MODAL LOGICS FOR TOPOLOGICAL SPACES by KONSTANTINOS GEORGATOS A dissertation submitted to the Graduate Faculty in Mathematics in partial fulllment of the requirements for the degree of Doctor of Philosophy The City University of New York  Abstract MODAL LOGICS FOR TOPOLOGICAL SPACES by Konstantinos Georgatos Adviser Professor Rohit Parikh We present two bimodal systems MP and MP  for reasoning about knowledge and e	ort Knowledge is interpreted as all true statements common to a set of possible worlds which represents our view E	ort corresponds to increase of information and trans

lates to a restriction of our view Such restrictions are parameterized by the worlds in our view and therefore are neighborhood restrictions The semantics of these logics consist of pairs of points and their neighborhoods In this spatial setting basic topo

logical and computation concepts are naturally expressed which make these systems ideal for studying computing knowledge by set theoretic means The system MP was introduced and proven complete for the class of sets con

taining arbitrary neighborhoods by Larry Moss and Rohit Parikh In this thesis MP  an extension of MP is introduced and proven complete for various class of spaces closed under unions and intersections among them topological spaces We also iv present necessary and sucient conditions under which a Kripke frame can be turned into a set theoretic model of ours Among our results is the nite model property and decidability for MP  In addition we present the algebraic models of these systems and discuss further work v  os os o to my parents vii Contents Abstract iv Acknowledgements vi  Introduction   Two Systems MP and MP  

 Language and Semantics 

 MP and MP    A Semantical analysis of MP    Stability and Splittings   Basis Model   Finite Satisability   Completeness for MP    Subset frames  viii  On the proof theory of MP    Canonical Model   Joint models  The Algebras of MP and MP    Fixed Monadic Algebras   Generated Monadic Algebras   Further Directions  References  ix List of Tables  Axioms and Rules of MP   x Chapter  Introduction In this thesis we shall present two logical systems MP and MP  for the purpose of reasoning about knowledge and eort  These logical systems will be interpreted in a spatial context and therefore the abstract concepts of knowledge and e	ort will be dened by concrete mathematical concepts Our general framework consists of a set of possible worlds situations scenarios consistent theories etc A state of knowledge is a subset of this set and our knowledge consists of all facts common to the worlds belonging to this subset This subset of possibilities can be thought as our view Thus two knowers having distinct views can have di	erent knowledge This treatment of knowledge agrees with the traditional one       expressed in a variety of contexts articial intelligence distributed processes economics etc  Chapter  Introduction Our treatment is based on the following simple observation a restriction of our view increases our knowledge This is because a smaller set of possibilities implies a greater amount of common facts Moreover such a restriction can only be possible due to an increase of information And such an information increase can happen with spending of time or computation resources Here is where the notion of e	ort enters A restriction of our view is dynamic contrary to the view itself which is a state and is accompanied by e	ort during which a greater amount of information becomes available to us Pratt expresses a similar idea in the context of processes   We make two important assumptions Our knowledge has a subject We collect information for a specic purpose Hence we are not considering arbitrary restrictions to our view but restrictions parameterized by possibilities contained in our view ie neighborhoods of possibilities After all only one of these possibilities is our actual state This crucial assumption enables us to express topological concepts and use a mathematical set theoretic setting as semantics Without such an assumption these ideas would have been expressed in the familiar theory of intuitionism      As Fitting points out in  Let hG R ji be a intuitionistic propositional model G is intended to be a collection of possible universes or more properly states of knowledge Thus a particular  in G may be considered as a collection of physical Chapter  Introduction  facts known at a particular time The relation R represents possible time succession That is given two states of knowledge  and of G  to say R is to say if we now know  it is possible that later we will know  Considering neighborhoods and inevitably points which parameterize neighborhoods the important duality between the facts which constitute our knowledge and the possible worlds where such facts hold emerges The other assumption is that of indeterminacy Each state of knowledge is closed under logical deduction Thus an increase of knowledge can happen only by a piece of evidence or information given from outside Our knowledge is external a term used by Parikh to describe a similar idea in   This fact leads to indeterminacy we do not know which kinds of information will be available to us if at all and resem

bles indeterminacy expressed in intuitionism through the notion of lawless sequence see    where not surprisingly topological notions arise To illustrate better these simple but fundamental ideas we present the following examples  Suppose that a machine emits a stream of binary digits representing the output of a recursive function f  After time t  the machine emitted the stream  The only information we have about the function being computed at this time Chapter  Introduction  on the basis of this nite observation is that f  f   f   As far as our knowledge concerns f is indistinguishable from the constant func

tion  where n   for all n After some additional time t ie spending more time and resources  might appear and thus we could be able to dis

tinguish f from  In any case each binary stream will be an initial segment of f and this initial segment is a neighborhood of f  In this way we can ac

quire more knowledge for the function the machine computes The space of nite binary streams is a structure which models computation Moreover this space comprises a topological space The set of binary streams under the prex ordering is an example of Alexandrov topology see   A policeman measures the speed of passing cars by means of a device The speed limit is  kmh The error in measurement which the device introduces is  kmh So if a car has a speed of 	 kmh and his device measures  kmh then he knows that the speed of the passing car lies in the interval    but he does not know if the car exceeds the speed limit because not all values in this interval are more than  However measuring again and combining the two measurements or acquiring a more accurate device he has the possibility of knowing that a car with a speed of 	 kmh does not exceed the speed limit Note here that if the measurement is indeed an open interval of real line and Chapter  Introduction  the speed of a passing car is exactly  kmh then he would never know if such a car exceeded the speed limit or not To express this framework we use two modalities K for knowledge and   for e	ort Moss and Parikh observed in   that if the formula A KA is valid where A is an atomic predicate and  is the dual of the   ie     then the set which A represents is an open set of the topology where we interpret our systems Under the reading of  as possible and K as is known the above formula says that if A is true then it is for A possible to be known ie A is armative Vickers denes similarly an armative assertion in  an assertion is armative i it is true precisely in the circumstances when it can be armed The validity of the dual formula  LA A

where L is the dual of K ie L  K expresses the fact that the set which A represents is closed and hence A is refutative meaning if it does not hold then it is possible to know that The fact that armative and refutative assertions are repre

sented by opens and closed subsets respectively should not come to us as a surprise Chapter  Introduction  Armative assertions are closed under innite disjunctions and refutative assertions are closed under innite conjunctions Smyth in  observed rst these properties in semi decidable properties Semi decidable properties are those properties whose truth set is re and are a particular kind of armative assertions In fact changing our power of arming or computing we get another class of properties with a similar knowledge theoretic character For example using polynomial algorithms armative assertions become polynomially semi decidable properties If an object has this prop

erty then it is possible to know it with a polynomial algorithm even though it is not true we know it now Does this framework su	ers from the problem of logical omniscience Only in part Expressing e	ort we are able to bound the increase of knowledge depending on information external knowledge Since the modality K which corresponds to knowledge is axiomatized by the normal modal logic of S	 knowledge is closed under logical deduction However because of the strong computational character of this framework it does not seem unjustied to assume that in most cases as in the binary streams example a nite amount of data restricts our knowledge to a nite number of relevant formulae Even without such an assumption we can incorporate the e	ort to deduce the knowledge of a property in the passage from one state of knowledge to the other Chapter  Introduction  We have made an e	ort to present our material somewhat independently How

