. The first one concems lhe rewríting of formulas. Why that? Simple. When processing a branch, there are three things that we can do, supposing we take the (reasonable) strategy of processing non-branching formulas (like true conjunctions, false implications...) first: (i) we can gothrough the branch and look fora true conunction. If there is none, go again and look for a false implication. If there is none, etc. (ii) we can go through the branch and look for the first wff which is a true conjunction or a false implication or... (iii) we can rewríte wffs on parsing, eliminating ali boolcans but, say, conjunctions. For instance, avb is equivalent to (-.ai-*>. So instead of creating a new node with v (a, b) , we create a node with -i (-a, -b). It uses the same place, and we don't have to care for disjunctions anymore. By the way, I'll also be using positive number to represent true wffs, and negative numbcrs for false ones. Thus, finding 97 in a branch means we have Ta there, whilst finding i means we have a false conjunction. The side effects of this approach are two. First, we don't check anymore if some node is present. For instance, if we had a formula like a«b -> atb. we will end up having a tree with twice the same node, as following: " f / \ / \ / \ a b * b flg . 2 6 And it should be clear why: take the formula -*& a. On parsing, we will have a conjunction between a positive wff a and a negative one-we have to store them in difTeient places, so there is no need to (actually one cannot) check whether some subformula is already there. And second, as I explained before, equivalences are being left out-since a<->b is equivalent to, say, -><ai-&) t-.(bi-,a> one would have to copy the two subformulas somewhere else, because they occur with different signs.52 10.3 tableau.h Let us now discuss how we implemente lhe tableau procedure. Before we dive into (parts of) the code of lhe tab leau o function, let us lalk a liule while about the way things are supposed to work. In lhe semantics for ZP5, we leamed lhat, since the knowledge branch is S5, that every knowledge fotmula has the same value in ali worlds, lhe same holding (witness monoclustered models) for belief formulas. So we don't have in principie to caie in which world a modalized formula holds, or noL This is the reason why I introduced a new siack, mod, which is like prf (where we store lhe branch), only with modalized formulas, to begin with. But not only do modalized formulas hold in every world. Suppose we B u t o f c o u n c one can use th is co p y m echanism . S ee lhe function cqpy() on th e program in the nex t chap ter, w h ich co u ld be used to «ccom ptish this. 162 A lübUauldte theorem prover fo r ZPS have a mie Kp wff: then p is also true in every world. So, when we process Kp, we won't add p to p r f , but equally to m o d . The only wffs which go to p r f are those whose truth is not "universal", so to speak, or those which lead to branchings. I explain: suppose we have k tavb) true. Then avb is true in every world. But we must be careful not to confuse this with either a is tnie in every world, or b is true in every! What would happen if we would add avb lo the m o d stack and process it lhere. So I decided to put wffs like avb tfnto p r f, where we process it in every existing world, if needed. Suppose that we have the worlds 0 ,1 and 2. We begin adding avb to p r f wilh index 2, and lhen store in a field in ALT lhal lhere is also 2 worlds more to be checked, if needed. If we don't get a contradiction with avb on world 2, we "backlrack" and try again, adding it lo p r f wilh index 1. If somewhen a contradiction is found, the olher altematives don't need lo be considered anymore. Speaking of contradictions, some words about how we find one. First, we have in p r f that ali wffs havean index (in the field p r f |x ] (2|, x being a line). Thus, finding a and -a in p r f such thal the index of a is equal to the index of -a means that is some world an atom is getting both truth and falsity, hence we have a contradiction. But another possibility is when we have an atom in m o d - like processing a true Kp and adding p to m o d . So, if we find a -p in m o d , or, for that maller, in PRF with any index, we also have contradiction. Or not? Would that it were so simple. Remember, we are also putting belief wffs in m o d . To see why this constitutes a problem, suppose one has -a in world 0 (laken lo be lhe inilial one), and Ba true in m o d : a is added to m o d . Now if we say Ihis is a contradiction, we'll end up having Ba->a true. So here is where lhe special array b l f comes into picture: there we stoie for each wff whether it has a "belief antecedent" or not. (Or, to put il more precisely, whelher the reduced formula belongs lo an open world, or not.) Thus, when adding a to m o d because Ba is true we make sure that we put a 1 inlo BLFiiocation of a). The contradiction function also checks for this, so we won't have problems. But let us take a look at parts of lhe code. tableau {) short gt, vai, done; done ■ vai 0; bckt_ptr CNULL; % gt is a loop variable, which we use while we are looking for a special wff (say, looking for a true conjunction); vai denotes lhe current number of valuations (worlds), and done is there lo indicate whelher we are through wilh lhe tableau construclion or not. They are set to 0 on the beginning, and bcktpt r (the counter in b c k t) to c n u l l . Now we enter the Main Tableau Loop: while not done, or as long as there are complex formulas in a branch, we try to reduce them. while ( Jdone ) { if ( CONTRADICTION ) ( if { bcktptr == CNULL ) SUCCEED else restore state (); ) 163 C hapter 10 First thing we do in a branch is to check where there is a contradiction. In this we call two functions: first, iw_contrad(), which examines whether there is an "inter_world" contradiction, that is, it begin looking the atoms in m o d . If not, we call a "normal" contradiction, contrad (l, which proofs only the atoms in p r f . Now, supposing we found a contradiction, we have to look whether there is another branch in store which we need to consider. If yes (bcktptr Is greater than c n u l l ), we restore the State as it was before the branching, and go on. If not, this one was the last (or only) branch, so we are done and ta b le a u o retums 1. Now suppose we didn't find a contradiction. So we have to look for complex formulas which we are going to process. We first l o o k f o r a true conjunction in m o d (this is what this macro does). Suppose we find one (that is, f o u n d holds) at location qt. Now we have to examine which kind of formulas this conjunction's subformulas are. First we examine the left one: if it's modal (k b...) then we add it to m o d . If this is also true of lhe right one, same procedure. Now in this case we have to remove the otd wff from p r f , what can be accomplished in two ways: if gt is lhe bottom line of the branch, we just reduce lhe ptf_ptr by one. Else we set g t 's operator to USD, to indica te there is nothing worth looking in this line.53 But we