Completeness of a Firstorder Temporal Logic with TimeGaps  Matthias Baaza   Alexander Leitschb Richard Zachbc a Institut fur Algebra und Diskrete Mathematik E Technische Universitat Wien A Vienna Austria b Institut fur Computersprachen E Technische Universitat Wien A Vienna Austria c Group in Logic and the Methodology of Science University of California Berkeley CA

	  USA Abstract The rstorder temporal logics with   and   of time structures isomorphic to   discrete linear time and trees of  segments linear time with branching gaps and some of its fragments are compared The rst is not recursively axiomatizable For the second a cutfree complete sequent calculus is given and from this a resolution system is derived by the method of Maslov   Introduction In recent years various temporal logics have been studied and applied to the de scription and analysis of dynamic properties of programs  The investigations have focussed on discrete linearly ordered wellfounded temporal structures be cause temporal states can then be identied with program states It turns out that the rstorder logics corresponding to this semantics are not recursively axiomati zable if   henceforth always and   nexttime are present in the language It is possible to characterize the set of natural numbers by    U x  where U x holds for exactly one domain element at each state and is determined by a recursion in   see   This incompleteness result is based on a standard model of linear time if similarity types are allowed  one can obtain completeness results for rst order temporal logic relative to classes of models of linear time see   With a change in the semantics branching time gaps  however a complete rstorder logic can be obtained this is the subject of the present paper Our proof of completeness can be carried over to several types of futureoriented temporal operators see 

there may be problems however if future and pastoriented operators are present simultaneously For simplicity we consider here only languages with   and   as the only tem poral operators and constants as the only function symbols We compare the logic of discrete linear time TL to the logic of discrete linear time with branching time gaps TB In both logics the semantics of the temporal operators are as usual a formula  A is true at a time point t i A is true at every time point  t a formula  A is true at t i A is true at t   The dierence lies in the admitted time structures for TL this is the class of structures order isomorphic to  We call such a structure an segment In such a segment there is always an earliest point for every point there is a unique next point and every point can be reached from the earliest point by passing nitely often to the next point For TB the admitted   to appear in Theoretical Computer Science    Corresponding author Email addresses fbaazleitschg	logictuwienacat zach	mathberkeleyedu  structures are isomorphic to possibly innitary wellfounded trees of segments There is always a unique earliest next point in time but also points after the gap which cannot be reached by successively passing on to the next point which are initial states in the next segments themselves etc We give a sequent calculus for TB which is shown to be cutfree complete by an extension of Schuttes reduction tree method The rules of the calculus constructed are not analytic in the sense that the formulas in the premises are not proper sub formulas of the conclusion Therefore cutfree proofs in general lack the subformula property a property essential for usual methods of proof search The completeness proof shows however that we can salvage a large part of analyticity enough to be able to construct a resolution system for the logic Every valid sequent has a cutfree proof which uses only formulas A and  A where A is a subformula of the endsequent Exploiting this property we construct a complete resolution method for TB using the method of Maslov   In a sense then the investigations of TB can also be seen as a case study in a how far the completeness proof of Schutte can be carried and b how to over come mild forms of nonanalyticity It also sheds some light on necessary conditions for the resolution calculus to be sound completeness is not problematic  The paper is organized as follows In Section  the semantical structures un derlying the logics TL and TB are introduced and a proof of nonaxiomatizability of TL is sketched In Section  we present the sequent calculus LB for TB The completeness proof for LB is presented in Section  Section  contains some re marks comparing fragments of TL and TB The resolution system for TB is developed in Section  Finally we conclude with a discussion of the signicance of the completeness result for future applications  Firstorder Temporal Logics We consider the following rstorder language free variables a b c a     bound variables x y z x     constant symbols f  g h f     predicate symbols of arbitrary arity P  Q R P     propositional connectives      quantiers   and the temporal operators   always    next time  Formulas are built up from the symbols as usual The sometime operator  is introduced by denition A     A If A    n B where i is either   or   then   n is called the temporal prex of A The semantics of a rstorder temporal logic is dened as follows Denition  Let T be a denumerable partially ordered set T belongs to the class L of linear discrete orders i it is order isomorphic to  it belongs to the class B of linear discrete orders with branching gaps i it is order isomorphic to a wellfounded tree of segments Denition  Let T be L or B and let Frm	L be the set of formulas over some rstorder temporal language L A structure K for L is a tuple hT fDigiT  fSigiT i where T T  Di is a set called the domain at state i Di Dj if i j Si is a function mapping free variables and constant symbols to elements of Di and nary predicate symbols to functions from D n i to fg We dene the valuation functions Ki from Frm	L to fg as follows Let A be a temporal formula and not and or impl be the truth functions for negation conjunction disjunction and implication respectively  A  P t      tn Ki	A  Si	P

  A   B Ki	A  not	Ki	B

 A  B C Ki	A  and	Ki	B Ki	C

  A  B C Ki	A  or	Ki	B Ki	C

 A  B  C Ki	A  impl	Ki	B Ki	C

 A  x B	x Ki	A   if Kidx	A	d

  for every d Di and   otherwise  A  x B	x Ki	A   if Kidx	A	d

  for some d Di and   otherwise A   B Ki	A   if Kj	B   for every j  i and   otherwise  A   B Ki	A   if Ki 	B   and   otherwise A formulaA is satised in a temporal structure KK j A iK	A  A is valid in a class of temporal structures T  T j A i every K  hT fDigiT  fSigiT i with T T satises A Denition  The logic of linear discrete time TL is the set of all formulas A Frm	L st L j A The logic of linear discrete time with branching gaps TB is the set of all formulas A Frm	L st B j A Example  In TL the formula   A    A is valid In TB however only   A    A holds The other direction   A    A does not hold in general as can be seen by evaluating the formula on the countermodel K  h   fDigi fSigii where S	A   and Si	A   for i   i   The semantics considered here is usually called initial semantics Normal se mantics is dened via truth in all states not only in K We will need the following lemma later on Lemma  Let A be a formula  j A i A is true in every world in every temporal structure  j A i j  A 	 j A i j  A Proof  If trivial Only if Let T be a temporal structure in which A is not true at a state i Consider T   fj T j j  ig T  is also a temporal structure and since our logics contain no operators acting backwards in time A is true at state i in T  if it was true in state i in T  But i is the initial state in T   If by the truth denition of   Only if immediate by 