ever knowledge of basic modal logic as in   or  is strongly recommended The language and semantics of our logical framework is presented in Chapter  In the same Chapter we present two systems MP and MP  The former was introduced in   and was proven complete for arbitrary sets of subsets It soon became evident that such sets of subsets should be combined whenever it is possible to yield a further increase of knowledge or we should assume a previous state of other states of knowledge where such states are a possible Therefore the set of subsets should be closed under union and intersection Moreover topological notions expressed in MP make sense only in topological models For this reason we introduce an extension of the set of axioms of MP and we call it MP  In Chapter  we study the topological models of MP  by semantical means We are able to prove the reduction of the theory of topological models to models whose associated set of subsets is closed under nite union and intersection Finding for each satisable formula a model of bounded size we prove decidability for MP  The results of this chapter will appear in  In Chapter  we prove that MP  is a complete system for topological models as well as topological models comprised by closed subsets We also give necessary and sucient conditions for turning a Kripke frame into such a topological model In Chapter  we present the modal algebras of MP and MP  and some of their properties Finally in Chapter  we present some of our ideas towards future work Chapter  Two Systems MP and MP  In section  we shall present a language and semantics which appeared rst in   In section   we shall present the axiom system MP introduced and proven sound and complete with a class of models called subset spaces in   and the axiom system MP  introduced by us which we shall prove sound and complete for among other classes the class of topological spaces   Language and Semantics We follow the notation of   Our language is bimodal and propositional Formally we start with a countable set A of atomic formulae containing two distinguished elements  and  Then the language L is the least set such that A L and closed under the following rules  Chapter  Two Systems MP and MP   L L L    K L The above language can be interpreted inside any spatial context Denition  Let X be a set and O a subset of the powerset of X ie O PX such that X O We call the pair hX Oi a subset space A model is a triple hX O ii where hX Oi is a subset space and i a map from A to PX with i  X and i  called initial interpretation We denote the set fx U x X U O and x Ug  X O by X O For each U O let U be the set fV V O and V  Ug the lower closed set generated by U in the partial order O  ie U  PU O Denition  The satisfaction relation j M  where M is the model hX O ii is a subset of X O L dened recursively by we write x U j M instead of x U  j M  x U j M A i x iA where A A x U j M if x U j M and x U j M x U j M  if x U  j M x U j M K if for all y U y U j M x U j M   if for all V U such that x V x V j M Chapter  Two Systems MP and MP   If x U j M for all x U belonging to X O then is valid inM  denoted byM j We abbreviate   and K by  and L respectively We have that x U j M L if there exists y U such that y U j M x U j M  if there exists V O such that V  U x V and x V j M Many topological properties are expressible in this logical system in a natural way For instance in a model where the subset space is a topological space iA is open whenever A KA is valid in this model Similarly iA is nowhere dense whenever LKA is valid cf   Example Consider the set of real numbers R with the usual topology of open inter

vals We dene the following three predicates pi where ipi  f g I  where iI     I where iI   ! Q where iQ  fq q is rational g There is no real number p and open set U such that p U jKpi because that would imply p  and U  f g and there are no singletons which are open A point x belongs to the closure of a set W if every open U that contains x intersects W  Thus belongs to the closure of  ! ie every open that contains Chapter  Two Systems MP and MP  

has a point in  ! This means that for all U such that U  U jLI therefore Rj LI Following the same reasoning Rj I  since belongs to the closure of   A point x belongs to the boundary of a set W whenever x belong to the closure of W and XW  By the above belongs to the boundary of   and Rj LI 

LI A set W is closed if it contains its closure The interval iI     is closed and this means that the formula  LI   I  is valid A set W is dense if all opens contain a point of W  The set of rational numbers is dense which translates to the fact that the formula  LQ is valid To exhibit the reasoning in this logic suppose that the set of rational numbers was closed then both  LQ and  LQ  Q would be valid This implies that Q would be valid which means that all reals would be rationals Hence the set of rational numbers is not closed    MP and MP The axiom system MP consists of axiom schemes  through  and rules of table  see page   and appeared rst in   The following was proved in   Theorem  The axioms and rules of MP are sound and complete with respect to Chapter  Two Systems MP and MP   Axioms  All propositional tautologies

 A  A A  A for A A                    K   K K  K  K KK   KL  K   K        K  LK   K  L Rules  MP K KNecessitation    Necessitation Table  Axioms and Rules of MP  Chapter  Two Systems MP and MP   subset spaces We add the axioms  and  to form the system MP  for the purpose of ax

iomatizing spaces closed under union and intersection and in particular topological spaces A word about the axioms most of the following facts can be found in any intro

ductory book about modal logic eg  or  The axiom expresses the fact that the truth of atomic formulae is independent of the choice of subset and depends only on the choice of point This is the rst example of a class of formulae which we are going to call bipersistent and their identication is one of the key steps to completeness Axioms  through  and axioms  through  are used to axiomatize the normal modal logics S and S respectively The former group of axioms expresses the fact that the passage from one subset to a restriction of it is done in a constructive way as actually happens to an increase of information or a spending of resources the classical interpretation of necessity in intuitionistic logic is axiomatized in the same way The latter group is generally used for axiomatizing logics of knowledge Axiom  expresses the fact that if a formula holds in arbitrary subsets is going to hold as well in the ones which are neighborhoods of a point The converse is not sound Axiom  is a well known formula which characterizes incestual frames ie if two points  and  in a frame can be accessed by a common point  then there is a point Chapter  Two Systems MP and MP    which can be accessed by both  and  It appeared in the equivalent form in           and was proved sound in subset spaces closed under nite intersection Obviously our attention is focused on axiom   It is sound in spaces closed under nite union and intersection as the following proposition shows Proposition  Axioms  through  are sound in the class of subset spaces closed under nite union and intersection Proof Soundness for Axioms  through  is easy For Axiom   suppose x U jK  LK  Since x U jK  there exists Ux  U such that x UxjK and since x U jLK   there exists y U and Uy  U such that y UyjK  We now have that Ux  Uy  U we assume closure under unions Thus x Ux  UyjK y Ux  UyjK x Ux  Uyj and y Ux  Uyj Therefore x U jK  L Chapter  Two Systems MP and MP   With the help of axiom  we are able to prove the key lemma  which leads to the DNF Theorem page  and this is the only place where we actually use it Any formula sound in the class of subset spaces closed under nite union and intersection which implies the formula note the di	erence from axiom   K  LK   K L where     and    are theorems can replace axiom   Chapter  A Semantical analysis of MP  In this chapter we prove nite model property decidability and strong reduction of the theory of topological models to that of subset spaces closed under nite union and intersection The latter was a conjecture in   All these are proved semantically without using any complete axiomatization for these models ie MP  and in fact preceded the results of the next chapter The approach in this chapter seems unrelated to the one of next chapter We are able to relate both in the last section  Stability and Splittings Suppose that X is a set and T a topology on X In the following we assume that we are working in the topological space X T  Our aim is to nd a partition of T  where a given formula retains its truth value for each point throughout a member  Chapter  A Semantical analysis of MP   of this partition We shall show that there exists a nite partition of this kind Denition Given a nite family F  fU  Ung of opens we dene the remainder of the principal ideal in T  generated by Uk by Rem FUk  Uk    Uk Ui

