Hähnle, Reiner

Ibens, Ortrun

KITopen

Improving Temporal Logic Tableaux usingInteger ConstraintsReiner Hahnle and Ortrun IbensUniversity of KarlsruheInstitute for Logic Complexity and Deduction Systems Karlsruhe Germanyfreinerortrungiraukade  IntroductionIn this position paper we present some ideas that aim to improve analytic tableau for temporal logics with the ultimate goal of reviving the interest in usingthem for temporal logic satisability checking Although tableau formulationsfor several propositional temporal logics exist 	 these are not used much inpractice	 because the tableau size becomes intractable already for quite smallformulas Moreover	 checking tableau closure is complicated and expensive toimplement For practical purposes	 usually automata theoretic approaches arepreferred  It might	 however	 still be interesting to have competitive tableauformulations as will be pointed out in Section The ideas for the research reported here were stimulated by 	  where anextension of analytic tableau with linear constraints led to an ecient mixedinteger programming formulation of a tableau system for nitely and innitelyvalued propositional logics The usefulness of linear constraints for manyvaluedlogics comes from the fact that the set of truth values can be identied witha subset of the integers	 hence manyvalued connectives can be characterizedusing integer constraints If we consider	 say	 a discrete linear temporal logicwithout past time operators	 then the always	 sometimes	 etc operators realizecertain properties of the natural numbers when Kripke worlds are identiedwith numbers and the linear order on the worlds with the successor function Itis tempting	 in our opinion	 to try using linear constraints over world variablesin order to give more concise tableau systems for such logics than it is possiblewithout the use of constraints In this note we give one possible formulation of aconstraint tableau system for one of the simplest propositional temporal logics	namely for discrete linear temporal logic with future always and sometimes Basic DenitionsLet L  hL  i be a propositional language of linear temporal logicLTL with negation	 disjunction	 conjunction	 always	 and sometimes denedas usual over a nonempty set of propositional variables L Call S  fk x     xnjk n  IN  x     xn  S g the set of signs overa nonempty set S  of world variables disjoint with L  The language of signedlinear temporal logic SLTL is L   fs  js  S   LgAn SLTLInterpretation v maps each propositional variable p  L  to theset of time points p holds at and each world variable s   S  to a time pointThe extension to formulas f  L and to signs s  S is immediateLet s    L 	 then s   is satisable if there is an SLTL interpretation vsuch that vs  v The extension to sets of SLTLformulas is immediateC  fs IN  sjs s  Sg is the set of SLTL constraints Usually we omitthe subscript of  We make use of the standard abbreviations   etc Constraint Tableau CalculusWe assume the reader is familiar with tableau calculi for propositional logics	 andin particular with uniform notation for propositional modal formulas as denedin  Just recall that type  formulas are conjunctive	 type  are disjunctive	type  are universal or always	 and type 	 are existential or sometimesIn contrast to classical propositional tableaux	 nodes consist of sets of signedformulas rather than single formulas Moreover	 we must accomodate linear constraints Hence	 a tableau node K is a pair hCi  L 	C A nodeK  hCiis satisable if  has a model v which as well solves CIn Table  we give correspondences between premises and conclusions of tableau rules for all types of compound formulas The superscripts      in the rules are to be understood as markers indicating that no logical rule is to beapplied anymore to the formula thus superscribed This is a crucial dierenceto conventional temporal tableau systems	 and we will see how completeness isrestored by dierent means There are four types of tableau rules which will beexplained in the followingLogical tableau rules can be specied as in Table  Note that taking thedisjoint union in the premise implies that the main formula of the premise doesnot occur anymore in the conclusion The letter k in the 	 rules is a new variablefrom S  that occurs not on the current branch Informally	 the conclusion of a	 rule says that there is some point in the future when  comes trueTable  Correspondence between premises and conclusions of temporal constrainttableau rules	      s     s   s  s     s   s  s   s   s     s     s   s  s     s   s     s    s   s    s   s   s     k c ks   s k   k  s    s k   k  The role of the linear constraints is to record the information on possiblemodels for each branch in a more concise form than in conventional tableauxTable  Temporal tableau rules for compound formulas	hf g Cihf    g  Cihfg Cihf g  Ci hfg  Cihfg Cihf  g  Cihfg Cihf kg  fc k CgiThe justication is that every class of counter models can in fact be characterizedby a set of linear inequations with integral solutions One link between formulasand constraints is constituted by the 	 rules as we have seen	 another link isobtained from the observation that	 obviously	 a formula and its complementcannot be true simultaneously at the same time point This concept is formalizedin the following conict rulehfs   t  g  Cihfs   t  g  C  fs  tgiAs a matter of fact	 there are other possible constraints one could generate	for instance	 s  t from s   and t    etc	 but these are being subsumed bythe synchronization rule below which we need for completeness anywayIn conventional temporal tableau systems such as in  two more ingredients are needed for which we have no substitute so far First	 for completenessone must have the information that an alwaysformula does hold in all subsequent states Usually this is achieved by reintroducing alwaysformulas in theconclusion of a rule and thus allowing them to be applied more than once As aconsequence	 in order to obtain an eective