 If Let T be a structure where A is false in the initial world Consider T   T   with    and S   S The addition of a state before the initial state does not change the truth of formulas in T  But in T   A is false in the initial world Only if immediate by     Remark   The logics we consider dier from the ones in the literature in that we do not use global and local variables but the interpretation of predicate symbols can vary over the states This is more in keeping with the tradition in quanticational modal logics However by using the Barcan formulas for   and   denable two sortedness and other expressible concepts most eects of global and local variables can be simulated Another minor dierence is in the denition of   Krogers   is dened via truth in all later worlds in Krogers logic our   can be dened by  A A his   can be expressed by   A in TL As indicated in the introduction the logic TL is not axiomatizable This was shown for the original formulation of Kroger by Szalas  and Kroger  Two bi nary function symbols have to be present for the results to hold If the operator until is also present or if quantication over local variables is allowed then the empty  signature suces as was shown by Szalas and Holenderski  and Kroger  re spectively Following Szalas  and Kroger  we sketch a proof for the incompleteness re sult for TL with equality where the signature contains two binary function symbols equivalently two ternary predicate symbols Let  designate the successor function and the constant the number zero Consider the formula axiomatizing the predicate U  U  	x

  U x  U y  x  y   In every model U x represents exactly the set of natural numbers If the lan guage is expressive enough we can write down the usual axioms for addition and multiplication eg Robinsons Q  A sentence of arithmetic is true in the natu ral numbers i its relativization to U x follows in TL from these axioms The nonaxiomatizability of TL thus follows from Godels Incompleteness Theorems  A Sequent Calculus for TB In the standard denition a sequent is an expression of the form A      Ak  B      B where the Ai and Bj are rstorder formulas For the purpose of completeness proofs it is more convenient to use instead innite sequents see eg Takeutis book  Ch   More precisely the completeness theorem requires a generalization of nite sequences of formulas to countably innite wellordered sequences We will use this more general notion of sequents and indicate the use of nite sequents explicitly Let  be a countable possibly nite wellordered sequence If  is order iso morphic to the wellordered set of numbers via a mapping st i  Ai for i then we write   Ai i Denition  A sequent is an expression of the form   where and  are countable wellordered sequences of rstorder temporal formulas Denition  The sequence Ai i is called a subsequence of Ai i if and there exists an orderpreserving  mapping   If the sequences are nite and  f     ng then is of the form fi      ikg f     ng A sequent    is called a subsequent of   if  and  are subsequences of and  respectively Denition  Let Si i be a sequence of sequents st Si  i  i for i  Then the sequent S  i i  i i is called the union sequent of Si i Note that the order type of i i is characterized by the property if i  j and i j are the wellordered sets of numbers corresponding to i and j respectively then all elements of i are smaller than all elements of j The validity of nite sequents is dened as usual A      Ak  B      Bk is valid in TL TB i A     Ak  B     Bl is valid in TL TB  A nite sequent is provable if it has a derivation in a suitable calculus The concepts of provability dened for nite sequents originally can be ex tended to the innite case via the usual compactness condition Denition  A not necessarily nite sequent S is called provable if there exists a nite subsequent of S which is provable  It is only a matter of convention that we use the term provable for innite sequents as LB works only on nite sequents This convention is however of es sential advantage in completeness proofs In our completeness proof we do not need the semantics of innite sequents particularly we do not speak about semantic

compactness ie about the property that an innite sequent is valid i there exists a nite subsequent which is valid  As basis for the sequent calculus LB for TB we take a variant of Gentzens calculus LK for classical predicate logic The rules of LK are wellknown and can be found in eg  We use a weakening friendly formulation of the rules The side formulas in the premises of the rules right  left  and left are not required to be identical eg  A    B    A B instead of  A  B  A B LB consists of the rules of LK plus the following rules for   and   A  A    A    left  A    A    A  right        nex    A     A nec Note that LB like LK is dened for nite sequents only If is A      An then   denotes the sequence  A      An similarly for    The notations   and   can be extended to innite sequents in a straightforward way eg  	Ai i   Ai i  Note that unlike the rules of LK the rules  left and  right are not analytic ie the subformula property does not hold  The rule nex works on the left and right sides of the sequent simultaneously but is ana lytic and nec is context dependent It is clear that nec corresponds to the necessitation rule common in Hilbertstyle modal calculi When using rules with two auxiliary formulas in one premise ie right or  left

 the inference is admitted even if only one formula is actually present implicit weakening  Alter natively we could have split the rule into two in a similar way as the right and left rules Otherwise the notion of proof is the standard one cf  Ch x   In particular recall that initial sequents are of the form A  A A any formula

and cutfree provable means having a proof not containing an application of the cut rule The sequent appearing at the root of the proof tree is called endsequent Proposition  If a sequent is LBprovable then a nonempty subsequent is provable without weakenings Proof This is easily seen by induction on the length of the proof and is due to the special formulation of the rules   Example 	 We give an LBproof of the formula   A    A A A  A A  left   A  A nex A A  A A  left   A  A nex  A  A  left  A   A nec   A    A nex   A  A   A  right   A   A contrleft    A    A right  Note that on the right branch of the proof we introduced  A twice on the left hand side of a sequent This is necessary because of the way nex introduces   in all formulas of the sequent Theorem   LB is sound for TB ie every nite LBprovable sequent is valid in TB Proof It is sucient to prove the soundness of the LBrules The soundness of the LKpart is proved as usual The soundness of the rules  left and  right follows from the recursion equivalence of  A and   A  A in the TBsemantics The soundness of nex follows from Lemma 	 and from the fact that   distributes over the propositional connectives eg  	A B is equivalent to  A  B  The soundness of nec follows from Lemma 	  from the TBequivalence of  A and   A from the distributivity of   over  and from the fact that  	A  B implies  A   B   If we look closely at the rules of LB we notice that  left and  right are not strictly analytical Therefore it is convenient to extend the usual notion of subformula Note that we have disjoint sets of free and bound variables A term is dened as usual but subject to the restriction that it may only contain free variables if also bound variables are allowed to occur we speak about semiterms Similarly we distinguish between formulas and semiformulas The concept of strict subsemiformula represents the intuitive notion of subformula while the denition of semiformulas takes care about the nonanalytic behaviour of   and   Denition  Let F be a formula The set ssf	F of strict subsemiformulas of F is dened as ssf	F  fFg 	F  where 	F    fFg if F is atomic ssf	A if F  A for f   g ssf	A  ssf	B if F  A B for fg ssf	A	x