Ui Proposition  In a nite set of opens F  fU  Ung closed under intersection we have Rem FUi  Ui    UjUi

Uj

for i   n Proof Rem FUi  Ui  S Ui Uh

Uh  Ui  S Ui Uh

Uh  Ui  Ui  S UjUi

Ui We denote S UiF Ui with F  Proposition If F  fU  Ung is a nite family of opens closed under inter section then a RemFUi  Rem FUj  for i  j Chapter  A Semantical analysis of MP   b Sn i  Rem FUi  F ie fRem FUigni  is a partition of F  We call such an F a nite splitting of F  c if V  V Rem FUi and V is an open such that V   V  V then V Rem FUi ie Rem FUi is convex Proof The rst and the third are immediate from the denition For the second suppose that V F then V RemF T VUi Ui Every partition of a set induces an equivalence relation on this set The members of the partition comprise the equivalence classes Since a splitting induces a partition we denote the equivalence relation induced by a splitting F by F  Denition  Given a set of open subsets G  we dene the relation  G on T with V  G V if and only if V   U  V  U for all U G  We have the following Proposition The relation  G is an equivalence Proposition  Given a nite splitting F  F F ie the remainders of F are the equivalence classes of  F  Proof Suppose V  F V then V  V Rem FU  where U   f U  j V  V  U U  F g Chapter  A Semantical analysis of MP   For the other way suppose V  V Rem FU and that there exists U  F such that V   U  while V  U  Then we have that V   U  U  U  U F and U  U  U ie V   Rem FU  We state some useful facts about splittings Proposition  If G is a nite set of opens then ClG  its closure under intersec tion yields a nite splitting for G  The last proposition enables us to give yet another characterization of remainders every family of points in a complete lattice closed under arbitrary joins comprises a closure system ie a set of xed points of a closure operator of the lattice cf   Here the lattice is the poset of the opens of the topological space If we restrict ourselves to a nite number of xed points then we just ask for a nite set of opens closed under intersection ie Proposition  Thus a closure operator in the lattice of the open subsets of a topological space induces an equivalence relation two opens being equivalent if they have the same closure and the equivalence classes of this relation are just the remainders of the open subsets which are xed points of the closure operator The maximum open in RemFU  ie U  can be taken as the representative of the equivalence class which is the union of all open sets belonging to RemFU  We now introduce the notion of stability corresponding to what we mean by a formula retains its truth value on a set of opens Chapter  A Semantical analysis of MP   Denition  If G is a set of opens then G is stable for  if for all x either x V j for all V G  or x V j for all V G  such that x V  Proposition  If G G are sets of opens then a if G   G and G is stable for then G  is stable for b if G  is stable for and G is stable for  then G   G is stable for  Proof a is easy to see while b is a corollary of a Denition  A nite splitting F  fU  Ung is called a stable splitting for  if RemFUi is stable for for all Ui F  Proposition  If F  fU  Ung is a stable splitting for so is F   ClfU U  Ung

where U F  Proof Let V F  then there exists Ul F such that Rem F  V  RemFUl eg Ul  T fUijUi F V  Uig ie F  is a re	nement of F  But RemFUl is stable for and so is RemF   V by Proposition a The above proposition tells us that if there is a nite stable splitting for a topology then there is a closure operator with nitely many xed points whose associated equivalence classes are stable sets of open subsets Chapter  A Semantical analysis of MP   Suppose thatM  hX T ii is a topological model for L  LetFM be a family of subsets of X generated as follows iA FM for all A A if S FM then XS FM  if S T FM then S  T FM  and if S FM then S FM ie FM is the least set containing fiAjA Ag and closed under complements intersections and interiors Let F  M be the set fSjS FMg We have F M  FM  T  The following is the main theorem of this section Theorem  Partition Theorem Let M  hX T ii be a topological model Then there exists a a set fF gL of nite stable splittings such that  F   F  M and X F  for all L  if U F  then U  fx U jx U jg FM and

 if is a subformula of then F  F  and F  is a nite stable splitting for where FM F M as above Proof By induction on the structure of the formula  In each step we take care to rene the partition of the induction hypothesis  If  A is an atomic formula then FA  fX g  fi ig since T is stable for all atomic formulae We also haveFA  F  M and XA  iA FM  Chapter  A Semantical analysis of MP 

 If   then let F   F  since the statement of the proposition is sym

metric with respect to negation We also have that for an arbitrary U F  U  U   If    let F   ClF  F  Now F  is a nite stable splitting containing X by induction hypothesis Observe that F  F  F   Now if Wi F  then there exists Uj F  and Vk F  such that Wi  Uj  Vk and Rem FWi  Rem FUj  Rem FVk eg Uj  T fUmjWi  Um Um F g and Vk  T fVnjWi  Vn Vn F g Since RemF  Uj is stable for  and Rem F  Vn is stable for  their intersection is stable for    by Proposition b and so is its subset RemF  Wi by Proposition a Thus F  is a nite stable splitting for containing X We have that F   FM whenever F   FM and F   F M  Finally W  i  U  j  V  k   Suppose  K Then by induction hypothesis there exists a nite stable splitting F   fU  Ung for containing X Let Wi  U  i  

Chapter  A Semantical analysis of MP   for all i f ng Observe that if x Ui Wi then x V j for all V Rem FUi and x V  since RemF  Ui is stable for  by induction hypothesis Now if V RemF  Ui Wi for some i f ng then x V j for all x V  by denition of Wi hence x V jK for all x V  On the other hand if V RemF  Ui  Wi then there exists x V such that x V j otherwise V  Wi  Thus we have x V jK for all x V  Hence Rem FUi  Wi and Rem FUi  Wi are stable for K Thus the set F  fRemFUij Wi  Rem FUig  fRem FUj Wj Rem FUj  Wjj Wj Ujg is a partition of T and its members are stable for K Let F be the equivalence relation on T induced by F and let F K  ClF   f Wi j Wi Rem FUig We have that FK is a nite set of opens and F   FK Thus FK is nite and contains X We have only to prove that FK is a stable splitting for K ie every remainder of an open in FK is stable for K If V  F V where V  V T  then there exists U  Ui or Wi for some i   n such that V   U while V  U  But this implies that V   FK V Therefore fRemF K Ug UFK is a renement of F and FK is a nite stable splitting for K using Proposition a Chapter  A Semantical analysis of MP   We have that FK F  M because Wi F M  for i   n Now if U F  then either UK  U or UK    Suppose    Then by induction hypothesis there exists a nite stable splitting F   fU  Ung for containing X Let F    ClF   fUi  Ujj   i j  ng