also have the olher possibilites. If lhe right subformula is not a modal one, we simply replace the conjunction wilh it. Similar if the right one is modalized and lhe left not. Now in case both are nol modals, we have to increase lhe poinler in p r f by one and put the left subformula there, replacing the conjunction wilh lhe right one. LOOK_FOR( UNO I I if < FOUND ) /* a true conjunction was found */ ( if < KB( OP< qt, L_SUB ) ) ) I T_ADD_2_MOD( LOC( gt, L S U B ) ); if ( KB< OP( gt, R SUB ) ) ) ( T_ADD_2_MOD( LOC( gt, R_SUB ) ); UPDATE_PRF; ) else < REPL_WITH [ R S U B ); ) ) else if < KB ( OP< gt, R SUB > ) ) ( T_ADD_2_M0D< LOC( gt, R_SUB ) ) ; REPL_WITH( L_SUB ); > else { ADD_TRUE< gt, R_SUB ) ; REPL_WITH( L_SUB ); } ) W orldng w ith dynam ic n g m o ry a llocatíon , e.g . w ith a linked lisl, you w ould ju sl rem ove Üie wfT: no " b lanks" betw een. 164 A tableau-like theorem p ro ve r f o r ZP S !f we couIdiTt find a true conjunction in pr f (only atoms and false conjuntions are there), the next case is l o o k m f o r ( u n d ), which does the same procedure as before, only looking in m o d . (See listing on Appendix Q Suppose now there is also no true conjunction in m o d : lhen we go back to p r f and try to find a false one. LOOK_FOR { -UND ); if ( FOUND ) /* a false conjunction found */ ( STORE_RIGHT_WFF/ save_state( qt ); if ( K B ( O P { gt, L_SUB )) ) [ F_ADD_2_M°D< LOC( gt, L_SUB ) >; } else { F_REPL_WITH( L_SUB >; ) ) If we succeed with the search, we store the right subformula (which hold in the olher branch) on b c k t . That means increasing b c k t p t r and also storing some informalion lhere: for instance, how many world lhere are, which value have the different poinlers ( p r f p t r , mod pt r, i i t p tr . . .) , which formula gave rise to Ihis branch, which modals were slill unprocessed, and so on. This is what s t o r e r i g h t wf f and save s t a te o accomplish; see lhe listing in Appendix C for the actual code. Now we check which kind of wff the left subformula is: if a modal, add to m o d , else replace the false conjunction in pr f wilh it. Now the next step, if there were no false conjunctions, il's processing knowledge and belief wffs, if any. So we l o o k_4_k b ( -k n w , -b e l )-first trying lhe false ones: I.OOK 4 KB ( -KNW, -BEL ) ; if ( FOUND ) /* a false modality was found * / < if< MOD(gt)(O) - -KNW ) ( MOD(gt)lO) - -U_KNW; if ( K B ( OP_M( gt, L_SUB )) ) { F_ADD_2_M0D( LOC_M( gt, L S U B ) ); } else í ADD_SCOPE( ++val, gt, -NEXT { prf_ptr ) ); if ( ! KB ( OP_M(gt, LSUB) ) ) { BLF(LOC_M(gt,L_SUB)) 1/ ) ) ) else /* - -BEL */ ( MOD [gt) [0] *= -U BEL; if ( K B ( OP_M( gt, L S U B )) ) ( • F_ADD_2_MOD{ LOCHI gt, L_SUB ) ); ) el se * { A D D S C O P E ( ++val, gt, -NEXT( prfptr ) ); ) 165 C hnpter 10 Having found something, we first check whelher is a false knowledge or false belief wff, and act accordingly. Suppose il is a false k n h : we replace the operator with -USD K, to show it was already processed (so we won't try it againt) We then look at the subformulas. If modal, add to modal, with the care of updating b l f if necessary. That is, the scope of this false k n h can be a closed world, so we have to indicate it. Else (left subwff is not modal), create a new world: Ihis is done wilh a d d s c o p e : v a i is increased, and we add lo p r f the false subwff wilh a new index. Then a l t is updated, as I mentioned in the beginning of this section. The case of belief is handled in a similar way. If lhere were no false knowledge or beliefs, lhen look for mie ones. L O O K 4 K B ( KNW, BEL ) ; if ( FOUND ) / * a true modal was found */ { if( MOD(gt)(0) - KNW ) ( MOD [gt) [0) UKNW; if ( OP_M( gt, L_SUB ) !- -UND > { T_ADD_2_MOD< LOC_M( gt, L_SUB J W } else < ADD_SCOPE< vai, gt, NEXT{ prfptr ) )? UPDATEALT; ) ) else /* - BEL */ ( MOD[gt](0) UBEL; if ( OP_M( gt, L_SUB > !- -UND ) I T_ADD_2__MOD ( LOC_M{ gt, L_SUB ) ); if < !K B ( OP_M( gt, L_SUB )) ) { BLF[LOC_M(gt,L_SUB)1 1; ) I else < ADD_SCOPE( vai, gt, NEXT ( prf ptr ) ); u p d a t e_a l t ; 1 } ) First we decide whether we're dealing with k n h or b e l , and set the operator lo "used". And we go to a similar song-and-dance as bcfoie, checking for what lhe subfoimula is to see where to put it and remeberíng to update a l t and b l f if need arises. Now suppose lhat after this long search we found just nothing: no contradictions, and no more formulas to be reduced in this branch. We are on our last hope: else if { OTHER_WORLDS ) < GONEXTWORLD; ) else done « 1; 166 A tableau-like theorem p ro ve r fo r 7P S o t h e r w o r l d s looks whclher a l t has anolher índices stored: if yes. go to lhe next altemalive. This results, as 1 mentioned above, of having added a false conjunction which is "universally true". We have to check each world lo see whelher we get a contradiction there. The loop continues. If there are no more worlds to be checked, (a certain field in a l t was set to 0) we are done. And that's iL 10.4 A working sesskm with TTP2P5 Now some examples from what happens when the program runs: let us type some formulas and see what happens: TABLEAU THEOREM PROVER FOR ZP5 Cesar A. Mortari VI.0, July 1990 Syntax: a..t (varlables), K, B, t, v, -> To exit type ';'<CR> Please type in a formula: Ka->a yes Time: 0 ms Ba->a no Time: 16 ms a->Ka no Time: 16 ms K(a->b)->(Ka->Kb) yes Time: 16 ms B(a->b)->(Ba->Bb) yes Time: 16 ms -iBa - > B->Ba yes Time: 0 ms -iK-iKa - > K-»K-»a yes Time: 16 ms Ka->KKa yes Time: 16 ms Ba->KBa yes Time: 0 ms 167 C hapler 10 Ba->BKa no Times 0 ms -»Ka->K-*Ka yes Time: 0 ma -tBa->K-iBa yes Time: 0 ms K (avb)->KavKb no Time: 16 ms KavKb->K(avb) yes Time: 16 ms 168 n n Implementation of the algorithm C, n.. A programming language that is sort oflike Pascal except more like assembly except that it isnt very much like either one, or anything else. It is either the best language available to the art today, or it isn't. RAY Sim a r d In this chapter Í'H present a C program which uses lhe lheorem prover for ZP5 from lhe last Chapler in order to implement the algorithm (see chapler 2) which decides whether some formula belongs to a minimal belief State or no t The algorithm itself is not very hard to implement, once we have a theorem prover for the logic in question. The program, which is called ALG. ZP 5, is an extension of TTP.ZP5. Th^ big change is lhat now we first type a wff, which describes everything that Angela believes, and then we are asked for anoiher one. The program then checks whelher lhe second formula belongs to Angela's belief state, given that she believes only the first one. The program calls the tableau procedure and retums an "yes" or "n o " . The main loop does the following things: • reads a formula (string) from lhe standard input; • parses lhe formula; • reads a second formula (string) from lhe standard input, and parses il; • runs lhe algorithm we discussed on chapler 2; and • prints lhe answer on the screen. 