if F  Qx A	x for Q f g The set sub	F of subsemiformulas of F is dened by sub	F  ssf	F  f  A j  A ssf	F g By sub	F we denote the set of formulas obtained from sub	F by replacing bound variables without matching quantier in each member of sub	F by free variables or constant symbols ie we obtain actual subformulas corresponding to the semi formulas   Completeness of LB The main result of this paper is the following theorem Theorem  LB is complete for TB Every nite TBvalid sequent S has a cut free LBproof from atomic axioms The proof requires some additional denitions and technical lemmata In order to emphasize the main lines of the argument we give a rough sketch of the proof in advance The proof uses a variant of Schuttes method of reduction trees as modied for intuitionistic logic with Kripke semantics by  Ch  x  It proceeds by exhibiting a countermodel for any given unprovable sequent in the following way Let us assume that S   is unprovable We rst generate a reduction tree  by reverse application of all the rules of LB except nex and nec This tree contains a branch B	S consisting of unprovable sequents only We form the union sequent of B	S and extract from it the subsequent  B   B consisting of all formulas of the form  A By reverse application of nex we arrive at the sequent B  B which is unprovable as well For this sequent we repeat the construction of a reduction tree By iterating this procedure we obtain an innite sequence N of reduction branches all of them containing unprovable sequents only Now we take the union sequent of the sequence of all sequents contained in these branches In turn we extract a subsequent  N   N consisting of all formulas of the form  A but with the following restriction  A is in  N only if it occurs in innitely many reduction branches of the sequence N  If  N is the empty sequence we have completed our construction and obtain a countermodel otherwise we continue as follows By construction  N   N is unprovable and so is any subsequent of the form  N   A for any formula  A occurring in  N  We then repeat the whole construction for all sequents  N  A note that these are unprovable too  This gives us a possibly innite and possibly innitary tree of innite chains of reduction branches containing unprovable sequents only This tree is contained in B and we obtain from it a countermodel for the original sequent S   Denition  The reduction tree R	S of a sequent S   is an innite innitary tree ie the nodes may be of innite degree st the set of nodes is a set of occurrences of sequents R	S is dened in stages as follows Stage R consists of S alone S is the root node of R	S

 Stage k   Suppose that the reduction tree Rk has already been constructed In order to construct Rk  we need some additional terminology Let B be a branch ie a maximal path starting from the root in Rk We call B closed if it is nite and its end sequent    contains an atomic formula which is contained in both  and  otherwise B is called open The free variables occurring in the sequents of a branch B are called the available variables of B if there are none pick any free variable and call it available Note that our sequents may be innite and thus there may be innitely many free variables even on a nite branch Since in the denition of Rk  there may be nodes of uncountable degree we need an uncountable supply of free variables note that this poses no problem as R	S is a semantic structure and not an actual proof tree  Constants occurring in S by construction no new constants are generated are treated like available variables The reduction applies to any top sequent ie leaf sequent of Rk The method is a generalization of the rstorder case which applies to        by extending it to the case of   For the time being we postpone treatment of   Concerning formulas with outermost logical symbols among        we proceed as in  We present only some typical cases and omit most of the details The principle is that of decomposing formulas according to their outermost logical symbol In order to avoid reducing formulas more often than needed we mark formulas as treated once the reduction has been applied to them In the rst step the root sequent contains only unmarked formulas So let us assume that S   is a leaf node of a branch B in Rk a Outermost logical symbol  left reduction

Let AiBi i be the subsequence of  consisting of unmarked formulas with outermost logical symbol  Then we dene S Ai Bi i   and add the edge S S to Rk Mark the thus reduced formulas Ai Bi i in S a Outermost logical symbol  right reduction Here let Ai  Bi i be the subsequence of  consisting of all unmarked for mulas with outermost logical symbol  Let 	S  f   Ci i j Ci  Ai or Ci  Big For every S 	S add S and the edge S S to Rk and  mark the formulas Ai Bi i therein Note that the node S has an uncount able degree in the new tree Rk  if is an innite ordinal We skip the denition for the other propositional connectives and refer the reader to  b Outermost logical symbol  left reduction Let   xi Ai	xi

 i be the subsequence of  consisting of all unmarked for mulas with outermost logical symbol  Let ai i be a sequence consisting of all free variables on the branch B from S to S Note that all sequents are countable and the length of B is nite thus is a countable ordinal again We dene S    Ai	aj

 j  i    and add S and the edge S S to Rk b Outermost logical symbol  right reduction Let   xi Ai	xi

 i be the subsequence of  consisting of all unmarked for mulas with outermost logical symbol  Create a sequence bi i of free variables which do not occur in any sequent constructed so far We dene S     Ai	bi

 i and add S and the edge S S to Rk Mark the formulas xi A	xi for i in the consequent of the new sequent S  The construction for  is completely symmetric to the case of  c Outermost logical symbol   left reduction Let  Ai i be the subsequence of all formulas in  which are unmarked and have   as outermost symbol Let S Ai  Ai i   and add S and the the edge S S to Rk Mark all formulas  Ai for i in  of S Note that like in the other cases the form of S is obtained by applying  left backwards c Outermost logical symbol   right reduction Let  Ai i be the subsequence of all formulas in  which are unmarked and have   as outermost logical symbol Let 	S  f   Ci i j Ci  Ai or Ci    Aig and add S and the edge S S to Rk for every S 	S  Note that like in case a above the degree of the node S in Rk  is uncountable provided is innite Finally mark the formulas  Ai for i in  of S As already indicated we do not introduce reduction rules for   here Suppose none of the reduction rules for       or   apply and the branch B from S to S

is open Then we simply add a copy S of S  and the edge S S to Rk Note that we work with occurrences of sequents not merely sequents The reduction therefore indeed produces a tree and not a cyclic graph

In order to guarantee that all formulas in the sequents are eventually processed we postulate a clockwise order in reducing          If we take the order as given we reduce   rst then  etc After having reduced   on all sequents we start with   again Since reduced formulas in right and left reductions are not marked these formulas can be reduced innitely often Without postulating such a clockwise order open branches would not dene countermodels in general By the above construction we obtain an innite sequence of trees which is monotonic Thus by taking the union over the sets of vertices and edges we obtain the limit tree R R is precisely the tree R	S we intended to construct Note that our construction if applied to formulas neither containing   nor   yields the familiar construction of a counterexample in classical predicate logic Indeed if A is such a formula which is not valid in the standard rstorder se mantics we obtain an innite open branch B representing a counterexample Our construction however is not completed so far In fact we may obtain open branches in R	S even for sequents valid in TB Note that in the construction of R	S itself we cannot obtain innite sequents provided the root sequent is nite But in some further constructions we will obtain innite sequents out of innite branches and apply the method of reduction trees to these sequents as well Let us illustrate the construction of R	S by a simple example cf also Example  Example  Let S be    A    A The tree R	S is given below     A   A   A A     A   A   A A    right red    A  A   right red   A    A R	S possesses two open innite branches As   A    A is TBvalid these open branches do not represent counterexamples On the other hand we will prove that for unprovable sequents there are always branches in the reduction tree con taining unprovable sequents only Take for example S   A    A We already know that S is not TBvalid R	S is the following tree consisting of one innite branch only   A   A  A   A   A  A    right red   A  A   right red   A    A It is easy to verify that the branch contains only sequents which are not valid in TB Clearly by soundness of LB these sequents are all unprovable In the case of LK nite sequents and an unprovable endsequent S we obtain a tree R	S with the following property If S is an unprovable sequent in R	S  then there is a successor of S in R	S which is also unprovable As R	S must be innite and its node degree nite there is an innite branch by Konigs LemmaThis innite branch consist of unprovable sequents only and represents a counterexample This argument obviously yields the completeness of LK In the case of innite sequents S there may be nodes in R	S of uncountable degree This phenomenon occurs if in a sequent S occurring in R	S  we have innitely many formulas containing an outermost logical operator with a binary reduction rule eg right or  right