where is the implication of the complete Heyting algebra T ie V  U W if and only if V  U  W for V U W T  We have that U  W equals XUW  ClearlyF  is a nite splitting containing X and F   F  We have only to prove that F  is stable for   But rst we prove the following claim Claim  Suppose U F  and U  F  Then U   U RemF  U  V  U RemF  U for all V RemF   U  Proof The one direction is straightforward For the other let V RemF   U  and suppose V U  RemF  U towards a contradiction This implies that there exists U  F  with U   U  such that V  U  U  Thus V  U  U  but U   U  U  But U  U  F  which contradicts U  F  V  by Proposition  Chapter  A Semantical analysis of MP   Let U  F  We must prove that RemF   U  is stable for   Suppose that x U j  We must prove that x V j  for all V  RemF   U  such that x V  Since x U j  there exists V T  with x V and V  U  such that x V j SinceF  is a splitting there exists U F  such that V RemF  U  Observe that V  U   U  U  so U   U RemF  U  by Proposition c By Claim  for all V  RemF   U  we have V U RemF  U  Thus if x V  then x V   U j because RemF  U is stable for  by induction hypothesis This implies that for all V  such that V  RemF   U  and x V  we have x V j  Therefore F  is a nite stable splitting for   Now Ui  Uj F M for   i j  ng hence F    F  M  Finally let U belong to F  and V  Vm be all opens in F  such that U  Vi Rem FVi for i   m Then x U j if and only if there exists j f mg with x Vj and x Vjj because x Vj  U j since Vj  U Rem FVj  This implies that U	   U	  U  m  i  V  i Chapter  A Semantical analysis of MP   Since U V   V  belong to FM  so does U	  and therefore U   U  U	  In all steps of induction we rene the nite splitting so if is a subformula of then F   F  and F  is stable for using Proposition a Theorem  gives us a great deal of intuition for topological models It describes in detail the expressible part of the topolocical lattice for the completeness result as it appears in Chapter  and paves the road for the reduction of the theory of topological models to that of spatial lattices and the decidability result of this chapter   Basis Model Let T be a topology on a set X and B a basis for T  We denote satisfaction in the models hX T ii and hX B ii by j T and j B  respectively In the following proposition we prove that each equivalence class under F contains an element of a basis closed under nite unions Proposition  Let X T  be a topological space and let B be a basis for T closed under nite unions Let F be any nite subset of T  Then for all V F and all x V there is some U B with x U  V and U RemFV  Proof By niteness of F  let V  Vk be the elements of F such that V  Vi for i f kg Since Vi  V  take xi V  Vi for i f kg Since B is a Chapter  A Semantical analysis of MP   basis for T  there exist Ux Ui with x Ux and xi Ui such that Ux and Ui are subsets of V for i f kg Set U   k  i  Ui  Ux Observe that x U  and U B as it is a nite union of members of B Also U RemFV  since U V but U  S

Vi for i f kg Corollary  Let X T  be a topological space B a basis for T closed under nite unions x X and U B Then x U j T  x U j B Proof By induction on  The interesting case is when    Fix x U  and  By Proposition  there exists a nite stable splitting F for and its subformulae such that F contains X and U  Assume that x U j B   and V T such that V  U  By Proposition b there is some V   U inF with V RemFV  By Proposition  let W B be such that W RemFV  with x W  So x W j B  and thus by induction hypothesis x W j T  By stability twice x V j T as well We are now going to prove that a model based on a topological space T is equiv

alent to the one induced by any basis of T which is lattice Observe that this enables Chapter  A Semantical analysis of MP   us to reduce the theory of topological spaces to that of spatial lattices and therefore to answer the conjecture of   a completeness theorem for subset spaces which are lattices will extend to the smaller class of topological spaces Theorem  Let X T  be a topological space and B a basis for T closed under nite unions Let M   hX T ii and M  hX B ii be the corresponding models Then for all M j  Mj Proof It suces to prove that x U j T  for some U T  if and only if x U j B  for some U  B Suppose x U j T  where U T  then by Corollary  there exists U  B such that x U  and x U j T  By Corollary  x U j B  Suppose x U j B  where U B then x U j T  by Corollary   Finite Satisability Proposition  Let hX T i be a subset space Let F be a nite stable splitting for a formula and all its subformulae and assume that X F  Then for all U F all x U and all subformulae of x U j T i x U j F  Chapter  A Semantical analysis of MP   Proof The argument is by induction on  The only interesting case to consider is when    Suppose rst that x U j F   with U F  We must show that x U j T   also Let V T such that V  U " we must show that x V j T  By Proposition b there is some V   U in F with V RemFV  So x V j F  and by induction hypothesis x V j T  By stability x V j T also The other direction if x U j T   then x U j F   is an easy application of the induction hypothesis Constructing the quotient of T under F is not adequate for generating a nite model because there may still be an innite number of points It turns out that we only need a nite number of them Let M  hX T ii be a topological model and dene an equivalence relation  on X by x  y i a for all U T  x U i y U  and b for all atomic A x iA i y iA Further denote by x  the equivalence class of x and let X   fx  x Xg For every U T let U   fx  x Xg then T    fU  U T g is a topology on X  Dene a map i  from the atomic formulae to the powerset of X  by i A  fx  x iAg The entire model M lifts to the model M    hX  T   i i in a well dened way Chapter  A Semantical analysis of MP   Lemma  For all x U and x U j M i x  U j M  Proof By induction on  Theorem  If is satis	ed in any topological space then is satis	ed in a nite topological space Proof Let M  hX T ii be such that for some x U T  x U j M  Let F  be a nite stable splitting by Theorem  for and its subformulae with respect to M  By Proposition  x U j N  where N  hX F ii We may assume that F is a topology and we may also assume that the overall language has only the nitely many atomic symbols which occur in  Then the relation  has only nitely many classes So the model N   is nite Finally by Lemma  x  U j N   Observe that the nite topological space is a quotient of the initial one under two equivalences The one equivalence is  F on the open subsets of the topological space where F  is the nite splitting corresponding to and its cardinality is a function of the complexity of  The other equivalence is X on the points of the topological space and its number of equivalence classes is a function of the atomic formulae appearing in  The following simple example shows how a topology is formed with the quotient under these two equivalences Chapter  A Semantical analysis of MP   Example Let X be the interval   of real line with the the set T  fg  f 

n  j n    g as topology Suppose that we have only one atomic formula call it A such that iA  fg then it is easy to see that the model hX T ii is equivalent to the nite topological model hX  T   i i where X   f x  x g

T    f fx  xg g and iA  f x  g So the overall size of the nite topological space is bounded by a function of the complexity of  Thus if we want to test if a given formula is invalid we have a nite number of nite topological spaces where we have to test its validity Thus we have the following Theorem  The theory of topological spaces is decidable Observe that the last two results apply for lattices of subsets by Theorem  Chapter  Completeness for MP  Open subsets of a topological space were used in   and in the previous section to provide motivation intuition and nally semantics for MP  But in this chapter we shall show that the canonical model of MP  is actually a set of subsets closed under arbitrary intersection and nite union ie the closed subsets of a topological space However these results are not contrary to our intuition for the following reasons the spatial character of this logic remains untouched The fact that the canonical model is closed under arbitrary intersections implies strong completeness with the much wider class of sets of subsets closed under nite intersection and nite union Now the results of the previous section allow us to deduce strong completeness in the sense that a consistent set of formulae is simultaneously satisable in some model also for the class of sets of subsets closed under innite union and nite intersection ie the  Chapter  Completeness for MP   open subsets of a topological space  Subset frames As we noted in section  we are not interpreting formulae directly over a subset space but rather in the pointed product X O The pointed product can be turned in a set of possible worlds of a frame We have only to indicate what the accessibility relations are Denition  Let X O be a subset space Its subset frame is the frame hX O R  RKi

where x UR y V  if U  V and x URKy V  if x  y and V  U If O is a topology intended as the closed subsets of a topological space we shall call its subset frame closed topological frame Our aim is to prove the most important properties of such a frame We propose the following conditions on a possible worlds frameF  hS R  Ri with two accessibility relations Chapter  Completeness for MP    R  is reexive and transitive