11.1 Data S tructu res and m a in () The macros I added to TTP.ZP5 are only a few of minor importancc. Wilh regard to global variables, the only changes are: short WFFfSTR LEN)(4), 169 Chapter 11 As one can see, w f f now has another line, where we store the modal degree of the formulas. And a global e was added, because we'll have now and then to add to w f f some new formulas. Remember, the algorithm states that a formula A belongs to Angela's belief State, when everything she believes is a , if believing A is a consequence of believing a and also o f lhe modalized subformulas o f A which have already been decided. So it's likely that we'll end up having to store these subformulas somewhere: suppose we get some subformula Kp of A which it doesn't belong to Angela's belief State. We have to build a conjunction of B a and -.Kp, before chccking whether this implies BA, and this means copying this Kp (now with a minus sign) somewhere else. Now the ma ln () function has a similar code as in the previous programs, but our goals here are other, so let us look at it. Basically there is again some startup informalion being outputed on the screen, after what one reads a wff (or V to end the program), and proceeds to parse it. Then a second wff is read in, and it goes on like: if ( formulai 0, tu, ti ) theWff(u) - '\0' ) ( printf("\nNow type the next wff:\nM); gets( theWff ); kl TickCount O ; lf ( formulai 0, tu, tj ) tt theWff[ul **- '\0' ) { a_pt r 1/ e wf_ptr; WFF(e)[0] BEL; WFF[e][1] ■ a_ptr/ p_ptr j/ if ( WFF(p_ptr1[3) > 0 ) for \ s 1; 3 <WFFlp_ptrJ13); + + s ) loop( s ); prf_ptr bckt_:ptr modjptr alt_ptr CNULL; e « p__ptr+l/ add_fi_psi( p_ptr ); prepare_tab( p_ptr, 0 ); for ( i 0; i < STR LEN; ++i ) BLFÍi) 0; if < tableau ()) ... After having succesfully parsed the first wff, we read lhe second one. If it is also OK, one adds B a to the w f f stack, and look which degree lhe second wff (pointed at by p_ptr) has. Starting with 1 and going until degree, we apply the algorilhm on subwffs of the wff in p ptr, as the algorilhm prescribes.This is what loop o does (we'll discuss it afterwards). Then we add a 1 or 0 to of w f f [ subwff] [2 ], for each examined subformula, in order to inform whether it belongs or not to Angela's belief slale. After having considered ali subwffs, we ran the algorithm on lhe main wff: we add lhe conjunctions of modalized subformulas (with a d d f i psi o ); the initial informalion for lhe tableau proof procedure is set (that is, B a , the wff we are examining, and so on, are added to the stacks p r f or m o d , depending on their main operator). Finally, tableau <) is called, and the answer is printed. We are now ready to consider more delails of lhe program. The other routines are in several files, which are included (with linclude <fJJe>) just before main o .They are 170 Im p lem en ta tion o f th e aigorithm linclude "macros.h" li nclude "prototypes.h" linclude "parser.h" linclude "tableau.h" linclude "subf.h" Everything is (almosi) the same; lhe new functions which implement the aigorithm were added . together in subf. h. A small change was made in parser. h: now the modal degree of a wff is automatically computed on parsing. (See listing on Apendix D.) The file tableau. h is the same. So let us look at what is new. 11.2 subf. h We have three functions in this file. The first one is íoop: void loop< s ) / * short s; { short p, j; p ■ a_ptr+l; while ( p <p_ptr ) ( prf_ptr " bckt_ptr mod_ptr ■ alt_ptr " CNULL; if ( WFFlp1[3] - s (& K B ( WFF[p|[0] ) ) I add_fi_psi( WFFlpHll \ t prepare_tab( p, 1 ); WFF(p)(2] tableau(); ) + + p ; ) ) Loop runs with an index s which denotes the degree which we are interested in. It makes a loop wilh index p from lhe place where our a is slored until p_pt r, which points to lhe second wff (lhe one we are trying to decided wheter ii's in Angela's belief stale). If lhe wff at p is a modalized one and its degree is equals lo s, we add to w f f its already decided modal subformulas (for degree 1 this is just nolhing), prepare lhe stacks and call tableau (). Then set h f f IpI m to whichever value (1 or 0) tableau () relums. And goxrn until ali wffs of degree s have been processed. Back lo lhe main loop, loop is calied again with s u , and so on. Let us now look at add f i psi o , which is responsible for adding lo h f f modal subformulas (of a certain wff) which have already been decided. vold add_fi_psi( p ) /* short p; ( short 1, r; if ( ATOMIC( WFF(p1(0) )) return; if < K B ( WFF(p|(0) )) { r = e; 171 C hapter I I 1 • copy I W F F l p H U ) .* + +e; if < WFF[p} ( 2 ) -1 ) WFF[e][0| PST ( WFF(pMO) )/ else WFF(eJ(01 NGT( WFFCPI(01 ){ WFF(e)[1) 1; ♦+e; WFF(e)(0) UND; WFF(e](1) r; WFFÍeJ ( 2 ) e-1; add_fi_psl( WFF(plll) ); 1 else /* conjunction */ ( add_fi_psi( WFFIpJll) ); add_fl_psl( WFFlpl121 ); ) 1 lf the wff p points at is atomic, it has obviously no modal subwffs, so we exit. Else, if we are considering some knowledge of belief formula, make a copy of it in the w f f array (we have to work with copies, of course). And call add f i p s i o on the subformulas. If the formula being considered is a conjunction, call a d d f i_ p s i o on its subformulas. The other two function sin su b f . h are, first, copy {), which just does what its name says. (Why this is needed was already discussed. See its coding on Appendix C.) And the other is p r e p a r e t a b o , which just add wffs to the correct arrays (pr f or m o d) beforc calling tab leau . And that's it. 11J A working sesslon with ALG.ZP5 Now some examples from what happens when the program mns: let us type some formulas and see what happens: BELIEF STATE ALGORITHM FOR ZP5 Cesar A. Mortari VI.0, August 1990 Syntax: a..t (variables), K, B, C, v, ->, <-> To exit type 1 ;'<CR> Please type in your 'alpha': Now type the next wff: Ba yes Time: 16 ms Please type in your 'alpha': a&b 172 Im p le m e n ta iio n o f th e a lg o rith m C hapter 11 Now type the next wff: Bb no Times 16 ms Please type in your- 'alpha': avb Now type the next wff: -.Ba yes Time: 16 ms i Please type in your 'alpha': avb Now type the next wff: ->Bb yes Time: 16 ms Please type in your 'alpha': avb Now type the next wff: B(avb) yes Time: 16 ms 174 Final remarks A conclusion is simply lhe place where someone gol tired ofthinking. A few words to conclude this work. We set out with lhe goal of characlerizing belief states in cases where agenls have only partial informalion about some domain. In lhe course of the investigation, we saw lhat there were different altematives which enabled us to reach this goal. Not every one of lhem worked, or worked equally well, for every logic, bul a nice feature was lhat we obtained an algorilhm with which lo characterize lhe belief states. This lead us to the olher two paris of Ihis work, which dealt, first, with decision procedures via valuation semantics for modal and epistemic logics, and second, with implementations of these procedures. What could we now say more about our original goal? As lhe old saying goes, every solution immediately raises more questions than we had former. So il is not surprising lhat Ihis should be the case here too. So let me mention some open