This paper gives an overview of the system with a special focus on the new features of 3 T A P Version 4.0, including: efficient completion-based equality reasoning, methods for handling redundant axiom sets, utilization of pragmatic information contained in axioms to rearrange the search space, and a graphical user interface for controlling 3 T A P and visualizing its output

Bernhard Beckert

Martin Sulzmann

Peter Oel

Reiner Hähnle

Beckert, Bernhard

Hähnle, Reiner

Oel, Peter

Sulzmann, Martin

TUbiblio

The Tableau-Based Theorem Prover TaP, Version 4.0

Haehnle, Reiner

KITopen

The Tableaubased Theorem Prover TAPVersion Bernhard Beckert Reiner Hahnle Peter Oel Martin SulzmannUniversity of KarlsruheInstitute for Logic Complexity and Deduction SystemsAm Fasanengarten   Karlsruhe GermanyfbeckerthaehnleoelsulzgiraukadehttpiwwwiraukadethreetapOverview TAP is a tableaubased theorem prover for manyvalued rstorder logics withsorts in the twovalued version with equality it is implemented in Prolog	This paper gives an overview of the system with a special focus on the newfeatures of TAP Version 	 including ecient completionbased equality reasoning methods for handling redundant axiom sets utilization of pragmaticinformation contained in axioms to rearrange the search space and a graphicaluser interface for controlling TAP and visualizing its output	 TAP has been developed at the University of Karlsruhe	 In  the projectstarted in cooperation with the Institute for Knowledge Based Systems of IBMGermany	 Since  the system is maintained and improved as part of a newproject at the University of Karlsruhe funded by the Deutsche Forschungsgemeinschaft DFG	Supported Logics Specication Syntax and Semantics TAP is able to handle full rstorder logics with any nite number of truth values	 Hierarchical treeshaped sorts attached to terms are supported	 In thetwovalued case special handling of the equality predicate is provided	 Currently versions for classical rstorder logic for a certain threevalued rstorderlogic  and for a sevenvalued propositional logic  are specied	 It is possible either to prove a theorem or to try to check the consistency of the axioms	The user species the input as a set of axioms and theorems contained in aknowledge base le the formulae do not have to be in any normal form	 ifthenand ifthenelse connectives can occur in the input to rearrange the searchspace such as to reect their particular pragmatics that diers from materialimplication	 Furthermore an ordering on constant and function symbols can bespecied in the knowledge base	Problems from the TPTP problem library  can be directly loaded	 Theyare automatically converted into TAP format	The Calculus TAP s calculus is based on free variable semantic tableaux   with Skolemfunctions	 Using free variable quantier rules is crucial for ecient implementationeven more if equality has to be handled	 They reduce the number ofpossibilities to proceed at each step in the construction of a tableau proof andthus the size of the search space	 When universal quantier rules are applied anew free variable is substituted for the quantied variable instead of replacingit by a ground term that has to be guessed	 Free variables can later be instantiated on demand when a tableau branch is closed or an equality is appliedto expand a branch	For replacing an existentially quantied variable by a Skolem term TAPuses the following rule The term needs not contain all the free variables on thecurrent branch rather it suces to include just the free variables occurring inthe quantied formula this has been proven to be sound in 	 In additionthe same function symbol is used for skolemizing formulae that are identical upto variable renaming 	 This more subtle rule for existential quantiers leadsto an easier implementation and can yield shorter proofs	To avoid unnecessary instantiations two kinds of free variables are distinguished namely universal and rigid variables	 Informally speaking a variable isconsidered to be universal in a formula  if a copy of  could be deduced andadded to the current branch without causing any branching of the tableau	 Otherfree variables are considered to be rigid	 Free variables marked as universal donot become instantiated when used in a branch closure or equality application	This feature allows TAP to nd shorter proofs	A nave approach to theorem proving in manyvalued logic would build aseparate tableau for each nondesignated truth value in order to refute a formula	 TAP uses the concept of truth value setsassigns introduced in  see  for details	 As a consequence TAP needs to build merely one tableau even inthe manyvalued case	Another feature of TAP is the generation of local lemmata   If a tableaurule generates branch extensions with common models then lemmata can beadded to extensions in order to make them logically disjoint	 In the twovaluedcase only disjunctive rules rules in Smullyans  terminology are of thistype	 Its extensions and resulting from an application to say   havethe models satisfying both and in common	 Thus if a local lemma is added the result are logically disjoint extensions and  anotherpossibility is to add to the rst extension	 TAP uses a depthrst strategy to search for proofs it proceeds by closingindividual branches of a tableau one after the other	 When a substitution is foundthat closes a branch it is applied to the whole tableau and then the provertries to close the next branch	 If later on a branch cannot be closed observing alimit on the number of copies of universally quantied formulae that may be usedon each branch or a limit on the length of branches backtracking is initiatedand other closing substitutions are searched for	 Dierent closing substitutionsfor a single branch are the only choice points in TAP s proof procedure allother indeterminisms in semantic tableaux like choosing the next formula to beexpanded are resolved using fair deterministic heuristics and strategies	Equality HandlingA special background reasoner is used for handling equality in TAP  Mixed Eunication problems