tableau system	 nodes N all whoseformulas have occurred already in a previous node M are looped back to thisnode M such that temporal tableaux really are directed graphs with a uniqueroot We intend to combine both rules	 the restart rule on alwaysformulas andthe loop back rule	 into a single rule called synchronization rule The rationalebehind it is to synchronize alwaysformulas so that they are available in thepresent state This has a similar eect as the restart rule	 but because in theconstraint part of a node we can accumulate information from dierent timepoints	 a nite number of applications of this rule is sucientIn slight abuse of notation let now  denote the unsigned part of an alwaysformula	 and  the unsigned part of its correspondent according to Table let   L   fpjp  L g	 ie  is a literal Then we can write down thesynchronization rule as followshfs  	 t  g  Cihfs  	 t  g  C  fs  tgi j hfs  	 t   s   syng  C  fs  tgiThe syn marker prevents indenite application of the synchronization rule	because the synchronization rule can only be applied to formulas without a synmarkerBut there are examples where the synchronization rule must be applied to aformula which itself is a synchronization result see Fig  So we need a rulewhich activates formulas with a syn marker	 called activation rule The idea isto activate a literal s  syn   if a literal s      together with an SLTLinterpretation v which maps vs  vs  cannot be found	 and to activate analways formula s  syn   if an always formula s      together with anSLTLinterpretation v which maps vs  vs  cannot be found There is aneective test for the applicability of the activation rule	 but the details must beomitted due to lack of space Let s  syn be a formula to be activated Thenthe activation rule is simply the followinghfs  syng  Cihfs  g  CiA node K  hCi in our tableau is contradictory if C is unsolvable Atableau is closed if every branch contains a contradictory node A formula   Lis valid i the tableau with root hfs  g fs  gi	 s  S 	 is closed In Fig we give a tableau proof of  p  phfs   p s   pgfs  gihfs k   p s  ps   p g fk     gihfs k  p s k   p   s l  p s   p g fl     gihfs l  p s k   p     gfl  k   gihfs  k  ps   p    gfk     gihfs k  ps l  p   gfk  l   gihfs k  psyn   gfk     gihfs k m  psyn   gfm     gihfs k m  p   g f  gihfs  k m  p s k   p    gfm     gihfs k m  p s k  p   gfm     gihfs k m  psyn   gfm     gihfskm  psyn skm  p   gfk m  km   gihfs l  psyn    gfl  k   gihfs l  psyns l  p   gfl  l   giFig  We omitted the rst steps which are purely propositional	 The next two stepsare straightforward  and  rule applications	 Next follow two applications of the synchronisation rule that split the proof into three branches	 The rst and the third ofthese can be closed after a conict rule application	 The second branch can be closedafter the activation rule and again the synchronization rule are applied	 Due to spacerestrictions only such formulas and constraints are displayed that are newly generated or that are needed in that step	 The formulas between which a synchronizationtakes place appear as the rst two formulas in the left leaf which is generated by thesynchronization rule	 ConclusionIt is fairly easy	 if somewhat tedious	 to prove that our rules preserve satisability Completeness	 however	 is a much tougher question	 and the proof is quiteinvolved This is one of the reasons why the present note is merely a positionpaper The other reason is that so far we did not show that the idea of incorporating linear constraints into temporal logic tableaux is in some sense a realimprovement over the current state of the art On the other hand	 we would liketo point out at least a few advantages that might be gained from it  Better performance if compared to conventional temporal tableaux Admittedly	 this is not obvious from the present appearance of the rules	 but thereis ample room for optimisation	 for example	 in the synchronisation rules  Implementing realtime logics like the ones suggested in   Extend the use of linear constraints to a translation of temporal logic satisability into integer programming as it has been done in  for manyvaluedlogic This would allow easy amalgamationwith other nonstandard featureslike multiple truth values needed	 for instance	 for hardware verication atthe switch level  or nonmonotonicity  in a homogenous framework  Perspective of a constraintbased approach to nonclassical deduction if integrated with ideas from   Lifting to restricted rstorder versionsReferences	 R	 Alur and T	 A	 Henzinger	 Realtime logics Complexity and expressiveness	 InProc th LICS pages 	 IEEE Press 		 C	 Bell A	 Nerode R	 Ng and V	 Subrahmanian	 Mixed integer programming methods for computing nonmonotonic deductive databases	 JACM to appear 		 J	 Burch E	 Clarke K	 McMillan and D	 Dill	 Sequential circuit verication usingsymbolic model checking	 In Proc th Design Automation Conference DAC 	pages  		 M	 DAgostino and D	 M	 Gabbay	 Labelled refutation sustems A case study	 InProc Second Workshop on Theorem Proving with Tableaux and Related MethodsMarseille	 Technical Report MPI Saarbrucken 		 M	 C	 Fitting	 Proof Methods for Modal and Intutionistic Logics	 Reidel Dordrecht		 R	 Hahnle	 A new translation from deduction into integer programming	 InJ	 Calmet and J	 A	 Campbell editors Proc Int Conf on Articial Intelligenceand Symbolic Mathematical Computing AISMC Karlsruhe Germany pages 	 Springer LNCS  		 R	 Hahnle	 Automated Theorem Proving in MultipleValued Logics Oxford University Press 		 R	 Hahnle and W	 Kernig	 Verication of switch level designs with manyvaluedlogic	 In A	 Voronkov editor Proc LPAR St Petersburg pages 	 Springer LNAI  		 P	 Wolper	 Temporal logic can be more expressive	 Information and Control  	

Improving temporal logic tableaux using integer constrains

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Improving temporal logic tableaux using integer constrains

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