 It is however still possible to prove the existence of an innite branch containing unprovable sequents For this purpose we will use a generalization of Konigs Lemma due to Takeuti  Denition  Let be a set and fWigi be a family of sets indexed by  If f Q i  Wi and   then f is called a partial function over  with domain domf    If domf  then f is called total If f and g are partial functions st dom f  D domg and f	x  g	x for all x D then we call g an extension of f and write f  g and f  g jn D  Theorem  Takeuti  p f Let be a set and fWigi be a family of nite sets Let P be a property of partial functions over st  P f holds i there exists a nite subset N st P f jn N holds  P f holds for every total f  Then there exists a nite subset N st P f holds for every f with N domf  Lemma 	 Let R	S be the reduction tree of a possibly innite unprovable se quent S Then R	S has a branch B	S containing unprovable sequents only Such a branch is called a reduction branch of R	S  Proof We have to show that in R	S  a sequent S is unprovable i there exists a successor S of S st S is unprovable Equivalently If all successors of a sequent node S are provable then S itself is provable Using transnite induction on trees by ordering trees according to the standard subset relation we derive from If S is unprovable then there exists an in nite reduction branch in R	S every maximal nite branch must end in a provable sequent  Thus by  every path leading to an unprovable sequent can be ex tended  Note again that the degree of some nodes in R	S may be uncountable but branches in R	S are always countable Thus it remains to prove Case  S is of degree  The rule used for the reduction of S has only one premise eg right  left   left  Then S has only one successor S Let us assume that S is provable By denition of provability of innite sequents there exists a nite subsequent S of S  which is provable too Now let B      Bm be the formulas in S obtained by reduction using some rule let us call it   Then by repeated application of  on the Bi combined with contractions and exchanges we obtain a nite subsequent S of S  which is provable too the proof of S can be easily extended to a proof of S Case  S is of degree   possibly of uncountable degree The rule corre sponding the reduction of this node must be binary  eg left   right  By denition of a reduction tree the successors of S must be of the form    Cjii i or Cjii i   where for all i we have ji f g depending on which of the two subformulas occurs on position i Moreover for every sequence ji i there exists a successor corresponding to this sequence In the argument to follow it does not matter whether the rule under consideration is a left or a right rule Thus we restrict attention to the case where  is a right rule and the reduced sequent is    Cjii i Now let Wi  f g for every i and f denote functions in Q iWi  f g  Let us assume that all successors of S are provable Then to every suc cessor S of S there corresponds exactly one f f g Thus if S corresponds to f we write S  Sf  Since Sf  is provable there exists a nite subsequent S f  of S f  which is provable too This means for every total f see Deni tion  there is a nite subsequent S f  of S f  st S f  is provable Hence for S     Cjii i and every f f g  we obtain a nite provable subse quent S f  of the form f  f  Cjii i  where   is a nite subset of  Let    fi      ing be an arbitrary nite subset of and let f f g  Then we call the nite sequence of formulas   Cfi i       Cfinin   selected for f if there are nite subsequences f  f of  respectively stf  f  Cfii i  is provable By the explications above there are such subsequences for every f  Hence there exist selected sequences for every total f  In order to apply Takeutis theorem we have to dene a property P of partial functions over R We choose P f  n  i      in domf Cfi i       Cfinin is selected P f obviously satises both conditions  and  of Theorem  Thus Takeutis theorem applies and there exists a nite set   fr      rg st if  domf then P f holds We dene F  ff j domf  g Then F is a nite set and P f holds for all f F  But this means that for every f F there exists s      sk R  domf st   Cfs s       Cfsksk  is selected ie there exists a nite subsequence f  f of  st f  f  Cfs s      Cfsksk is provable Now the set f g is isomorphic to f gf g the set of all binary sequences of length  Thus for every such binary sequence  i      i there exist nite subsequences    of  st S     Ci r       Cir is provable We see that the Ci r       Cir for i      i f g f g  B are exactly the reduction formulas obtained from the reduction of the nite subsequent S   Cr       Cr where  is the union sequence of   B and  is the union sequence of  B  By repeated application of the binary rule  under consideration we can derive S from the sequents S   Together with the respective LBproofs of the S we obtain a proof of S But S   is a nite subsequence of S and thus S is provable   Note that in order to prove lemma  we made use of the compactness of the provability concept which holds by denition  We did not use semantic com pactness of the logic TB and do not even claim that TB is indeed compact So far we know that for unprovable sequents S there must be an innite branch containing only unprovable sequents ie a reduction branch in R	S  In our next step we pass the ordinal  in our construction and obtain innite sequents out of nite ones note that if S is nite then R	S contains only nite sequents  The basic idea is to construct innite unprovable sequents out of reduction branches and iterate this procedure innitely often Denition   Let S be an unprovable sequent and B be a reduction branch in R	S  Let S be the union sequent of B see Denition  and S  Ai i   Bj j be the subsequent of S  consisting of all formulas in S with outermost logical symbol   Let S be Ai i  Bj j This is the sequent S   stripped of its outermost  s  Then S is called the successor of S wrt B Lemma  Let S be an unprovable sequent and B be a reduction branch in R	S

and let S be the successor of S wrt B Then S is unprovable Remark  By lemma  we know that R	S must have a reduction branch thus the assumption of the lemma can always be fullled and S exists  Proof Let S be Ai i  Bj j Assume by way of contradiction that S is provable By denition of provability there is a nite subsequent SAi       Aik  Bj       Bj of S  which is LBprovable But from S we can derive in one step  using nex  the sequent S   Ai       Aik   Bj       Bj  Since S  is the successor of S wrt B by Denition  S  is a nite subsequent of the union sequent U B of B Thus if B  Si i there exists a nite initial segment B   S      Sn of B with S   S and so that the union sequent U B  of B contains S   Let left	   denote the set of all formulas in  and right	   denote the set of all formulas in  By construction of R	S we have that left	Si left	Sj