 R is an equivalence relation  R R  RR   ending points F has ending points with respect R  ie for all s S there exists s S such that for all s S if sR s then sR s  extensionality condition For all s s S if there exists s S such that sR s and sR s and for all t S such that tRs there exist t t S such that tRs tR t and tR t and for all t S such that tRs there exist t t S such that tRs t R t and tR t then s  s  union condition For all s  s S if there exists s S such that sRR s  and sRR s then there exists s S such that for all t S with tRs  then tR Rs  or tR Rs  intersection condition For all fsigiI  S Chapter  Completeness for MP   if there exists s S such that siR s for all i I then there exists s S such that for all ftig  S with tiRsi and tiR t for all i I and some t S then tiR Rs  The frame F is strongly generated in the sense that there exists s S such that for all s S sRR s We have the following observations to make about the above conditions Con

ditions  to  and  are rst order while the intersection condition is not The extensionality condition implies the following for all s s S such that sR s and sRs then s  s which implies that R R is the identity in S In view of the extensionality condition the relation R  is antisymmetric So we can replace condition  with the condition that R  is a partial order Now we have the following proposition Proposition  If X T  is a topological space then its closed topological frameFT satis	es conditions  through  Proof Let R   R  and R  RK Conditions    are straightforward For each x V  X T the pair x

T xU U is its ending point with respect R  and condition  is satised The extensionality condition represents the set theoretic Chapter  Completeness for MP   extensionality of the space The union and intersection condition is satised because T is closed under nite unions and innite intersections respectively Finally FT is strongly generated by x X for any x X The above proposition could lead to the consequence that topological models are possible worlds models in disguise But the following theorem shows that this is not the case There is a duality Theorem  LetF  hS R  Ri be a frame satisfying conditions  through  Then F is isomorphic to a closed topological frame FT  Proof We shall construct a topological space X T  and a frame isomorphism f from F to FT  Let X  f s j s S is an ending point of F g and T  f Ut j t S g  fg We also let s Ut if there exists s  such that sR s and s Rt Note that using conditions    we can show that if s Ut implies s Ut  then Ut  Ut  Chapter  Completeness for MP   and by the extensionality condition Ut  Ut  if and only if tRt  Therefore the above settings are well dened It only remains to show that T is closed under innite intersections and nite unions For the former we must show that T iI Uti belongs to T  for Uti T i I If T iI Uti  we are done If not then there exists s Uti for all i I This by denition implies that there exist fsigiI such that siRti and siR s Now intersection condition applies and let s be as in condition  We shall show T iI Uti  Us   For the left to right subset direction let r T iI Uti This implies by denition that there exist frigiI  S such that riRsi thus riRti and riR r for all i I By the intersection condition riR Rs and therefore r Us   For the other subset direction let r Us   Then there exists r S such that rR r and rRs Condition  implies that there exist frigiI  S such that riRti and riR r thus r T iI Uti for all i I Therefore T iI Uti  U  s T  We can prove similarly that Ut  Ut  U  s s  as in the union condition using the union condition and condition  Let f be the map from S to X T dened in the following way fs  s Us  where s is the ending point of s in F  Chapter  Completeness for MP   The map f is a frame isomorphism If s Ut X T then there exists s S such that sR s and sRt We have fs  s Ut and f is onto Let fs  fr  s Ut for some s s r t S We have that sR s rR s sRt and rRt By extensionality property s  r and f is bijective Now observe that tR s if and only if Us  Ut if and only if t UtR t Us if and only if ftR fs

where t is the common ending point of t and s in F  We have also tRs if and only if Ut  Us if and only if t UtRKs Us if and only if ftRKfs Therefore f preserves the accessibility relations in both directions and is a frame isomorphism Note that in the above denitions we could have used equally well the equivalence class of s S under the equivalence induced by the symmetric closure of R  instead of the ending point of s in F  The above proofs show that the crucial conditions are conditions  through  and if we are to strengthen or relax the union and intersection conditions we get accordingly di	erent conditions in the lattice of the set of subsets of the space The same holds for condition  We only used this condition to show Chapter  Completeness for MP   that there exists a top element ie the whole space and satisfy the hypothesis of the union condition If we do not assume this condition the union of two subsets will belong to the set of subsets if they have an upper bound in it We state this case formally without a proof because we are going to use it later Proposition   Let X O be a subset space closed under in	nite intersections and if U V O have an upper bound in O then U  V O Then its frame FO satis	es conditions  through   A frameF satisfying conditions  through  is isomorphic to a frame FO where X O as in    On the proof theory of MP We shall identify certain classes of formulae in L  This approach is motivated by the results of Chapter  In fact these formulae express denable parts of the lattice of subsets see section  Denition  LetL   L be the set of formulae generated by the following rules A  L  L  L  L   K L  Let L  be the set fK Lj L g Formulae in L  have the following properties Chapter  Completeness for MP   Denition  A formula ofL is called persistent whenever    is a theorem see also   A formula of L is called antipersistent whenever  is persistent ie     or equivalently   is a theorem A formula of L is called bipersistent whenever       or equivalently    is a theorem Thus the truth of bi persistent formulae depends only on the choice of the point of the space while the satisfaction of persistent formulae can change at most once in any model We could go on and dene a hierarchy of sets of formulae where each member of hierarchy contains all formulae which their satisfaction could change at most n times in all models All the following derivations are in MP  Axioms  through  # see table at page  Proposition  All formulae belonging to L  are bipersistent Proof We prove it by induction ie bi persistence is retained through the appli

cation of the formation rules of L   If A is atomic then A is bi persistent because of axiom   If   then is bi persistent by induction hypothesis IH and the fact that bi persistence is a symmetric property with respect negation Chapter  Completeness for MP    if  K then we have the following  K  K  by IH

  K    K by Axiom     K   K  K   K by   therefore is bi persistent  If   then we have       by IH

        by IH           in S                 by   Chapter  Completeness for MP   A faster semantical proof would be the initial assignment on atomic formulae extends to the wider class of L $ This implies that formulae in L  dene subsets of the topological space Formulae in L  have similar properties as the following lemma show Lemma  If is bipersistent then K is persistent and L is antipersistent Proof  L L by Axiom 

L L by bi persistence of  Similarly MP K  K We prove some theorems of MP  that we are going to use later Lemma  If is bipersistent then MP      Proof The one implication is straightforward by normality For the other        by bi persistence of

     in a normal system The following is the key lemma to the DNF Theorem and generalizes Axiom  Lemma  For all n MP K

Chapter  Completeness for MP   where i are bipersistent Proof By induction on n For n   MP K LK   K L

follows by Axiom  and bi persistence of and  Suppose that the lemma is true for n  k For n  k !   K LK   LK k LK k   K L  Lk LK k  by IH