problems, or rather, some directions to further investigalions. First, it is still open whether one can find a reasonable characlerization melhod for lhe logics Z, ZG and ZP via trying to find lhe smallest slable sei. Chances are small, because of lhe infinile number of modalities, but it would be nice to have a definite answer. Second, one should also investigale whether formulas like B(-.IG4->fl) really behave like default ruies, which was one of lhe motivalions to use logics of knowledge and belief. I left this question untouched, because a thorough research on it would constitute by itself anolher dissertation. Supposing thus we have an affirmative answer, one could lhen investigate, with respect to the different EDL-systems, the resulling default logics. Third, until now, as we have seen, I have kept to the case in which we consider only one agent, but the most inleresting situations would be of course the ones involving more, interacting agenls-as, for instance, in a distributed system. A strong suspicion, not lo say certainty, is lhat things will be a lot more complicaled, witness Halpem and Moses' remarks with respect lo a multi-agent, SS-based knowledge logic. And fourth, what happens if agents are not ideal, eilher not logically omniscienl, or not fully inlrospective? Working on this problem presupposcs first of ali lhe existence of EDLogics wilh respect to which agents have lhese desired characleristics, and, as I had opportunity to mention, we are far from having, for instance, reasonable non-omniscient logical systems. So an important direction for further research is the development of "more realislic" logics of knowledge and belief-a lopic which particularly inlerests me, and which I prelqnd to consider in future works. Appendices and References A l. ZS and Z5* are the same logic: (Z5 C Z5») 1. KKA -> BKA 2. KA -» KKA 3. KA -> BKA (Z5* C Z5) 1. KA -> A 2. BKA -* BA 3. KA -> BKA 4 . KA -» BA A2. p and p* are equivalent: (p =>p*) 1. BBA -> -iB-iBA 2. B-BA -> -.BBA 3. KB-.BA -> K-.BBA 4 . B-.BA - > KB-iBA 5. -.BA -» B-.BA 6. -BA -> KB-.BA 7. BA -> BBA 8. BA -» -.B -üA 9. B-iBA -> -.BA 10. KB-.BA -> K-.BA 11. -.BA -> K-.BA Some derivations One should not clulter one's mind wilh trivialilies. G. HARMAN, Change in View m 4 1.2 TR 1, RB, m* 2.3 TR db 1.Transp 2. RK, k P 5b 4.5 TR 4b 1.7 TR 8, Transp, DN 9. RK. k 6,10 TR 179 A p p en d ix A <p* =>p) 1. BA -» BBA 4* 2. BBA -> -iB-iBA 3. BA -> -.B-J3A 1 J TR 4. -,B-«BA -» K-iB-iBA p* 5. BA -» K-,B-nBA 3,4 TR 6 . B A - > B iB i4 5 » 7. -iB-iBM -» Bi4 6, Transp 8. K-iB-tBA -» KBA 7, RK, k 9. BA -> KBA 5.S TR A3. K ZCP BA <-> ->K-iKA 1. KA -> BA m 2. -*BA -> ->KA I , Transp 3. K(-.BA -> --.KA) -> (K-iBA -> K-.KA) * 4. K(-iBA -> -iKA) 2, RK 5. K-.BA -> K-.KA 3,4 MP 6. -.K- .KA > .K .BA 5, Transp 7 . -.BA -* K-.BA p* 8. -.K-.BA -> BA 7, Transp, DN 9. -.K-.KA -> BA 6,8 77? 10. K-iKA -» B-.KA m 11. -.B-JKA -> -.K-iKA 10, Transp 12. BKA-> -<B-.KA db 13. BKA -> -.K-.KA 11,12 TR 14. BA - > BKA c 15. BA -> -.K-.KA 13,14 TR 16. BA <-> -.K-.KA 9,75, D /o A4. Hz c p B(BA -> KA) 1. B(KA -> (BA .-> KA)) -> (BKA -> B(BA -> KA)) *<2. B(KA -> (BA -> KA)) Taul, RB 3. BKA -> B(BA KA) 12 MP 4. BA -> BKA c 5. BA -> B(BA -♦ KA) 3,4 TR 6. B(-.BA -> (BA -> KA)) -> (B-.BA -+ B(BA -> KA)) k>> 7.' B(-iBA -» (BA -> KA)) Taul, RB 8. B-.BA -> B(BA -> KA) 6,7 MP 9. -iBA -> B-iBA 5b 10. -.BA -» B(BA -* KA) 8,9 TR 11. BA v -,BA -> B(BA -♦ KA) 5,10 Taul 12. B(BA -> KA) 11, Taul, MP 180 3 ) d ) GTT.Z5 Listings Real programmers dont comment their code. h was hard to write, it should be hard to understand. BI. F i l e : DAcros.h / * --------------------------------M A C R O S -------------------------------------------- * / Ideflne STR LEN 256 / * length of strlngs */ 1define MAX LINES 64 f * max nr. of lines ln a table *{ •define MAX_R0MS 30 / * max nr. of rows ln a table */ 1define CNULL (-1) /» a "polnter" to NIL */ Ideflne UND 5 /* conjunction */ Ideflne ODR 6 /* dlsjunction */ Ideflne IMP 7 / * lmpllcatlon */ Ideflne NEG 8 / * negatlon */ Ideflne EQÜ 9 /* blcondltlonal */ Ideflne KNH 10 / * knowledge */ Ideflne BEL 11 / • belief */ Ideflne P_AT0MIC< c > (c >» 'a' &fi c <- '1;') /* atoms on lnput */ Ideflne ATCWICÍ c ) (c > BEL) / * lnternally, atoms are averythlng but connectlves */ Idefine INIT NODE(a,b,c) WFF{wf_ptrl(0) a; WFF{wf_ptr][1J b; WFF(wf_ptr)(2) c / * -----------------------------G LO BA LS -------------------------------------- */ short HFFISTRLEN](3J, /* stores the tree representation of the wff */ wf j>tr, /* polnter to current location ln WFF * / TBL{MAX_ROWS) [MAX LINES); /* the GTT// char theWffISTR_LEN); /* input formula */ long kl,k2,k3,k4; /* for time measurement */ B2. F i l e : prototypes.h / « ------------------------------f u n c t i o n p r o t o t y p e s ------------------------------------» / /* - parser.h - * / lnt formulai short, short *, short * ); lnt formandorl short, short * , short * ); lnt rest_fd( short, short, short *, short * I; 181 A p p en d ix B t int rest_form( short, short, short *, short * ); lnt forml( short, short *, short * ); short get_place{ short ); int unlfy { short, short ) ; /* - table.h - */ short make_table( vold ); vold copy_llnes( short, short ); vold , prlnt_table( short, short ); vold dlsplayWFF( vold ); vold ccpy_lines( short, short ); lnt aiphaNE ( short, short, short, short, short, short ); int ganvneNE( short, short, short, short, short ); lnt accesK( short, short, short, short ); int accesB( short, short, short, short ); lnt deltasNE( short, short, short, short, short ); lnt betasNEl short, short, short, short, short ) ; int gamnaP( short, short, short, short, short, short ); lnt alphaPt short, short, -short, short, short, short vold spllt_lines( short, short, short ); B3. F i l e : p a r s « r . h / * ----------------------PARSER FUNCTIONS----------------------*/ /* The parser - formula() - processes the input string and store it lf a wff) in the WFF array, In a treellke representation. This parser ls based ln a Prolog one developed by Franz Guenthner. The overall strueture ls the sane, but of course we adapted lt to our data structures here. */ formula( xi, xo, * ) /*---------------V short xi, *xo, *z; » ( short xn, zn; if ( form_and_or{ xl, *xn, czn ) ) { lf < rest_form{ xn, zn, xo, z ) ) return 1; else return 0; ) else return Q; I form_and_or( xl, xo, zo ) /*-------------------- */ short xl, *xo, *zo; í short zl, xn; lf ( form_l( xl, 4xn, tzi ) ) < if ( rest_fd( xn, zl, xo, zo ) ) return 1; ) else return 0; rest_fd( xl, ti, xo, to ) f *-------------------- */ short xl, zi, *xo, *zo; í short z, 1, old_ptr; old_ptr * wf_ptr; lf < theWff(xl) - (t form_and_or( xl+1, xo, «z )) GTT.ZS Listings < INIT_NOOE( UND, ri, r ); i get_place( wf_ptr-l ); if ( i - CNULL ) •to wf_ptr++; else- ( *zo i; wf_ptr ■ old^ptr; ) return 1; else if ( theWff(xl) - 'v' t t form_and_or( xi+1, xo, tz )) 1 1NITJN0DE{ ODR, zi, z); i get_place( wf_ptr-l ); if ( 1 ~ CNULL ) *ro " wf_ptr++; else { *20 " i; wf j»tr " old_ptr; ) return 1; else l *xo xi; *20 - *i; retum 1; ) rest_form{ xi, zi, xo, zo > /*----------------------•/ short xi, zi, *xo, *zo; 1 short z, oldjptr, 1; old_ptr wf_ptr; if (( theWff(xi) - ) tt ( theWff(xl+l| - •>• ) tt formula( xi*2, xo, tz )) í INITNODE( IMP, zi, z ); i get_place( wf_ptr-l ); if ( 1 - CNULL ) *zo •* wf_ptr++; else ( *zo 1; wf_ptr old_ptr; ) return 1; ) e l s e if (< theWff[xi] - '<• ) tt ( theWff(xl+1) - > tt ( theWfflxl+2) =« •>' ) tt formula( xl + 3, xo, tz )) INIT_NODE< EQU, zi, Z ) ; 1 = get_place( wf_ptr~l ); if ( i - CNULL ) *zo « wf_ptr++; else ( *zo 1; wfjptr *» old_ptr; ) return 1; ) else 1 *xo xl; *zo • zi; return 1; 183 A ppend ix B I form_M xl, xo, to ) / *---------------- */ short xl, *xo, *10; ( short f, 1, old_ptr; old_ptr wf_ptr; lf ( (unslgned chac) theHfffxll -» (unslgned chac) •-»• > { lf ( form_l( xl+1, xo, *f )) ( INIT_NODE( NEG, f, CNULL); 1 • get_place( wf_ptr-l ); lf ( 1 - CNULL ) •zo wf_ptr++; else ( *zo 1; wf_ptr old_ptr; ) return 1; ) eis® return 0; ) lf { theWff|xl] - 'K*) I lf (form_l( xl+1, xo, tf )) [ INIT_N0DE< KNW, f, CNULL)? 