and right	Si right	Sj for  i j n Hence left	U B

 left	Sn and right	U B

 right	Sn  In other words S    is a nite subsequent of Sn S    is provable and thus Sn is provable too But this is impossible because S is a reduction branch Hence S must be unprovable   Denition  Let S  be an unprovable sequent A nexttime sequence is an innite sequence of reduction branches Bi i st B  is a reduction branch of S  and for every i   Bi is a reduction branch of a successor Si of Si  wrt Bi  All variables occurring in Bi are available for the construction of Bi  ie for the reductions left and right  Note that by Lemma  nexttime sequences exist for all unprovable sequents This is easily seen by induction Example  We construct a nexttime sequence N S  corresponding to the sequent S  A   A The following sequence is a reduction branch in R	S  B S   A   A  A   A  A   A  A   A    The union sequent of B  is  A   A  A    A Therefore the successor of S  wrt B  is SA  A  A For S we obtain a reduction branch of the form BS A  A   A  A   AA  A   A    with the union sequent  A   AA  A   A The successor of S wrt B is S  S  A  A  A In general Si  S  A  A  A and Bi  B for i   The sequents in a nexttime sequence represent necessary conditions for a se quent S to be true If S is true at time point  then S  is true at point  S at  etc But these conditions are not sucient Let us look at the sequent S   A   A We know that S  is not TBvalid Let us assume that S  is true at time point  Then the sequent SA  A  A is true at time point  and at time k we would have A  A   A being true According to our semantics there is a counterex ample to the sequent S at every time point k  But recall that at time point  we may set A to false Note that  is not a successor ordinal Thus in order to construct counterexamples to sequents we have to jump across time gaps this jump will be performed via reverse application of the necessitation rule Denition  Let S be an unprovable sequent A gapjump tree G	S for S is a tree with nodes consisting of nexttime sequences satisfying the following conditions  The root of G	S is a nexttime sequence of S   Let N be a nexttime sequence in G	S corresponding to a sequent S Then NNi  i  are edges in G	S if the Ni are constructed as follows Let N  Bi i and     be the union sequent of N ie    is the union sequent of the union sequents of the Bi  Let  N be the subsequence of all formulas in  with outermost logical symbol    N is a subsequence of   obtained in the following way delete all formulas in  except formulas of the form  A where  A occurs in the right hand sides of innitely many successor sequents in N  Thus we obtain a sequent of the form  N   N   N   Ai i and dene a nexttime sequence Ni for every  N  Ai i  provided  N  Ai is unprovable If  N is empty then N is a leaf in G	S  In the denition of the nexttime sequence Ni all free variables available for the construction of N are available for the construction of Ni too for the left and right reduction in the reduction branches  If  N is empty then according to denition  the node corresponding to N must be a leaf But even if  N is nonempty it might be the case that the sequents  N  Ai are provable and thus do not dene new nexttime sequences We will see in Lemma  that such a case cannot occur Example  We construct a gapjump tree with root N S   where N S  is the nexttime sequence of Example  For N  Bi i  we had Bi  B for i   and BA  A  A A  A   A  A   AA  A   A    B  starts with   A   A so we obtain as union sequent of N   AA A   A  A     A  A    where formulas if we do not use contraction may be repeated innitely many times Note that in all the successors Si  wrt Bi we have Si   A  A   A for i  and thus  A occurs innitely often on the right side Hence  N   N    A   A We have to consider only the single sequent S  A A The only edge leaving N S  in G	S  is N S   N S

 A nexttime sequence N S for S is easily obtained We construct the reduction tree for S and nd a reduction branch B with successor sequent SA  A  It is immediately clear that the successor sequents will be repeated innitely often A nexttime sequence for S is N S  Bi i where B    A A  A   A  A A    B   A  A  A  AA  A    Bi  B  for all i   N S is a leaf node of the gapjump tree since there is no formula of the form  A occurring in innitely many successor sequents on the right hand side in fact there are no such formulas at all  In dening the gapjump tree we have constructed a sequent of the form S N   N where  N contains only formulas appearing innitely many times S can be extracted from the nexttime sequence N  which is also a node in the tree If the consequent of S is empty then clearly N is a leaf in the gapjump tree Otherwise we obtain sequents of the form  N  Ai where  Ai occurs in  N  We call S  the  extract of N and every sequent S N  A for  A in  N a right reduct of S  The term right reduct should not be confused with the  right reduction of S which has a dierent form

 Lemma  Let N be a node in a gapjump tree G	S for unprovable S and S be the  extract of N  If the consequent of S is not empty then every right reduct of S is unprovable Remark  A consequence of this lemma is that every right reduct denes a nexttime sequence and thus a successor node of N  Proof Let S N   N be the  extract of N st  N is not empty and let  A be a formula in  N  Assume by way of contradiction that S N  A is provable By denition of provability there is a nite subsequent S N    N of S  st S is provable N is either empty or A alone If N is empty then S  is a subsequent of the union sequent of N recall that N is a nexttime sequence  We show that there exists a successor sequent Si of a branch Bi  in N st   N is a subsequence of the antecedent of Si Let  C be a formula in   N   C occurs in some sequent SC in a reduction branch B in N  By denition of a reduction branch   C must occur in almost all descendents of SC and thus also in the union sequent  left reduction  By denition of a successor sequent the successor wrtB must contain  C in the antecedent Moreover  C must occur in all further successor sequents in N  As   N is a nite sequence there must be a successor sequent Sj of a reduction branch Bj  st  N is subsequence of the antecedent of Sj  But then Sj would be provable which contradicts Lemma  If N  A then like in the case where   N is empty above we obtain a successor sequent Si in N st  N is a subsequence of the antecedent of Si By denition of a  extract the formula  A must occur in innitely many Sj s Observe that  N is a subsequent of the antecedents of all Sj for j  i Therefore there must be a k st   N   A is a subsequent of Sk If S   N  A is as we assumed provable then so is S N   A by application of the rule nec  Since S  is a subsequent of Sk Sk were provable too again contradicting Lemma    Corollary 	 If N is a leaf in a gapjump tree then the consequent of the   extract of N is empty Proof Assume that the  extract S of N were not empty Then S would have right reducts Any such right reduct S is unprovable by Lemma  But then there would be a nexttime sequence N S for S and and an edge NN S

   Proof of the Completeness Theorem  We have to show that nite unprovable sequents are not valid More precisely if S is a nite sequent which is unprovable in LB then there exists a TBinterpretation K for S which falsies S Let G	S be a gapjump tree for S We dene the following TBinterpretation K  hT fDBgBT  fSBgBT i where  T is the set of all occurrences of reduction branches in the nexttime sequences

in G	S  For the remaining part of this proof we use the letter B for occurrences of branches and B for the branch corresponding to B Moreover we introduce the following partial order If B and B are two occurrences within the same nexttime sequence i i then there are i j  st B  i and B  j  We set B  B if i  j and B  B if j  i Clearly B  B for i  j If BB are occurrences in dierent nexttime sequences N  N  which are nodes of G	S

then B  B i there is a path from N to N  in G	S  Evidently the order type of  is in T    For every B T  DB is the set of all free variables V B occurring in B Note that by the denitions of a nexttime sequence  and of a gapjump tree   V B V B if B  B Thus we obtainDB DB  for B  B second condition of Denition 