K L  Lk LK k   K L  Lk Lk  by Axiom    K L  Lk Lk   K L  Lk Lk  by Lemma   K LK   LK k LK k   K L  Lk Lk  by   All formulae of L  can be expressed in terms of bi persistent persistent and antipersistent formulae by means of the following normal form Chapter  Completeness for MP   Denition   is in prime normal form PNF if it has the form K

n i  Li where  i L  and n is nite

 is in disjunctive normal form DNF if it has the form Wm i  i where each i is in PNF and m is nite To keep the notation bearable we shall omit the cardinality of nite conjunctions and disjunctions writing eg W i i instead of Wn i  i Suppose that is a formula in the following form  i  i  Li  j Kji A

where i i j i L  We shall call such a form conjunctive normal form CNF Using the distributive laws we get the equivalent formula  k   lk ilk

 mk Limk

 nk K jnk ink A Since L  is closed under negation and conjunction and K distributes over conjunc

tions we can express the above formula in the following form  k  k K  k

 mk Lmkk 

where k k  mk k belong to L  So is equivalent to this formula which is in DNF Therefore DNF and CNF are e	ectively interchangeable up to equivalence Chapter  Completeness for MP   We now give the formal analogue of the Partition Theorem Theorem  DNF For every L there is eectively a in DNF such that MP  Proof By induction on the logical structure of   If  A where A is atomic the result is immediate because the set of atomic formulae belongs to L  and A is in PNF  Suppose that   Then by induction hypothesis is equivalent to a formula in DNF which implies that is equivalent to a formula in CNF and by the above discussion is equivalent to a formula in DNF  If    then is equivalent to a disjunction of two formulae in DNF ie is itself in DNF  If  K then is equivalent to a formula in CNF and hence is equivalent to a formula of the following form  i K  i  Li  j K j i A

since K distributes over conjunctions Now  since the formula K  K K  K is a theorem of S	 the above formula is equivalent to  i  Li   Ki  j Kji A A

Chapter  Completeness for MP   which is in CNF  If   then by induction hypothesis is equivalent to a formula of the form   i  i Ki  j L j i A Since  distributes over disjunctions in every normal system the above formula is equivalent to  i   i Ki  j Lji A By Lemma   it is equivalent to  i  i   Ki  j L j i A A  Using theorems of S for   and S for K and Lemma  formula  implies  i  i Ki  j L

Ki  j i A   By Lemma  formula implies formula  and hence they are equivalent Ob

serve now that Ki belongs to L  and since L  is closed under conjunctions the last formula is in DNF This is the only step of the proof which makes use of Axiom   Thus is equivalent to a formula in DNF This completes the proof The DNF theorem is the most important property of MP  An immediate corol

lary is that as far as MP  is concerned we could have replaced the   modality with Chapter  Completeness for MP   K since the formulae in normal form are dened using these two modalities Almost all subsequent proof theoretic properties are immediate or implicit corollaries of the DNF Theorem We close this section with the following proposition which together with Axiom  shows that   is equivalent to   Proposition  For all L MP     Proof Since see  p S  f         g     

we have only to show that MP     

where is in prime normal form For that consider the following derivation in MP      K

Vn i  Li by Lemma  

Vn i  Li

Chapter  Completeness for MP    K

Vn i  Li and  Li are bi persistent    K

Vn i  Li  Canonical Model The canonical model of MP  is the structure C  S fR  RKg v

where S  fs  L js is MP  maximal consistentg

sR t i f L j  sg  t

sRKt i f L jK sg  t

vA  fs SjA Sg

Chapter  Completeness for MP   along with the usual satisfaction relation dened inductively sj C A i s vA s  j C  sj C  i s  j C sj C i sj C and sj C sj C   i for all t S sR t implies tjC sj C K i for all t S sRKt implies tjC We write C j if sj C for all s S A canonical model exists for all consistent bimodal systems with the normal axiom scheme for each modality as MP and MP  We have the following well known theorems see  or  Theorem  Truth Theorem For all s S and L sj C i s Theorem  Completeness Theorem For all L C j i MP We shall now prove some properties of the members of C  The DNF theorem implies that every maximal consistent theory s of MP  is determined by the for

mulae in L  and L  it contains ie by s  L  and s  L  Moreover the set Chapter  Completeness for MP   fK LjK L sg is determined by s L  alone this is the K case of the DNF theorem The following denition is useful Denition  Let P  L  We say P is an L  theory if P is consistent and for all L  either P or  P  Let S  L  We say S is an L  theory if S is consistent and for all L  either S or  S Hence s L  is an L  theory and s L  is an L  theory What about going in the other direction When does an L  theory and L  theory determine an MP  maximal consistent theory When their union is consistent because in this case there is a unique maximal extension To test consistency we have the following lemma Lemma  If P and S are an L  and L  theory respectively then P S is consistent if and only if if P then L S Proof Suppose that PS is not consistent then there exists P and fLigni   S such that MP n i  Li  

Chapter  Completeness for MP   which implies since K distributes over conjunctions MP n i  Li  K Therefore L S and L  S The other direction is straightforward because  L It is expected that since L  and L  theories determine MP  maximal consistent sets they will determine their accessibility relations as well Proposition  For all s t S a sR t if and only if i t if and only if s where L 

ii if L t then L s where L  b sRKt if and only if K t if and only if K s where L  Proof For a right to left let t then by the DNF Theorem has the form  i  i Ki  j Lji A

where   jk L  Then  has the form   i  i Ki  j L j i A

which is equivalent to  i  i Ki  j L

Chapter  Completeness for MP   as in the proof of the DNF Theorem Observe here that in the case where L  if K t then K t which implies that K s Thus by ai and aii  s Therefore sR t For the other direction ai is straightforward using the bi persistence of  For aii if L t then L s and use Lemma  to show that L s For b right to left we proceed as above Let K t then by the DNF Theorem it has the following form  i  Li  j K j i A

where i  j i L  Thus K s The other direction is straightforward by the denition of RK From the above proposition we have that for all s t S if sR t then s L   t L  and if sRKt then s L   t L  We write RKR  for the composition of the relation RK and R  ie if s t S we write sRKR t if there exists r S such that sRKr and rR t Similarly for R RK For the composite relation RKR  and R RK we have the following corollary of proposition  which we stay here without proof Corollary  For all s t S a sR RKt if and only if i if s then L t where L 

ii if L t then L s where L  b sRKR t if and only if if L t then L s where L  Chapter  Completeness for MP   We shall now prove that the canonical model C of MP  satises the conditions of Section  on page  Proposition  The relation R  is reexive and transitive Proof Holds in every system containing S Proposition  The relation RK is an equivalence relation Proof Holds in every system containing S	 Lemma  For all s t S if sR RKt then sRKR t Proof See   It is immediate using Axiom  The relation R  has ending points as shown in the following proposition Proposition  For each s S there exists s S with sR s such that for all s S if sR s then sR s Proof Let A  s L 