1 get_place( wf_ptr-l ); lf < 1 - CNULL ) *zo " wf_ptr++; else [ *io 1; wf_ptr ■ old_ptr; ) return (1); ) else return(0); ) lf ( theWff[xl] - 'B') { lf (form_l( xl+1, xo, *f )) { INIT_N0DE( BEL, f, CNULL); i * get_place( wf_ptr-l )? lf ( 1 -CNULL ) *zo » wf_ptr++; else f *to • 1; wf_ptr " old_ptr; ) return(1); t else return(0); ) lf ( theWff[xl] - M' > í Lf t formulai xl+1, xo, Cf )) < lf ( theWff(*xo) - ')« ) [ + + ( * x o ); *zo * f; return 1; ] else r e tu rn 0; 184 G TT.2 5 L is tin g s if ( P_ATCMIC( theWff[xi]) ) { i wf_ptr-l; while ( 1 !CNULL t( WFF(lj[0) !theWff[xi] ) - i; if ( i -CNULL ) [ NFF(wf_ptr][0] theWff(xi); *zo " wf_ptr++; ) else *zo « 1; *xo xi+1; return 1; ) else return 0; ) short get_place( ptr ) / *-------------------- */ short ptr; ( while < ptr 1CNULL ti WFF(ptr)[OJ 1WFFÍwf_ptr)[0J ) - ptr; if { ptr - CNULL ) return ptr; if < unify( WFF[ptr)(l), WFF[wf_ptr][1] )} < if ( WFF[WFF|ptr|[111(0] " NEG ) return 1; if < unify< WFFjptr)[2J, WFF[wf j>tr)|2) )) return ptr; else return< get_place( ptr-1 )); ) else return( get_place{ ptr-1 >); ) unifyl x, y ) /*---------- */ short x, y; { if ( WFF[x][0] !WFF[y)(0) ) return 0; else < if ( AT0MIC( WFF(x)[0] )) return 1; else I if ( lunify( WFF[x) U K WFF(y){l) ) ) return 0; else ( if ( unify( WFF(x)[2 ], WFF[y][2) ) ) return 1; else return 0; ) ) ) ) ) ) B4. F i l e : outputs.h /* ---Functions for prlntlng results----- */ vold displayWFFO /* - shows the contents of WFF - ---------------- ./ 185 A p p en á ix B short J; printf("\n"); for ( j 0; j < wf_ptr; ++J > { swltch( WFF|[0| ) ( case NEG : putchar( ' ;putchar(»-*);putchar l')*); break; case KNN : putchar(* (') .'putcharCK*) ;putchar ('J'); break; case BEL : putchar('(');putchar('B');putchar{')'); break; case UND: putchar(' l') íputcharC**) ;putchar{*)'); break; case ODR: putchar(' C^putcharCv^puteharf) break; case I»: putchar<'|•);putchar(*>');putchar(1)'); break; case EQU: putchar(' l•);putchar{'-');putchar(•]•); break; default: putcharí'l');putchar{ WFFfjMOJ >;putchar('J'); ) í printf C\n"); for ( 3 * 0 ; j < wf_ptr; ++j ) < swltch( OTF(jl(0] ) ( case NEG: case KNM: case' BEL: case UtO: case ODR: case If*>: case EQU: printf (11); break; default: putchar(•(•);putchar{' ') ;putchat(*]'>; 5 I printf("\n*>; for ( } ■ 0; j < wf_ptc; +*j ) swltch( WFFJj][0| > { case UND: case ODR: case IM*: case EQU: printf(-l%d)-,»FF!j)12)); break; defffult: putchar('[');putchar(' ');putchar(')'); ) ) void prlnt_table( rows, lines ) /*------------------------- ./ short rows, lines; I short 1, j; printf<"\n\n*** TABLE ***\n\n->; for ( j * 0; j < rows; ++J 5 186 G T T .Z S L is tin g s p r i n t f j ) ; p r in t f (■ V n * ) ; f o r ( j 0; J < rows; ++j ) ( p u t c h a r ( ; p u t c h a r ; p u t c h a r ( • - ' ) ; I p r l n t f ( " X o " ) ; f o r { i 0; i < l i n e s ; ++i ) I f o r ( j 0; J < rows; ++j ) p r l n t f C l % d | - , TB L[} ) |1 )) ; p r l n t f ( " \ n " ) ; J p r i n t f B 5 . F i l e : t a b l t . h short make_table() /*--------------- */ ( lnt 1, /* run variable */ row, /* current row in the table */ line, / * current line in the table */ ad_lines; /* in the modal operator cases, nr. of lines which were added */ TBL(0](0) « 1; /* Initialize first row, two lines (prop var) */ T8L[0)[1] 0; line 1; ad_lines ■ 0; / * MAIN TABIZ LOOP */ for { row 1; row <wf_ptr-l; ++row ) lf ( row >MAXROWS || line >MAXLINES ) I printf("\n\n***** ERROR : TABLE TOO LARGEI return 0; ) switch( WFF[row](0) ) /* swltch the connectives */ { case NEG : for ( i * 0; 1 <line; ++1 ) IBL(rowJH) 1TBL{WFF{row) (1) ) (1) ; break; case UND : for ( 1 ■ 0; 1 <line; ++1 ) TBL(cow)[i] TBLlWFFlrow)(1})[1] fifc TBL[WFF|rowJ(2)] (i); break; case ODR : for ( 1 0; 1 <« line; ++i ) TBL(row)(1J TBL(WFF(row][1]](i) II TBL[WFF(rowj[2]](i]; break; case 1MP : for ( 1 * 0; 1 <« line; ++1 ) TBL[rowJ[1) - !TBL(WFF(row](1]](i) II TBL(WFF[row](2]](1]; break; case EQU : for ( i * 0; 1 <« line; ++1 ) TBL(row](ij - (TBL(WFF(row](1]J[1] " TBL[WFF[row][2]|[1]); break; case KNW : ( for ( 1 * 0; 1 <line; ++i ) lf ( TBL[WFF[row]{1]](1] -= 0 > TBL(row][i] 0; else I lf ( ganmaNE( row, line, WFF(row][l], 0, i ) ) ( lf ( deitasNE( row, line, WFF(row](l], 0, 1 ) ) A p p en d ix 8 « spllt_lines( rcw, llne+l+ad_lines, i ) ++ad_lines; ) else * TBL(row|(l) 0; 1 else TBL(row)(lj 1; ) line line + ad__lines; ad_lines 0; break; ) case BEL : í for ( 1 0 ; 1 <line; ++1 ) lf < (alphaNM row, Une, WF(rowMÜ, 0, 1, 1 ) ) TBL(row)(1J 0; else ( lf ( alphaNEÍ row, line, WFF(row)(l), 0, 1, 0 )) ( lf ( betasNE( row, line, WFF(row)(l), 0, 1 ) ) { split_llnes( row, line+l+ad_llnes, i ); ♦+ad lines; i else TBL(row)(l) 0; ) else TBL[row)(1) 1* I line line + ad_llnes; ad_lines 0; break; ) default : /* Atomlc proposltions */ l copy_llnes( row, line ♦ 1 ); for ( i 0; 1 <* line; ++i ) TBL[row)(ij 1; line 2 * (line + 1) 1; while ( 1 <line ) TBL(row)(1++J 0; ) > ) return ((short) line+1); ) void copy_lines( row, lines ) / *--------------------------- ./ short row, lines; ( short i, j; for ( 1 0; 1 < lines; ++1 ) for ( J 0; j < row; ++;) ) TBL(j)(l + lines) TBLUUi); > alphaNEÍ r, 1 , am, b, currLine, value ) /*--------------------------------*/ short r, 1, am, b, currLine, value; ( while ( b <1 t t TBL[am) (b) !value ) ++b; if ( b > 1 ) return 0; /* else we're at a line with am=value */ if ( accesB( r, 1, currLine, b )) return 1; else return( alphaNE( r, 1, am, b+1, currLine, value )); 188 G T T .Z S L is tin g s J gamnaNE( r, 1, am, b, currLine } /*---------------------------*/ short r, 1, am, b, currLine; í whlle ( b <■ 1 i( TBL(am)(b) !0 ) /* b is where to begln the search */ ++b; if | b > 1 ) return 0; /* eise we're at a line with am=0 V lf ( accesKf r, 1, currLine, b )) /* b satlsf scope of cur_line */ return 1; else return( gann«NE( r, 1, am, b+lr currLine )); ) accesK( r, 1, v, vn ) / * - that ls, v<k,r>vn - -----------------------------*/ short r, 1, v, vn; i * short 1; i 0; whlle | 1 < r ii ( HFF[1)(0] 1KNW II TBL|l)[v) - TBL(it(vn] ) ti i WFFHKO) IBEL || TBLlWFF(iMUMvn) - 1 ) ) ++i; lf < 1 >r ) return 1; else return 0; accesB( r, 1, v, vn ) /* - that is, v<b,r>vn - •--------------------------- «/ short r , 1, v, vn; { short 1; 1 0 ; whlle ( 1 < r ii ( WFF[i|(0) !BEL || ( TBL(l)(v] - TBL|i)|vn) ii { TBL[1]fv] !1 !! TBL[WFF(1]{1))|vn| - 1 ) )) ii ( WFF [1J (0] !