 Denition of the evaluation function SB for B T Set SB	a  a for a DB note that elements of DB are available as constant symbols in the extended language  IfA is an atomic formulawe dene SB	A   if A occurs in the antecedent of a sequent occurring in B and   otherwise We have to show that this truth assignment is consistent ie that it is impossible that an atomic formulaA occurs in an antecedent and in a consequent of a sequent in B Thus let B  Si i By construction of a reduction tree we have sub 	Si sub	Sj for i  j see denition   In particular all atomic formulas occurring in the antecedent consequent of Si also occur in the antecedent consequent of Sj Thus if A occurs in the antecedent of Si and in the consequent of Sj it must occur in both sides in Sk for k  max	i j  This however contradicts the denition of reduction branches in a reduction tree Denition   as B would be closed So SB is consistently dened It remains to show that K as dened above is indeed a countermodel to S It suces to show the following If F is a formula occurring in the antecedent consequent of a sequent in a reduction branch B in a nexttime sequence N oc curring as a node in G	S

then KB	F   KB	F    Then by Denition  and by the niteness of S the implication corresponding to S is falsied in K as KB S   B being the rst reduction branch in the root of G	S

 Note that F cannot occur in an antecedent of S and in a consequent of S for two sequents S S in B The reasons are the same as for atomic formulas described above We prove by induction on the logical complexity of F  If F is an atomic formula logical complexity  then follows fromKB	F  SB	F and the denition of SB Suppose that has been shown for all formulas of logical complexity n Let F be a formula of logical complexity n   If the outermost logical symbol is not a temporal operator     then the reduction to the case n follows exactly the classical rstorder case see  Ch  x  It remains to handle the cases F   F  and F   F  for some formula F   F   F  Let us assume that F occurs in the antecedent consequent of a reduction branch B Because B is a branch in a nexttime sequence there is a succes sor S wrt B see Denition  on which the next reduction branch starts see Denition   By denition of successor F  occurs in the antecedent consequent of S which is the rst sequent of the successor branch B By the induction hypothesis KB  F    KB  F     By Denition  KB	F  KB	 F   KB  F   As F must occur on the same side as F  we conclude that holds for F   F   F  a F occurs in the antecedent of a sequent in a reduction branch B in the nexttime sequence N By the semantics of   we haveKB	F   i for all B st B B it holds that KB  F    So let us assume that F occurs in the antecedent of the sequent Si in B Then   F  must occur in the antecedent of a sequent Sj for some j  i By denition of a nexttime sequence N   F  must occur in the antecedent of the successor of B By induction  F  occurs in the antecedent of every sequent in every reduction branch in this nexttime sequence Hence  F  occurs in the union    and in the antecedent of the  extract of N  By denition of right reducts and the gapjump tree then   F  also occurs in the antecedents of all initial reduction branches in the successor nodes N  of N in G	S  By the same arguments as before we have that  F  occurs in the antecedent of every sequent in every reduction branch B  B Every reduction branch containing  F  in the antecedent of some sequent also contains F  in the antecedent of some sequent by  left reduction  Hence by the induction hypothesis KB  F    for all B  B and therefore KB	F   b F occurs in the consequent of a sequent in a reduction branch B in the nexttime sequence N By denition of  right reduction  which is binary there is a sequent in B which either contains in the consequent F  or   F  In the former case we have immediately by the induction hypothesis thatKB	F    and hence KB	F   Otherwise observe that the successor of B contains  F  in the consequent We have two cases i either all reduction branches  B in N contain   F  or ii some branch B contains F  in the consequent of some sequent The former holds if at every  right reduction of F in N the right premise lies on the reduction branch the latter if in some reduction the left premise does Case ii is handled as above For case i  observe that  F  occurs in the consequent of every successor sequent of branches B  B in N  Thus by denition of the  extract  N   N of N   F  belongs to  N  Then there is some right reduct of N of the form  N  F  By Lemma  this right reduct is unprovable and thus is the initial sequent of the rst reduction branch B of some successor node N  of N in G	S  By the induction hypothesis KB  F    Since B  B the semantics of   gives us KB	 F    This concludes the proof of and we have shown that K falsies S   Remark  If the original sequents may be innite in particular of unbounded logical complexity then we no longer have a wellfounded ordering on the sequents On the other hand the reduction steps which yield innite sequents in the proof keep the logical complexity of formulas occurring in the sequents bounded Hence if the starting sequent is of bounded logical complexity in particular if it is nite  we have a wellfounded order Otherwise the induction proof is problematic  TL versus TB It should be interesting to compare the two logics TL and TB A comparison from the viewpoint of expressibility would clarify the possible application of TB in a program specication and verication environment Such an analysis however would go beyond the scope of the present article An analysis from a logical point of view can be given more easily Here the comparison centers around the induction rule in propositional TL see   A B A  A A  B ind and the weaker necessitation rule of TB Proposition   The propositional fragment of TB is decidable  The fragment of TB without   is equal to S 	 The monadic fragments of TL and TB are undecidable  The fragment of TB without   is axiomatizable by LK plus nex  The fragment of TB without   is equal to the fragment of TL without    Proof  sub   is nite  LB without   collapses to the sequent calculus for S given in   Follows from the undecidability of monadic modal predicate logic see below  A cutfree proof can contain  left   right  or nec only if   occurs in the endsequent  By   a proof has to be found before jumping over the rst gap i one exists   In contrast to  above the monadic fragment of TB without   and hence by   the fragment of TL without   is decidable Proposition  It is decidable if a monadic temporal formula containing no  s is satisable Proof Note that   distributes over all propositional connectives Hence any formula F containing no  s is equivalent to a formula of the form W jKj where Kj  k Ejk  l Ajl Ejk   ek	x

i  e   ikLik	x

Ajl   al	x

i  a   ilLil	x

where Lijk is a negated or unnegated atomic formula F is satisable i Kj is sat isable for some j Consider the set K   	K  	K with  	K  f i  eke   ikLik	tk j kg 	K  f

i  ala   ilLil	tk j l k ek alg where tk are constant symbols and   v Lik	tl is considered as a propositional literal Lvikl K is satisable i K is satisable in classical propositional logic   So already the monadic fragments containing   but not   are undecidable It is worth to recapitulate the construction of the proof of Kripke  A binary predicate P x y can be encoded in monadic temporal logic as P 	x y  	P 	x  P	y

 Let F be a formula in the language of predicate logic and F  be obtained from it by replacing nary predicates P x      xn by 	P 	x    Pn	xn

 If F is valid then F  is too it being a substitution instance of F  If F is not valid then we construct a temporal countermodel for F  Let M be a rstorder structure in which F is not satised By the LowenheimSkolem Theorem we can assume M to be countable We can enumerate all ntuples of elements of the domainM using a function e Let T be  and Sj	Pi  fag i a is the ith component of the jth in e ntuple ofM  So 	P 	a    Pn	an

is true in h fDi  Mgi fSigii i M j P a      an  As remarked above the undecidability of the monadic fragments of TL and TB follows from the undecidability of dyadic predicate logic and the above construc tion We have two immediate consequences First the monadic fragment of TL with   is not even axiomatizable since we can replace the function symbols    by a unary a binary and two ternary predicate symbols These predicate symbols can in turn be replaced by temporal constructions of the kind used above so nonaxiomatizability follows from the nonaxiomatizability of the full logic see Section   A second interesting consequence is that already the fragment with only one monadic predicate symbol but including   is undecidable With some adjust ment to the construction of the countermodel in the proof above a binary predicate  can also be encoded by 	P x   P y