B  fLj L   L sg

C  fKj L  K sg Chapter  Completeness for MP   Now the set T  B  C is an L  theory T is consistent If not then there exist L  Ln B and K C such that MP L  Ln  K

and thus MP  L   Ln   K But the formula at the left of the implication belongs to s Therefore  K s so K s a contradiction Now for L  either L T or L T  Observe that if A then  L s and therefore L T  by denition So AT has a unique maximal extension by Lemma  call it s For all s S s included such that sR s we have that sL   sL   sL  and if L s then  L s so L s Thus sR s using proposition  Therefore s is the ending point of s The above proposition implies that s and s have a common ending point if and only if s  L   s  L  The one direction comes from proposition  while the other from the proof of the above proposition because the construction of the ending point of a maximal consistent theory s depends solely on s L  Proposition  The canonical frame of MP  satis	es the extensionality condition of Section  Chapter  Completeness for MP   Proof We have to prove that for all s s S if there exists s S such that sR s and sR s and for all t S such that tRKs there exist t  t S such that tRKs  tR t and tR t and for all t S such that tRKs  there exist t t S such that tRKs t R t and tR t then s  s Since s and s have a common child s L   s L  We have only to show that s L   s L  For that suppose that L s with L  then there exists t S such that tRKs and t By the hypothesis of the condition there exist t t S such that tRKs  tR t and tR t This implies that t  L   t L  so t and L s the other direction is similar Therefore s  s Proposition  The canonical frame of MP  satis	es the union condition of sec tion  Proof We have to show that for all s  s S if there exists s S such that sRKR s  and sRKR s then there exists s S such that for all t S with tRKs  then tR RKs  or tR RKs Let A  f K j K s   s L  g Chapter  Completeness for MP   and B  f L j L s   s L  g We shall show that T  AB is an L  theory It is clear that for all L  either L or L belongs to T  Suppose T is consistent If not there exist K A and fLigni   B such that MP   K

which implies MP   K

n i  LK i  Since fLigni   B there exist ftig n i   S such that either tiRKs  or tiRKs and Li ti for all i f ng Since K A we also have that K ti for all i f ng In particular K ti and LK  ti Now choose i f ng Since sRKR s  and sRKR s there exists tR ti and tRKR s  and tRKR s Therefore K t and Li t by proposition  a and corollary   Therefore  K

n i  LK i  t which is a contradiction This proves that T is an L  theory Observe here that we could have dened T for an innite number of si%s for an innite version of the union condition and still get an L  theory It would be the Chapter  Completeness for MP   rest of this proof that would not work If it did then the canonical model would have satised an innite version of the union condition Now the required s of the condition is any maximal extension of T  Suppose neither sR RKs  nor s R RKs Then by denition and corollary   it must be the case that there exist    s L  such that L   s  L  and L  s L  But then we have that for     s  L  L  s  and L  s but L T which is a contradiction Similarly for any t S such that tRKs  because T  t and the rest of the condition is satised Proposition  The canonical frame of MP  satis	es the intersection condition of Section  Proof We have to show that for all si S i I if there exists s S such that siR s for all i I then there exists s S such that for all ftig  S with tiRKsi and tiR t for all i I and some t S then tiR RKs  Let ftjigiIjJ all subsets of S such that for all j J  tjiRKsi and there exists t j  S such that t j iR t j  This class is not empty since fsigiI Let A  f K j   iIjJ t j i L  g Chapter  Completeness for MP   and B  f K j  iIjJ t j i L  g We shall show that T  A  B is an L  theory It is clear that either L or L belongs to T  Suppose T is consistent If not there exist K A and fLkgnk   B such that MP   K

which implies MP   K

n k  LK k  Each tji contains K because K t j  To see that suppose K  t j  Then there exists t S such that tRKt j  and  t Lemma  implies that there exist ftigiI  S such that tiR t and tiRKsi But  ti for all i I hence L T  a contradiction Now for each k   k  n choose tjki such that k t jk i  The choice of i is irrelevant since tji contain the same formulae in L  for all i I We now have K k t jk i and therefore  K

n k  LK k  si Therefore t is an L  theory Chapter  Completeness for MP   Now let s be any maximal extension of T  If tji L  then L T and thus L s and if L s L  then L tji  By Corollary   we have that t j iR RKs  for all i I and j J  Therefore the intersection condition is satised Corollary The canonical frame of MP  is isomorphic to a subset frame FOc where Xc Oc is a subset space closed under in	nite intersections and if U V Oc have an upper bound in Oc then U  V Oc Proof By Proposition  and Propositions  through  By the construction of Theorem  Xc consists of the ending points of the mem

bers of the domain of the canonical model We dene the following initial assignment ic iA  f s j A s g In this way the model M  hXc Oc ici is equivalent to the canonical model as a corollary of frame isomorphism Corollary  For all s S and L we have s if and only if s UsjM Denition  A subset X of S the domain of the canonical model C  is called K closed whenever if s X and sR t or sRKt then t X Chapter  Completeness for MP   The intersection of K  closed sets is still closed therefore we can dene the smallest K  closed containing t for all t S We shall denote this set by St For t S we dene the model C t 

St Rt  

Rt K

vt

where Rt    R  jSt and R t K jSt  ie the restrictions of R  and RK to S t We shall call this model the submodel of C generated by t Proposition  The frame of a submodel C t is isomorphic to a closed topological frame Proof Observe that since the domain is K  closed the frame is strongly generated The rest of the conditions are inherited from the canonical frame Now the proposition follows from Theorem  Now as above we have the following corollary Corollary  A submodel C t is equivalent to a closed topological model It is a well known fact that a modal system is characterized by the class of gen

erated frames of the canonical frame Proposition The system MP  is strongly characterized by closed topological frames Chapter  Completeness for MP   Since the axioms and rules of MP  are sound for the wider class of subset spaces with nite union and intersection we also have the following Proposition  The systemMP  is strongly characterized by subset frames closed under nite unions and intersections Now by Proposition  and  Corollary  and Theorem  of Chapter  where we proved the equivalence of a topological model with the model induced by a basis closed under nite unions we have the following corollary Corollary The systemMP  is strongly characterized by open topological frames as well as subset frames closed under in	nite unions and intersections  Joint models In this section we are going to prove that the canonical model is strongly generated in the sense that there is a world in it which access every other using the relation RKR  This translates to the fact that the canonical model as a set of closed subsets has a greatest element a universe ie it represents a topological space The usual way to proceed in this case see  is to prove a rule of disjunction but the question is which one In uni modal logics we use the primitive modality which determines the accessibility relation Here   determines the partial order but not surprisingly we must also use K It turns out that we do not need such a rule in full generality but Chapter  Completeness for MP   only with respect to bi persistent formulae What we want to prove is the following rule if MP K  K   Kn then MP i for some i   i  n

for    n L  Note that the disjunction rule does not hold for S	 In the following we shall assume that X  X Xn are disjoint This is with

out any loss of generality since we can always replace a topological model with an equivalent one using a distinct but homeomorphic topological space Denition  Let hX  T  v i hX T vi hXn Tn vni be a nite number of topological models Their joint model is hX T vi where X  n  i  Xi