« KNW || < TBL(i] [ vj - TBL|iJ|vn] )) ) ++1; lf < i >« r ) return 1; else return 0; ) deltasNE( r, 1, am, b, v ) /*------------------- */ short r, 1, am, b, v; i short i; 1 0 ; whlle ( 1 < r ü ( W F F [1 )[0 ) !KNW || TBL{i)|v) » 1 || gamnaP( r, 1, WFF(1)|1|, 0, v, am ) )} +♦1; lf ( 1 >r ) return 1; else return 0; betasNE( r, 1, am, b, v ) /*-------------------- ./ short r, 1, am, b, v; short i; i 0; while ( 1 < r ii ( W F F [ 1 | ( 0 ) !* BEL I I l B L | l ) ( v ) - I I I alphaP( r , 1 , W FF| 1 ) 1 1 ) , 0 , v , am ) H ++i; 189 A p p en d ix B if « i >r > return 1; else r e tu r n 0 ; garrmaP ( r, 1, aq, b, v, am ) /*-----------------------*/ short r, 1, aq, b, v, am; I while I b <i ti TBL[aq)(bj I0 ) ++b; if ( b > 1 ) return 0; /* else we're at a line with aq-0 */ if ( TBL[am)(b] - 1 c c accesK( r, 1, v, b )) return 1; else return( gamnaP( r, 1, aq, b+1, v, am )); ) alphaPI r, 1, aq, b, v, am ) /*-----------------------./ short r, 1, aq, b, v, am; i while ( b <1 cc TBL(aqüb) !0 ) ++b; if ( b > 1 ) return 0; /* else we're at a line with aq*0 */ if ( TBL(am|(bl *■ 1 (( accesB( r» 1, v, b )) return 1; else return( alphaP( r, 1, aq, b+1, v, am )); ) void split_lines( r, 1, v ) /*----------------------*/ short r, 1, v; i short j; for 1 j 0; j < r; ++j ) TBLíjJHl TBLÍJJM; TBL(rJtv] 1; TBL(r](1) 0; B 6 . F i l e : m a i n . c GTT.25 -Version 1.0 Cesar Mortari - May 1990 TTiis program lrplements the construction of a generalized truth-table for the epistemic-doxastic logic Z5. GTT INCLUDES • llnclude "rracros.h" llnclude •prototypes.h" llnclude "parser.h" llnclude "outputs.h" llnclude "table.h* mainO í short i, j, k, u; 190 G T T .Z 5 L ittin g s printf("\n* GTT BU1LDER FOR Z5 *"); printf ("W Cesar A. Mortarl *"); prlntf("\n* VI.0, May 1990 *"); m printf("\n\nSyntax:\n ) * a..t (variables), K, B, i , v, - > , <->\n" prlntf(*\nTo exlt type ';'<CR>\n"); printf("\nPlease type in a formula:\n"); for {;?) ( /* Soma lnitiallzatlons... */ wf_ptr 0? WFF(O)IO) CNULL; /* to take care of first Input atcxn...*/ gets( theWff ); /* input the formula to be (dis)proven */ lf ( theWff{01 -- *;') /* program ends... */ return 1 ; /* ELSE remove blanks from theWff */ t 0; j 0; while ( theWff(11 !- '\0• ) ( if ( theWff(i) !- ' ' ) theWffIJ++) theWff(i); ++1; ) theWff|j) - '\0'; /* time use in parsing */ /* make_table time */ / * prl nt the contents of WFF kl TickCount(); /* initialize time counting lf < foroula( 0 , t u , t i ) t t theWff( u ] - • \ 0 ' ) { k2 TickCount<); k make_table(); k3 TickCount{); displayWFFO; prlnt_table( wf_ptr, k ); k4 - ({ TickCountO k3 ) k3 - ((k3-k2)* 100)/6; k2 - ((k2-kl)M00)/6; prlntf("Parsing time: %ld ms", k2); prlntf("\tMake table: %ldms",k3); printf("VnOutput time: %ldms*,k4); prlntf {"\tTotal times %ld m s ^ ^ " , k2+k3+k4) 100) / 6; / * printing time */ else /* it was not a wff, so try again... */ prlntf(*\nSyntax error...\n\n"); 191 0 TTP.ZP5 Listings Lubarskfs Law qfCybernetic Enthomology: There is always one more bug. C l. F i l e : m a c r o s . h / * ------------MACROS------------------- */ •define STR_L£N 256 /* length of strings Ideflne CNULL (-1) / * a "polnter" to NIL Ideflne E**>TY <-2> •define LMK (-3) /* left marker */ Ideflne RMK (-4) /* right marker */ •define L SUB 1 •define R SUB 2 •define EQU 5 /* blconditlonal */ Ideflne UND 6 /* negation */ •define KNW 7 /* knowledge * / •define BEL e /* belief */ •define U KNW 9 /* knowledge, used */ •define U BEL 10 /* belief, used * / •define USD 11 •define P_ATOMIC( c ) Ideflne ATCMIC( c > •define NEXT{ 1 ) Ideflne NEXT_M( 1 ) Ideflne OP< 1, x ) Ideflne OP_M( 1, x ) Ideflne LOC( 1, x ) Ideflne LOC_M< 1, x ) •define KB( x ) •define INIT_NODE(a,b,c) •define F_MOD< x ) •define T MOD! x ) ( c >« *a' tf c <- 't' ) ( C > USD I1 c < -USD ) WFF|PRF[1] [ 1] ] [ 0 ] w FFiMoommnoi WFF[WFF[PRF[1][11)1*11(01 WFF(WFF(MOD[il (UHxinOl WFF[PRF(1)(1])(x) WFFlMODflimjlxl ( x . . KNW | | x WFF[wf_ptr](0] WFF [wf_ptrl[11 WFF (wf_ptr|[21 - < MODÍxJIOl { MOD[x)(0) -KNW |I MOD(xl[OJ KNW |I MOD[x|(01 - * -BEL ) BEL » •define SUCCEED Ideflne LOOK_FOR( x ) { done ■ 1; return 1; ) gt prf_ptr; \ 192 T T P 2 P 5 L is tin g s Idefine •define tdeflne tdeflne Idefine Idefine Idefine Idefine Idefine Idefine Idefine •define •define Idefine m Idefine Idefine LOOK M FOR( X ) while ( gt 1CNULL tt PRF(gt)(Q) !x ) \ - gt gt modjptr; \ while ( gt ICNULL tt MOD[gt](0] L O O K 4 K B ( x , y ) gt " mod_ptr; \ while ( gt ICNULL «t MCO|gt][0) !x tt MOD(gtJlO] !y ) FOUW gt !ADD_TRUE( i, X ) R£PL_WITH( x > F_REPL_WITH( X ) ADD_MQDAL( 1, X ) ADD_SCOPE( ã , 1, O ) T A D D 2 M D D ( X ) F_ADD_2_M0D( X ) TH_REPL_WITH( 1, x ) STORE RIGHT WFF ++pcf_ptr; \ PRF(prf_ptr](2) -PRF(1](2); \ PRF(prf_ptr)(l) LOC( 1, x ); \ PRF(prf_ptr){0) NEXT{ prf_ptr ) PRF|gC)(l) LOC( gt, x ); \ PRFlgtj[0] NEXT( gt ) PRF(gt)(l) LOC( gt, x ); \ PRFlgt)(0) - -NEXT( gt ) ♦+prf_ptr; \ PRF (prf J)tr| (2] 0; \ PRF(prf_ptr|(1) LOC_M< i, x ); \ PRF(prf_ptr)[0| NEXT{ prf_ptr ) ++prf_ptr; \ PRF(prfjptr)(2) * a; \ PRF(prf_ptr)(1) LOC_M( 1, 1 ); \ PRF(prf_ptr][0) * o ++mod_ptr; \ KX) [mod_ptr) (11 ■ x; \ KX>[mod_ptr) (0) NEXT_M( mod_ptr ) ++mod_ptr; \ MDD|mod_ptr)111 " x; \ MDO|mod_ptc)[0] - -NEXT_M( mod_ptr ) MOOUJ |1] MOD(lJ[0] LOCM ( 1, NEXT_M( 1 ' KNW; ++bckt_ptr; \ lf ( KB( OP( gt, RSUB )) BCKT(bckt_ptr](3) else \ BCKT|bckt_ptr][3] -PRF(gt](2]; \ BCKT(bckt j>tr)(0) gt; \ BCKT[bckt_ptcj (2J WFF (PRF IgtJ (11 H21; BCKT(bckt_ptr](lj - -WFF(BCKT[bckt_ptr](2]]|0] lf { gt prf_ptr ) - prf_ptr; \ else PRF(gt](0] USD ALT(++alt_ptr] * vai; \ ALT|++alt_ptr] prf_ptr-l; \ ALT(++alt_ptr] mod_ptr; \ ALT(++altjptr] bckt_ptr; \ ALT(++alt_ptr] prf_ptr; \ ALT(++alt_ptr] PRF(prf^ptr)|1);\ ALT(++alt_ptr] PRF(prf_ptr](0] OTHERWORLDS ALT(alt_ptr-6] > 0 GO_NEXT_WORLD --ALTlalt_ptr-6]; \ prf_ptr ALT(alt_ptr-5]; \ modjatr ALT[alt_ptr-4]; \ bckt_ptr ALT(alt_ptr-3]; \ PRF(ALT|alt_ptr-2]|(0] ALTlaltjptr-1]; \ PRF|ALT(alt_ptr-2]1(11 ALT|alt_ptr]; \ PRF(ALT(alt_ptr-2)|(2] = ALT|alt_ptr-61 UPDATE_PRF UPDATE_ALT S93 Â p p en á ix C Idefine COOTRADICTION iwcontrad( mod_ptr, prf_ptr ) II contrad( prf_ptr J fdefina NON CONTRAD p2 í CNULL CS ( PRF|p2H0J + PRFipHOJ !0 II PRF(p2][2) !PRF[p)|2J > GLOBALS S h o r t HFFÍSTR LENJ( 3 ) , PWISTRLENH3], ALT[STRLENj, BLFfSTR LEN], BCKT ÍSTRLEN]HJ, M0D(STR_LEH1(21, v f _ p t r , p r f _ p t r , m o d _ p t r , a l t _ p t r , b c k t _ p t r ; c h a r t h e W f f ( STR_LEN ]; l o n g k l , k 2 ; */ / * stores the tree representation of the wff */ /* where to save states in branching */ /* pointer to current location In WFF /* Input formula * / f * for tlroe measurement V C2. F i l e : prototypes. h /* -----------FUNCTION PROTOTYPES ------- /* --parser. h ----- */ lnt formula( short, short *, short * ); lnt form_and_or< short, short *, short * ); int rest_fd( short, short, short *, short * ); lnt rest form( short, short, short *, short * ) int forml ( short, short *, short * ); /* -tableau.h----- */ lnt tableau( vold ); lnt contrad( short }; lnt lw_contrad( short, short J i vold save state( short ); vold restore state( vold ); /* --maln.c----- */ int maln( vold ); C3. F i l e : parser. h /* ----------------------PARSER FUNCTIONS • formula( xl, xo, * ) /*---------------- */ short *1, *xo, *t; { short xn, zn; lf ( form_and_or< xl, txn, tzn 1 1 { if ( rest__form{ xn, zn, xo, z > ) return 1; else return 0; í else return 0; 194 T T P .Z P 5 L is tin g s fonn_and_or( xl, xo, zo ) /*-------------------- */ short xl, *xo, *zo; ( short zl, xn; lf ( form_H xl, txn, fizl ) ) { t lf ( rest_fd( xn, zl, xo, zo ) ) return 1; ) else return 0; í rest_fd{ xl, zl, xo, zo ) /* short xl, zl, *xo, *zo; ( short z; lf { theHff(xl) ~ £& form_and_or< xl+1, I INITJ*X)E( UND, Zl, Z ); * zo wf_ptr++; return 1; ) else lf ( theWff(xl] - 'v' form_and_or( xl+1, < INIT_NODE( -UND, zl, z); / * avb WFF (zl)(0] - -(WFFlzl|(0)); WFF(z)(0) - -(WFF(z)(0)); *zo ■ wf_ptr++; return 1; ) else ( *xo xl; *zo zl; return 1; ) rest_form< xl, zl, xo, zo ) /*----------------------./ short xl, zl, *xo, *zo; ( short z; lf (( theWff(xl) - ) i i ( theWff(xl+1] it formulai xl+2, xo, tz )) I INIT_NODE( -UM), zl, z ); /* a->b WFF(Z)IO) - -(WFF(z)(0)); * zo ■ wf_ptr++; return 1; ) else lf (( theWff(xl1 -- '<• ) i i ( theHff(xl+1) &t ( theHff(xl+2] - '>' ) i i l ) else ( INIT_NODE( EQU, zl, z ); *zo wf_ptr++; return 1; *xo xl; *zo zl; return 1; form l ( xl, xo, zo ) /* xo, tz )) xo, &z )J - «a4-Ja) * / - •>' ) - -,(ac-4>) */ formula( xl+3, xo, &z )) 395 A p p e n d ix C short xi, *xo, * z o ; í short f; if ( (unslgned char) theWff(xi) •* (unslgned char) ) ( lf ( forml| xi+1» xo, *f H ( WFF[f) (0) - -( WFF(f]ÍO| ); *zo f? return 1; ) else return 0; ) lf ( theWff[xlj - 'K' ) ( if (form_l< xl+1, xo, *f )) I INIT_N0DE( KNW, f, CNULL); *zo wfjptr++; return(1); ) else return 0; ) if ( theWff[xi] - 'B' ) { if <form_l( xi+1, xo, *f )) ( INIT_NOOE( BEL, f, CNULL); *zo • wf_ptr*+; return 1; ) else return 0; ) if ( theWff [xi] - M ' > í if ( formulai xi+1, xo, *f )) I if ( theWff(*xo| - •)* ) I ++(*xo); *20 » f; return 1; ) else return 0; I } if ( P_ATCMIC( theWff[xi]) ) t WFF(wf_ptr)[0] theWff(xi); *zo ■ wfjptr++,* *xo xi+1; return 1; ) else return 0; C4. File: tableau. h / « -------------------------PROVER STUFF----------------------*/ /* tableau is the function whoe does the job. te long as there are branches cjoslng and backtracking points, it will run. It stops (returning zero) when some branch is kept open - i.e., no contradictions and no more wffs to split - or (then returning one) when ali branches led to contradictions and there is nothlng more to do (no more stored branches). 196 T T P .Z P 5 L is tin g s ( short gt, vai, done; done vai 0; bckt_ptr CNULL; whlle < Idone ) ( /*dabug(); */ lf < CONTRADICTION ) lf ( bckt_ptr -CNULL ) /* and nothing ls stored on BKCT; I.e., */ SUCCEED /* no more branchlng... * / else restore_state(); ) else { LOOK_FOR( UND ); if { FOUND ) /* a ture conjunction, found */ i if ( KB( OP{ gt, L_SIB ) ) ) I T_AOD_2_MOO( LOC( gt, L_SUB ) ) ; lf ( KB( OP( gt, RSUB ) ) ) ( T_ADD_2_MOO( LOC{ gt, RSUB ) ); UPDATE_PRF; ) else l R£PL_WITH( RSUB ); » ) else if ( KB( OP< gt, R_SUB ) ) ) í T ADD 2 MOO ( LOC( gt, R_SUB ) ); REPLWITH( LSUB ); I else { ADD_TRUE( gt, R_SUO ); REPLWITH( L_SUB ); ) í else { LOOKJ4FOR ( UND ); lf ( FOUND ) í if { OP_M< gt, L SUB ) !- -UND ) í lf ( OP_M( gt, R_SUB ) !- -UND ) ( T_ADD 2M0D( LOC < gt, L_SUB ) ); TM_REPL_WITH ( gt, R_SUB ); ) else { ADDMODAL( gt, RSUB ); 3W_REPL_WITH( gt, L_SUB ); ) ) * else lf ( OP_M( gt, RSUB ) !- -UND ) i ADD_MODAL ( gt, L SUB ),* • 1M_REPL_WITH( gt, RSUB ); ) else ADD_MODAL( gt, LSUB ); ADD_MODAL( gt, RSUB ); tableau() 1 m A p p en d ix C ( ) else 5 LOOKJFOR( -UND ); lf ( FOUND ) /* a false conjunction found */ ( STORE_RIGHT_WFF; savestate ( gt ) ,g if ( KB( OP( gt, L_SUB )> ) F_ADD_2_MDD( LOC( gt, L_SUB ) ); > else í F_REPL_WITH( L_SUB }; í i else / * now we have modals */ I LOOK_4_KB( -KNW, -BEL ); lf { FOUND ) { if( MOD(gt)(0) - -KNW ) í NDD(gt)(01 - -U KNW; if ( KB( OP_M( gt, L__SUB )) ) ( F_ADD_2_MDO( LOC_M{ gt, L__SUB ) } ; ) else I fcDD SCOPEl ++valr gt, -NEXTl prf_ptr ) ); lf 7 !KB( OP_M(gt,L_SUB)) ) í BLF[LOC_M(gt,L_SUB)] 1; ) I í else /* -- -BEL */ i MDDlgtJ(OJ - -U BEL; lf < KB( OP_M< gt, LSUB )) ) 5 F_ADD*_2_M0D< LOC_H( gt, L__SUB > ); I else i ADD SCOPE( ++val, gt, -NEXT( pcf_ptr ) ) t } i ) else I LOOK_4_KB( KNW, BEL ); if ( FOUND i / * a true pi was found */ I lf ( MGD(gt) (0) - KNW ) i MODlgtl101 ü_KNW; if < OPM( gt, LSUB ) !- -UND ) í T A D D 2 M O D ( LOC_M{ gt, L_SUB ) ); ) else I ADD_SCOPE{ vai, gt, NEXT( prf_ptr ) ); UPDÃTEALT; ) ) else f * -*■ BEL */ { MOD(gt] [01 U_BEL; if ( OP_M( gt, LSUB ) ?» -UND ) ( 198 T T P .Z P 5 L is tin g s contrad( p ) /*--------- ./ short p; < short p2, 1; ) < ) else T_ADD_2_M00( LOC_M( gt, LSUB ) ); lf í !KB( OP_M( gt, L_SUB >) ) { BLF[LOC_M{gt,LSUB)j 1; I ADD_SCOPE( vai, gt, NEXT< prf_ptr ) )? UPDATE ALT; ) í sisa lf ( 0THERJI0R1DS ) { G0__NEXT__M0R1D; ) alsa whlle ( p ]CNULL «« 1ATCMIC1 PRF{p)(0) )) -p; lf ( p - CNULL ) return 0; p2 p-X; whlle( NON_CONTRAD ) -p2; lf ( p2 !CNULL) return 1; else return( contrad( - p )); lw_contrad( m, p ) /*-------------- */ short p; { short p2, 1; whlle ( m !CNULL !ATCMIC( MOO|m)|OJ )) -m; lf ( m CNULL ) return 0; p2 m-1; whlle ( p2 !CNULL 6t ( MOD|p2)(01 + MOD[m](0) !0 ) ) -p2; lf ( p2 !■ CNULL) return 1; p2 p; whlle< p2 !CNULL <& { PRf[p2|(0) ♦ M0D(m){0) !0 II ( ( BLF (PRF (p2)(1)J 1 |( PRFlp2)12J 0 ) ti BLF[MOO(m]|11J - 1 ))) --p2; lf ( p2 !CNULL) return 1; else return( lw_contrad( - m, p )); 199 A p p en d ix C void save_state( r ) /*------------*/ short r? í short 1; ' ++bckt_ptr; BCKT(bckt_ptr][OJ LMK; BCKT(bckt_ptr){1} -PRF[r)(0); BCKT[bckt_ptrj í21 PRF[rí(l); BCKT[bckt_ptr)(3] PRF(r](2); lf < r ~ prf_ptr cc KB( OP( r, R_SUB )> ) --prf_ptr; /* bota is last line on PRF */ /* now save th® modals */ lf ( mod_ptr > CNULL ) for ( 1 0? 1 <« mod_ptr? +♦! ) í ++bckt_ptr; BCKT[bckt_ptr)(0) USD; BCKT{bckt_ptrJ[1] 1; BCKT(bckt_ptr][2] M00[l][0); BCKT[bckt_ptr) f 3) -M0D(i][l]; ) ++bckt_ptr; BCKT(bckt_ptr)[0) RMK; BCKT(bckt_ptr][1) prf_ptr; BCKT[bckt_ptr](2] mod_ptr; BCKT(bckt_pt r)13] alt_ptr; vold restore_state () /*------------ */ prf_ptr BCKT(bckt_ptr)(1]; mod_ptr BCKT|bckt_ptrJ[2]; alt_ptr BCKT[bckt_ptr)(3]; - bckt_ptr; whlle ( BCKT(bckt_ptr| 10] !U * ) { lf { BCKT[bckt_ptr)J0) - USD ) ( MOD(BCKT[bckt_ptr)[1)1(0) BCKT(bckt_ptr)[2]; MOD(BCKT[bckt_ptr)(1)1(1) BCKT(bckt_ptr)[3]? ) else ( PRF(BCKT[bckt_ptr)(0)1(0) • BCKT(bckt_ptr)(1); PRF[BCKT[bckt_ptrj(0))(1) BCKT(bckt_ptr)(2); PRFlBCKT[bckt_ptr)(0)1l21 BCKT(bc*t_ptr)(3); ) - bckt_ptr; ) - bckt_ptr; lf < KB( BCKTtbckt_ptr]tH > ) < PRF(BCKT[bckt_ptr)[0]](0) USD; +«nod_ptr; MOD|mod_ptr](0) BCKT[bckt_ptr)(1); MOD(íBod_ptr] (11 • BCKT(bckt_ptrU21; í else ( PRF(BCKT[bckt_ptr][0]][0] BCKT[bckt_ptr)(1); PRF(BCKT[bckt_ptr][0]](1] BCKT(bckt_ptr)(2); ) lf ( bckt_ptr > 0 ) ( BCKT(bckt_ptr)(1) BCKT(bckt_ptr-l)[1]; BCKT(bckt_ptr) (2) BCKT(bckt_j5tr-l) [2]; BCKT|bckt_ptr) (3) BCKT(bckt_ptr-l)l3); BCKT(bckt_ptr-l)(0) BCKT(bckt_ptr)[0]; BCKT(bckt_ptr-l){l) BCKT[bckt_ptr+l][1]; BCKT(bcktj)tr-l][2] BCKT(bckt_ptr+l][2]; 