 We do not know however whether the corresponding fragment of TL is still not axiomatizable Even without a deep analysis it is obvious that propositional TL is decidable by embedding it into the the monadic second order logic of one successor of Buchi  A decision method based on a similar reduction method as the one used here forTB can be found in  For the same reason the quantied propositional variant of TL is decidable We do not know whether quantied propositionalTB is decidable Note that even though propositional TB    equals S the propositionally quantied logics dier Hence the result of Kremer  III ie that propositional S is recursively isomorphic to secondorder logic is of no help here We conjecture however that quantied propositionalTB is not axiomatizable as well In summary we have the following situation TL TB propositional decidable decidable monadic w!o   equal and decidable monadic not axiomatizable undecidable quantied propositional decidable not axiomatizable" full rstorder not axiomatizable axiomatizable  Resolution for TB A practical consequence of the cutfree completeness of LB is the ability to construct a resolution calculus The exact relationship between cutfree proofs in sequent calculus and resolution proofs has been investigated at length by Mints   This relationship is also the starting point for very fruitful investigations into resolution systems and strategies for other nonclassical logics eg linear logic see   The resolution procedure for TB works as follows The formula F to be proved  F to be refuted is translated to clause form via translation rules based on the calculus LB The translation is structure preserving and the literals have the form   A	a      an  where A is the subsemiformula corresponding to this literal and a      an are free variables or constant symbols A clause is an expression of the form C where C is a set of literals A clause may carry a variable restriction denoted Ca meaning that a resolution involving C is only allowed if a does not occur in the resulting clause and if a is not substituted into The rules are the resolution and factoring rules plus two rules corresponding to the nec and nex

rules By Lemma  and replacement of free variables with constant symbols we can assume that F is closed and does not start with   or   Denition  Let F be a semiformula and let       n be all the constant symbols and bound variables without matching quantier in order of occurrence Then the code of F is dened as F        n  where F  is an nary predicate symbol and  is a canonical renaming mapping       n to new free variables The axiom set Ax F is dened as the smallest set satisfying the following Let P       n and P       n be two atomic subsemiformulas of F with the same predicate symbol Then the clause f P 	      n  P 	      n g Ax F  where i  i with  the renaming as above and  a most general unier of       n and       n  The clause translation Cl	F is the following set of clauses Cl	F  S fCF A j A sub	F g  Ax F  f F        ng  where CF A is given by the following table  A occurrence CF A

 B pos f B   B  Bg  B neg fB  Bg ff  B  Bg Here a stands for x in the code for A	x   is the same for all literals in a clause in CF A  and positive and negative occurrences are dened as usual Note that there are no translation rules for formulas with outermost symbol   just as there are no introduction rules without restrictions for   in LB This is clear since there is no relation between A and  A which depends only on A Denition  The degree deg	A of a semiformula A is the number of occur rences of logical symbols except   and   in A The degree of a clause is deg	C 

 if C   maxfdeg	A j A C or  A Cg otherwise Note that maxfdeg	A j A sub	F n fFg g  deg	F  since F is assumed to be prexfree see the comments above  The resolution calculus for TB consists of the the following rules C  fA	a      an  A	b      bn g C  fA	a      an g fact C  f A	a      an g C  fA	b      bn g  C n f A	a      an g      C n fA	b      bn g   res where  is the most general unier of a      an and b      bn and it is assumed that the resolved clauses are variable disjoint ie by renaming variables  The resolution rule is subject to the following restrictions  deg	C C  min   deg	C  f Ag  deg	C  fAg

  if one of the two resolved clauses is restricted on the variable a then 	a  a and a does not occur in   C n f A	a      an g      C n fA	b      bn g   f A      A  B      B ga f  A       A   B       B ga nexr f  A       A  Bg f  A       A   Bg necr The application of the rules nexr and necr is restricted so that the resulting literals are still within sub	F  The calculus therefore depends on F we actually are giving a construction schema for resolution calculi for each F  The following should be noted about the variable restriction  Proposition  In any resolution inference a there is never a restriction on more than one variable in any one clause and b at most one of the two premises carries a restriction Proof a Resolution removes all restricted variables condition  above from the resolvent and the property holds of all input clauses b First of all restricted clauses are input clauses in Cl	F corresponding to positive occurrences of  or negative occurrences of  or are derived from them by applications of nexr  but not using other rules cf a

 A resolution inference with two premises which both carry restrictions would be up to leading  s of the form f A	x 	a b c  	x A	x ga fA	x 	b a c  	x A	x ga   f	x A	x  	x A	x g res Other constellations are ruled out by the degree restriction on resolution Since neither a nor a may be substituted into by the unier  they cannot stand opposite each other Instead they must unify with two other variables b and b respectively ie 	b  a and 	b  a But then the restricted variables would occur in the resulting clause violating condition  of resolution   Example  Consider the formula   	x A	x  x   A	x  The subsemi formulas are S   A	x S   A	x

A resolution proof is given by fS  S  g fS  Sg fS   S   ag fS   S  ag nexr fS  S  ag res fS  S  ag necr fS  b S g b fS S g res fS  S g res fS  S  g fS  g res fS  g  res Example  By contrast consider the formula F  P f  x P x  which is not valid Without the eigenvariable condition we would have the following deriva tion of the empty clause fF  eg fP x a  	xP xga fP x bP c bg fP c b F  bg fP x b F  bg res f 	xP x F  bg res f 	xP x F  dg fF  bg res  res For the resolution step res to work either 	a  b or 	b  a The former case is expressly forbidden in the latter case the restricted variable would appear in the resulting clause Theorem 	 The resolution calculus for TB is sound If  is derivable from Cl	F then j F   Proof We show how a resolution derivation  not using the goal clause f F g can be translated to an LBderivation Associate to each clause C in  the substitution C   where  is the original renaming of the bound vari ables and constants in subsemiformulas of F whose code occurs in C and  is the cumulative substitution of the subderivation in  ending in C In eect if A	x 	a is a literal in C then A	x C is the formula A	a  If  ends in a clause C f A  a       An an  B  b       Bm bm g we obtain an LBproof of SC A C      A  nC  B   C      B  mC  If C carries a variable restriction the restricted variable is bound by a weak quantier in SC  We argue by induction on the length of  h    consists of a clause C from Cl	F n f F g only If C Ax F  say C  f P  a  P  a g the sequent P a  P a is the corresponding axiom If C  CF A  where A sub	F  then we construct an LBproof of SC  We present here only some cases  C  f A B A Bg  The corresponding proof is A A B  B AB  A B right  C  f A	x 	a  	x A	x ga  The corresponding proof is x A	x  x A	x