T is the topology generated by the subbasis n  i  Ti and vA  n  i  viA for each atomic formula A As this construction was dened it brings us from topological models to topo

logical models and the accessibility relations between points and subsets in the old models are transferred to the new one We only add more by adding more subsets Observe that the truth assignments for the atomic formulae remain the same and that extends to bi persistent formulae Chapter  Completeness for MP   Proposition MP  provides the above rule of disjunction Proof By contradiction Suppose that none of    n is a theorem of MP  Since topological models characterize the system there are hX  T  v i hX T vi hXn Tn vni and x  x xn belonging to X  X Xn respectively such that xi  vii for   i  n Let hX T vi be the joint model of hX  T  v i hX T vi hXn Tn vni Then we have xi  vi and therefore x XjKi for all x X and   i  n Therefore K   K   Kn is not a theorem of MP  We can similarly prove a stronger disjunction property namely if MP K K   K   Kn then MP K i for some i   i  n

for    n L  Now we are able to prove the following Theorem  The canonical model of MP  is strongly generated Proof Let T  fKj MP L g  fLj MP  L g Chapter  Completeness for MP   The set of formulae T is an L  theory For consistency suppose that MP  L  L Ln

for some L  L Ln T  This implies that MP K   K   Kn

and because i L  for   i  n we can use the rule of disjunction and get MP i for some   i  n

which is a contradiction Now for any member of the canonical model s let S  T  fj s L g

ie T plus the bi persistent formulae of s The set S is consistent If MP L  L Ln  

where Li T for   i  n and s L  then MP L  L Ln  L But consistency of T implies that  is a theorem a contradiction Moreover S has a unique maximal extension by the DNF Theorem call it s So we have showed that for all s and t in the canonical model there exists s and t such that sR s and sRKt  This implies that in the canonical subset model the subset Us is the required universe Chapter  Completeness for MP   By Theorem  we complete the set of conditions of page  which turn the frame of the canonical model into a closed subset frame To summarize we have the fol

lowing corollary note that the canonical subset model is hXc Oc ici of Corollary  Corollary  The canonical subset model of MP  is a topological space Chapter  The Algebras of MP and MP  In this section we shall give a more general type of semantics for MP and MP  Every modal logic can be interpreted in an algebraic framework An algebraic model is nothing else but a valuation of the propositional variables in a class of appropriately chosen algebras We shall also make connections with the previous chapters  Fixed Monadic Algebras Interior operators were introduced by McKinsey and Tarski   Denition  An interior operator I on a Boolean algebra B  hB

  i is  Chapter  The Algebras of MP and MP   an operator satisfying the conditions Ia  b  Ia  Ib

Ia  a

IIa  Ia

I   To each interior operator I we associate its dual C  I the closure operator which satises Ca  b  Ca  Cb

a  Ca

CCa  Ca

C  Universal quantiers were introduced by P Halmos  Denition  A universal quanti	er on a Boolean algebra B is an operator satisfying the conditions a  b  a b

   To each universal quantier we associate its dual   the existential Chapter  The Algebras of MP and MP   quanti	er which satises a  b  a  b

a  a

 Denition  Let I be an interior operator on a Boolean algebra B Let IB  faja  Iag and CB  fajCa  ag ie the xed points of I and C respectively Let BI  IB  CB then BI  hBI

   i is a Boolean subalgebra of B Denition  A xed monadic algebra FMA B is a Boolean algebra with two operators I and satisfying Ia Ia A valuation v on B is a function from the formulae of MP to the elements of B such that vA BI where A is atomic

vK  v An algebraic model of MP is a FMAB with a valuation v on it We say is valid in this model i v   and valid in an FMA i it is valid in all models based on Chapter  The Algebras of MP and MP   this algebra Finally is FMA valid if it is valid in all FMA%s The notion of validity can extend to a set of formulae Observe that the important part of the algebra is the smallest subalgebra contain

ing BI and closed under the operators I and  Theorem  Soundness for FMAvalidity If a formula is a theorem of MP then is FMAvalid Proof Let hB vi be an algebraic model We must prove that for all axioms  v   First observe that in a Boolean algebra v     is equivalent to v  v Take for instance K   K We have that Iv Iv implies vK   v K implies vK   K   We leave the rest of verications to the reader Similarly for rules Theorem  Completeness for FMAvalidity If is FMAvalid then is a theorem of MP Proof The proof is actually the Lindenbaum construction We dene the following equivalence relation on L  if and only if MP  Chapter  The Algebras of MP and MP   We denote the equivalence class of with  and dene the following partial order on the set B of equivalence classes    if and only if MP  All the required properties of an FMA follow from the axioms and rules of MP If we dene the valuation on B with v   then we have    if and only if MP   Generated Monadic Algebras We shall now dene the algebraic models of MP  Denition  A generated monadic algebra GMA B is an FMA satisfying in addition CIa  ICa Ca b  Ca c  CCa Cb  Cc The concepts of algebraic model validity GMA validity are dened as for FMA%s We used the direct algebraic translation of MP  axioms but we could have dened Chapter  The Algebras of MP and MP   it with a di	erent presentation Observe that we only need CIa  ICa because the other direction is derivable see Proposition  We now have the following Theorem  Algebraic completeness of MP  A formula is a theorem of MP  if and only if is GMAvalid Proof We omit the proof since it is similar to Theorems  and  It is known that a modal algebra determines a general frame see  So in our case the canonical algebraic model of MP  ie its Lindenbaum algebra must determine a closed topological model actually its canonical frame We shall state only the interesting part of this correspondence the bijection on the domains The accessibility relations are dened in the usual way Theorem

There is a bijection between the set of the ultra	lters of the canonical algebra ofMP  and the pointed productX T where X	T  is the canonical topology of MP  The general theory of modal logic provides for yet another construction A frame determines a modal algebra In case of the canonical frame the modal algebra deter

mined must be isomorphic to the canonical modal algebra In our case this algebra which must be a GMA has a nice representation It is the algebra of partitions of Chapter  The Algebras of MP and MP   the topological lattice as it appeared in Section  A full account of this result and detailed proofs will appear elsewhere Chapter  Further Directions There are several further directions  Due to the indeterminacy assumption see Introduction MP  can be a core logical system for reasoning about computation with approximation or uncer

tainty

 A discrete version of our epistemic framework can arise in scientic experiments or tests We acquire knowledge by a step by step process Each step being an experiment or test The outcome of such an experiment or test is unknown to us beforehand but after being known it restricts our attention to a smaller set of possibilities A sequence of experiments test or actions comprises a strategy of knowledge acquisition This model is in many respects similar to Hintikka%s oracle see  In Hintikka%s model the inquirer asks a series of  Chapter  Further Directions  questions to an external information source called oracle The oracle answers yes or no and the inquirer increases her knowledge by this piece of additional evidence This framework can be expressed by adding actions to the language Preliminary work of ours used quantales for modelling such processes A similar work without knowledge considerations appears in    Since we can express concepts like armative or refutative assertions which are closed under innite disjunctions and conjunctions respectively it is very natural to add innitary connectives or xed points operators the latter as a nite means to express the innitary connectives This would serve the purpose of specifying such properties of programs as emits an innite sequence of ones see  for a relevant discussion An interesting direction of linking topological spaces with programs can be found in    Our work in the algebras of MP  looks very promising GMAs see  have very interesting properties A subalgebra of a GMA corresponds to a complete space and this duality can be further investigated with the algebraic machinery of modal logic see   

  or category theoretic methods  Axiom  forces monotonicity in our systems If we drop this axiom an ap

plication of e	ort no longer implies a further increase in our knowledge Any change of our state of knowledge is possible A non monotonic version of the Chapter  Further Directions  systems presented in this thesis can be given along the lines of    It would be interesting to consider a framework of multiple agents Adding a modalityKi for each agent i and assigning a di	erent set of subsets or topology to to each agent we can study their interaction or communication by set theoretic or topological means  From our work became clear that both systems considered here are linked with intuitionistic logic We have embed intuitionistic logic to MP or MP  and it would be interesting to see how much of the expressiveness of these logics can be carried in an intuitionistic framework  Finally in another direction Rohit Parikh considers an enrichment of the lan

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