200 T T P .Z P 5 L is tin g s BCKT[bckt_ptr-l J(3) BCKT[bckt_ptr+l)(3); BCKT[bckt_ptr)[OJ PMK; ) •1m - bckt_ptr; C5. F i l e : maln.c TTP.ZP5 - Verslon 1.0 Caaar A. Mortarl - July 1990 Tíila program implementa a tableau-llke theorem prover for the eplstemic-doxastlc logic ZP5. /* ------------------TTP.ZP5 INCLUDES llnclude "macros.h" llnclude "prototypes.h" llnclude "parser.h" llnclude "tableau.h" maln() { short i, j, u; printf CXn*******4**********4*** ***************** «************«*••) ; printf(*\n* TABLEAU THEOREM PROVER FOR ZP5 *"); printf("\n* Cesar A. Mortarl *■); printf("\n* VI.0, July 1990 *■); printf("Xo**********************************44*4****************"); printf("\n\nSyntax:\n a..t (variables), K, B, t, v, ->\n"); printf("\nTo exit type ';'<CR>\n"); printf("XnPlease type in a formula:\n"); for (;;) { /* Some initiallzations... */ wf_ptr 0; • prf_ptr bcktjptr mod_ptr altjptr CNULL; for ( i 0; 1 < STR_LEN; ++i ) BLF(l) 0; gets( theWff ); /* input the formula to be (dls)proven */ lf { theWff[0J -« ';') /* program ends... */ return 1 ; /* ELSE remove blanks from theWff */ i 0; 3 0 ; while ( theWff[i] !- 'NO' ) ( lf ( theWff[i) I- • • ) theWff(j++) theWff(i]; ++l; ) theWff[J1 - »\U'; kl « TickCountO; /* lnltialize time counting */ lf ( formulai 0, cu, ti ) t t theWff(u) *- '\0' ) ( if < KB< WFF(UJO) ) ) ( 201 j-ippertdia C ) else ++mod_ptr; M00(0)[0j - -WFF[ 1 ] (0 ]; MODlO)tl) 1? ++prf jptr? PRF(0)(0| - -WFF(1II0}; PRF(0)(1J 1? PRF[OJ[2) 0; ) lf ( tableau()) /* *11 branches were successfully closed... */ I k2 - <( TickCountO kl ) * 100) / 6 ; putchar( 'y1 )? putcharf ce* ); putchart *8* ); ) else /* some open branch tableau{) teturned zero *f K k2 - <( TickCount() kl ) * 100) / 6; putcharl 'n' ); putcharí 'o' ); } printf("\nTlmes %ld ms\n\n\n"„ k2); /* lt was not a wff, so try again... */ printf("\nSyntax error.. •\n\n*') ? ) else 202 i ALG.ZP5 Llstings 43rd Law ofComputing: Anything that can go wr Error: Segmentation violation Core dumped. Dl. F i l e : s u b f . h vold loop( s I /*----------- */ short s; ( short p, J; p aj>tr+l; whlle { p <■ p_ptr > { prf_ptr " bckt_ptr ■ mod_ptr " alt_ptr « CNULL; lf ( NFF(p|(3) - s t t KB( WFFlpl(0) ) ) { add_fl_psl( WFFlpHU 1; preparetab( p, 1 ); /*d_wff ();*/ lf ( tableau<) ) WFFlpJ(2) 1; else WFF [p] 12] 0; ) ++p; vold add_fljpsl{ p ) /*-------------*/ short p; 1 short 1, r; if ( ATOMIC< WFF(p)(01 I) return; lf { KB( WFFlpMOJ )) I r » e; 1 copy( WFF[pj(1) >; ++e; lf ( WFF(p)[2] - 1 ) WFF[e] [0] PST( WFF[p] (Oi ); else WFFIeMO) NGT( WFF(p] {0J ); WFF[e] [1] 1; ++e; WFF(e](0] UND; WFFlel(l) r; WFFle)(2] e-1; 205 A p p en d ix D add_fi_psi( HFF(p)(Ij ); ) else /* conjunction */ < add_fl_p8l( HFFípHU ); add_fi_psi ( WFF|p)[2J U ) short copy ( 1 ) short 1; { short 1, r; lf ( ATOMIC( WFF(1]ÍOJ ») { ++ej WFF[el [0] MFF(1)(0); return e; ) else tf í KB( WFF[1)[0) ) ) í 1 copy( MFFllJdl ); ++e; HFF5®Jt0] - «FFUMOJj * F F | e ) l U 1; return e; ) else f 1 copy{ HFF (1) [ 1J ); r copy( WFF(1J[21 >; ++e; WTI«U0] NFFtlHO); HFFÍeJUl 1; WFFfe)[21 r; return e; vold prepare_tab{ p, x | /*-----------------*/ short p; I short J; lf í MFF(«M0] - BEL > /* Ba ls alone ; no fl psl V [ ++mod_ptr; MO0[mod_ptrl[0) NFF[e][0); MCO[mod_ptrl[11 e; ) else { v+prf_ptr; PRF[prf_ptr)ÍOJ WFF(e)(0)| PRF[prf_ptrj( l j e; PRF[prf_ptr)(2] 0; I j copy { p ); lf ( x t t WFF(j][OJ < 0 ) WFFÍJl(O) - -WFF[jj[0J; ++e; WFFJe)[0] BEL; wfTteim 3; ♦+mod_ptr; HDD[mod_ptrl(0J - -BEL; M0D[mod_ptr](1) - ©; 204 A L G .Z P 5 L ii l in g t D2 F i l e m a i n . c ALG.ZP5 - Version 1.0 Cesar A. Moctarl - August 1990 Ihis program implements the aigorithm to decide whether some wff belongs to some belief state in the eplstemlc-doxastlc logic ZPS. '/ /* ALG.ZP5 INCLUDES */ linclude " macros.h" linclude "prototypes.h" linclude "parser.h" linclude "tableau.h" linclude "subf.h" maln() i short 1, j, u, s; printf("\n***************•*********************»*****»**********«); prlntf(*\n* BELIEF STATE ALGORITHM FOR ZP5 *"); prlntf("\n* Cesar A. Mortari *"); printf("\n* VI.0, August 1990 *•); printf printf<"\n\nSyntax:\n a..t (varlables), K, B, t , v, ->\n"); printf("\nTo exit type ,;,<CR>\n"); for <;;) I /* Some lnltlalizatlons... */ wf_ptr 0; prf_ptr bckt_ptr mod_ptr altj>tr a_ptr p_ptr CNULL; for ( 1 0; 1 < STR_LEN; ++1 ) WFF(1)(2| HFF(i)|3) BLF(i) 0; printf("\nPlease type in your 'alpha*:\n"); gets( theHff ); lf ( theHff(0] - ■;*) return 1 ; /* ELSE remove blanks from theHff »/ 1 0 ; j 0; whlle { theHff(11 l- *\0* ) ( if ( theHff (1) !■»••) theHff(J++} theHff(1); ++i; ) theHfflj) - *\0'; lf ( formula( 0, tu, d ) cc theWff(u] - •\0• ) lf ( formula( 0, tu, íj ) c« theWff|u) -= «NO* ) ( a_ptr » 1; /* add Ba to WFF */ e * wf_ptr; WFF(e)[0] BEL; HFF[e)[l] a_ptr; printf("\nNow type the next wff:\n"); gets( theWff ); kl TickCount(); /* lnltiallze time counting */ 205 A p p en d ix D p_ptr j; lf ( WFF[p_ptrJ [3J > 0 ) for ( s 1? 9 <MFF[p_ptr][3]; ++s ) loop( s ); prf_ptr ■ bckt_ptr mod_ptr " alt_ptr ■ CNULL; « p_ptr+l; add_fl_psi{ p_ptr í; preparetab( p_ptr, 0 ); for í i 0; 1 < STR_LEN; ++i ) BLF(l) 0; /*d_wff();*/ lf ( tableau ()» { k2 - (( TickCountO kl ) * 100) / 6; putcharj "y" ); putchar í *«' ); putcharj 'a® ); > else { k2 - {( TickCount() kl ) * 100) / 6; putchar( 'n' ); putcharf 'o' ); ) 5 prlntf("XnTlme: %ld ms\n\n\n*, k2); > else printf{'•\nSyntax error...\n\n*); ) ) 2 0 6 References A bibliography is usually a lisl ofdutl, dry books placed at the end ofadult tedious book, assuring lhe reader íhçit i f he reads lhe listed books he will be more bored slill. 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[Va85] V ard i, M. "A model-theoretical analysis of monotonic knowledge". Proceedings o f the Nimh International Joinl Conference on Artificial Intelligence, 1985, pp. 509-5 Í2. [Va86] VARDI, M.Y. "On epislemic logic and logical omniscience". In [Ha86a], pp. 293-305. [XW83] XIWEN, M. Sl WEIDE, G. "W-JS: a modal logic of knowledge*'. Proceedings o f the Eighth International Joint Conference on Artificial Intelligence, 1983, pp. 398-401. 211 C urricu lum V itae I was bom in 19S7 in Santa Maria, Rio Grande do Sul, Brazil, where I lived until 1 was fifteen. In 19731 moved to Florianópolis, capital of (he State of Santa Catarina, where I began, in 1976, to study Philosophy at the Federal University of Santa Catarina (UFSC). Afler concluding my studies in 1978,1 moved in the following year to Campinas, São Paulo, where I began doing research in Logic for a master degree at the State University of Campinas (UNICAMP). 1 got my M.A in Logic and Philosophy of Science in 1983, with a disseitation called "Valuation Semantics for some systems of temporal logic". Even during my studies there I began teaching at the UFSC, mosüy Logic, sometimes Philosophy of Mathematics. Soon I got tenure at the Philosophy Department, and 1 remained there until the end of 1983, when I got a special leave of absence to come lo TUbingen and work for a Ph D. at the Philosophy Faculty.