 C  fA	x 	a  	x A	x g  The corresponding proof is A	a  A	a

left  C  ff  A  Ag The corresponding proof is   A   A  A   A  left h   We distinguish cases according to the last inference in  Let N denote the negative and P the positive set of literals in a clause and N and P its translations respectively  The last inference in  is a resolution where the premises do not carry a variable restriction C N  P  fA a g C N   P   f A b g  N   N   P  P  res By induction hypothesis we have LBproofs   of N  P  AC a and AC  b N    P    The unier  does not substitute into eigenvariables of  or  We obtain a proof   N  PA a     A b N    P   NN    PN   cut  The last inference in  is a resolution where one premise contains the restricted variable a  N P  fA	a ga  N   P   f A	b g  N   N   P  P  res By Proposition  a resolution involving restricted variables can only take this form By induction hypothesis we have LBproofs   of N  P  A	a

 and x A	x N    P    The unier  does not substitute into restricted variables ie eigenvariables of    Since a is restricted we have 	a  a and 	b  a so b cannot occur in the resulting clause Hence it satises the eigenvariable condition We obtain a proof   N  PA	b  N P x A	x  right    x A	x N    P   NN    PP   cut  The last inference in  is fact  N  P  f	  A a    A b g  N N  f	  A a g fact By induction hypothesis we have a proof  of N  P  A a  A b or A a  A b N  P  Since there are no restrictions on variables we can rename b via 	bi  ai in  With contraction we obtain a proof of N  PA a or A a N  N   The last inference in  is necr  Add a nec inference to the LBproof  The last inference in  is nexr  Add a nex inference to the LBproof   If there were a resolution proof of  which does not use the goal clause f F g then we could translate that into an LBproof of the empty sequent  Such a proof of course is impossible Hence any resolution derivation of  must use the goal clause f F g By the degree restriction the last inference in such a derivation must be a resolution between fF g and f F g A resolution derivation of fF g can as above be translated into an LBproof of  F  Remark  Observe that the degree restriction on the resolution rule is necessary for soundness Otherwise eg P   P would have the following proof f P   P g fP  P   P g fP g res f P g nec f P   P g f  P  P   P g f  P g res  res In fact a formula  F has a refutation without degree restriction i j F  but j F is not equivalent to j F in contrast to   and   cf Lemma   Theorem  The resolution calculus for TB is complete If j F then  is deriv able from Cl	F  Proof We give for each LBproof  of a sequent  F  a resolution proof of  from Cl	F  By Theorem  we can assume that  is cutfree analytic that its axioms are atomic and by Proposition  that it contains no weakenings Let    be a sequent in  As can easily be seen a formula A occurs positively negatively in    i it occurs positively negatively in F  Furthermore every formula A in  corresponds to exactly one subsemiformulaA of F  which can be determined by tracing the formula A downwards through  We translate  to a resolution proof  of fF g by induction on its subproofs  If  ends in    then  ends in  N  P	 where the semiformulas whose codes occur in     are those subsemiformulas of F corresponding to the formulas in    There is no variable restriction on the last clause in  We present here some cases    is an axiom Translate P a  P a to a clause f P  a  P  a g where P P

is the subsemiformula of F corresponding to the left right P a  This clause is in Ax F 

  ends in a contraction on a formula A By induction hypothesis we have a resolution proof of  N  P  fA a  A b g without restriction of variables A is the subsemiformula of F corresponding to A Apply fact    ends in right By induction hypothesis we have resolution proofs ending in  NP	fA g and  N  P	   fBg The clause f A B A Bg is in Cl	F  We obtain a resolution proof f A B A Bg   N  P  fAg  N  P  fA B Bg   N   P	   fBg  N   N   P  P	   fA Bg   ends in left By induction hypothesis we have a resolution proof ending in f A	x 	a g   N  P	g The clause fA	x 	b  	x A	x g is in Cl	F  We obtain a resolution proof  f A	x 	a g   N  P fA	x 	b  	x A	x g f 	x A	x g   N  P   ends in right By induction hypothesis we have a resolution proof of    fA	x 	a gg The clause f A	x 	b  	x A	x gb is in Cl	F  We obtain the resolution proof   N  P  fA	x 	a g f A	x 	b  	x A	x gb  N  P  f	x A	x g Note that the conditions on b in the right premise are met since a satises the eigenvariable condition   ends in  left By induction hypothesis we have a resolution proof of f A   Ag   N P	 The clauses fA  Ag and f  A  Ag are in Cl	F  We obtain a resolution proof  f A   Ag   N  P fA  Ag f   A  Ag   N  P f  A  Ag f  Ag   N  P   ends in nex Append a nexr inference to the resolution proof to obtain   ends in nec Append a necr inference to the resolution proof to obtain    Note that in the translation to resolution the restriction on the rules are all satised The uniers can be chosen so that only the variables in the clauses from Cl	F are substituted into Given a proof  of F we thus have a resolution proof  of fF g from clauses in Cl	F  By resolving with f F g Cl	F  we obtain    The translation above shows actually that a renement of resolution is complete namely where every resolution step has to involve at least one input clause ie a clause form Cl	F  The resolution method developed here diers signicantly from the resolution method of Robinson developed for classical clause logic hence the fact that input resolution is complete is not a contradiction to the wellknown fact that input resolution in the classical case is not complete  Conclusion We have seen how the passage from a nonaxiomatizable temporal semantics to an axiomatizable one is paralleled by an extension of the completeness proof of the propositional logic The point where the proof fails for TL is where a true formula starting with   is reduced even innitely often but no derivation can be obtained The extension of the semantics is prompted by this phenomenon and makes a complete reduction of the formula possible The reduction discussed here is very similar to Krogers completeness proof for propositional TL This prompts the question of how to extend similar propositional completeness proofs to the rstorder case by avoiding nonaxiomatizability of the standard semantics by extension of the semantics itself A candidate for such investigations would be eg innitevalued #Lukasiewicz logic It also prompts the question for a characterization of classes of formulas where a sequent calculus is complete for the original semantics say as those formulas where the reduction works It is quite natural to ask whether the predicate logic of linear time with gaps the structures being sequences of segments is axiomatizable or not let us call this logic TLG Indeed even the pure  part of TLG is not axiomatizable This result can be obtained by reducing the problem to the nonaxiomatizability of the innitevalued Godel logic with truth values from the set f   n jn N  fgg  fg However the proof of this result is quite involved placing it outside the scope of this paper It will be presented elsewhere Another problem which has not been addressed in depth so far is the corre spondence between temporal logics discussed here and number theory The proof of nonaxiomatizability of TL by reduction to arithmetic and the induction rule of propositional TL suggest that there is a close relation This suggestion is sup ported by our result the semantics of TB is a nonstandard semantics similar to nonstandard models of arithmetic Viewed this way it is not as surprising that TB would have a complete axiomatization References

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