are extracted from tableau branches and passed on to thebackground reasoner that employs the completionbased method from  tosolve mixed Eunication problems and thus to close tableau branches	The equality reasoner can be invoked multiply during the construction of asingle tableau branch	 After a futile try to nd a unier the data computedby the background reasoner is reused for later calls in particular it is reusedfor dierent extensions of the branch	 We call this feature of TAP incrementalequality reasoning 	Search Space RestrictionsOne possibility to restrict the search space is to avoid putting redundant axiomson the tableau in the rst place	 This is particularly useful with huge axiomatizations where only a small subset of the axioms is actually needed to provea given theorem	 Only formulae that potentially take part in closing a tableaubranch are fetched from the knowledge base	To decide which axioms are redundant and which might be needed to close abranch TAP employs a method that is an extension of the well known techniqueof computing links connections between atomic subformulae whether a formulais linked to an atom on the branch depends on the equalities both on the branchand in the knowledge base in addition sorts have to be taken into concern	Another method for restricting the search space is pruning tableau branchesIf a tableau rule application generates several subbranches and in the followingone of the subbranches can be closed without using any formula created by thatrule application then the other subbranches could be closed the same way andcan thus be pruned removed	 This technique is essentially what has beencalled condensing in  although we control it dierently	User InteractionApart from commands given in the Prolog shell TAP can be controlled via agraphical user interface GUI based on XWindows	 The GUI allows the user to The minimalistic tableau	based theorem prover leanTAP  that consists of only lines of Prolog code is an implementation of the same basic search strategyleanTAP is due to the elimination of all overhaed quite ecient and easy to un	derstand On the other hand it is missing most of the extensions of the calculusand additional features built intoTAP  like equality handling powerful heuristicsfor rearranging the search space a graphical user interface etcload and compile preprocess knowledge bases to change all parameters andswitches to specify the theorem to be proven and to start the prover	 Duringthe proof search the tableau tree is displayed and continually updated	 The usercan dene spy points to stop the prover for example whenever a branch is closedor after a certain number of tableau rule applications	 When the prover stopsthe user can navigate through the tableau to inspect the partial proof that hasbeen constructed	Integration of Automated and Tactical Theorem ProvingThe aim of integrating tactical and automated theorem provers is pursued in ajoint project with a research group in tactical theorem proving for formal software verication	 A system that integrates the respective latest versions of TAPand the Karlsruhe Interactive Verier KIV   has been built	 Here TAPis used to prove problems from twovalued rstorder logic with equality thathave been generated by KIV	 The main issues that have to be dealt with in theseproblems are their redundancy and their size up to one thousand axioms	 Someof TAP s features such as sorts orderings incremental completionbased equalityreasoning ifthen connectives and restricted formula fetching have proved to becrucial in order to deal with those problems	Availability TAP is implemented in SICStus Prolog with a small part of portable C	 Partsof TAP s compiler module are written using the Unix tools Lex and Yacc resp	Flex and Bison	 The design is as modular as possible therefore it is not toodicult to port TAP to other architectures and to add new features	 TAP is available via the World Wide Web and via anonymous ftp	 To obtainfurther information or to download the source code and the users manual  usethe TAP home page httpiwwwiraukadethreetap	 Alternativelyconnect via anonymous ftp to sonjairaukade 			 and change tothe directory pubthreetap	References Bernhard Beckert A completion	based method for mixed universal and rigidE	unication In A Bundy editor Proceedings th International Conferenceon Automated Deduction CADE Nancy France LNCS  pages Springer  Bernhard Beckert Reiner Hahnle Karla Gei Peter Oel Christian Pape and Mar	tin Sulzmann The many	valued tableau	based theorem proverTAP  version Interner Bericht  Universitat Karlsruhe Fakultat fur Informatik  Bernhard Beckert Reiner Hahnle and Peter H Schmitt The even more liberalized	rule in free variable semantic tableaux In Georg Gottlob Alexander Leitsch andDaniele Mundici editors Proceedings rd Kurt Godel Colloquium KGC BrnoCzech Republic LNCS  pages  Springer  Bernhard Beckert and Christian Pape Incremental theory reasoning methodsfor semantic tableaux In Proceedings 	th Workshop on Theorem Proving withAnalytic Tableaux and Related Methods Palermo Italy LNCS Springer  Bernhard Beckert and Joachim Posegga leanTAP  Lean tableau	based deductionJournal of Automated Reasoning   Marcello DAgostino Dov Gabbay Reiner Hahnle and Joachim Posegga editorsHandbook of Tableau Methods Kluwer Dordrecht  To appear Marcello DAgostino and Marco Mondadori The taming of the cut Journal ofLogic and Computation   Melvin C Fitting FirstOrder Logic and Automated Theorem Proving Springernd edition  Reiner Hahnle Towards an ecient tableau proof procedure for multiple	valuedlogics In Proceedings Workshop on Computer Science Logic Heidelberg Germany LNCS  pages  Springer  Reiner Hahnle Automated Deduction in Multiplevalued Logics volume  of International Series of Monographs on Computer Science Oxford University Press Reiner Hahnle and Werner Kernig Verication of switch level designs with many	valued logic In A Voronkov editor Proceedings th International Conferenceon Logic Programming and Automated Reasoning LPAR St Petersburg RussiaLNCS  pages  Springer  Reiner Hahnle and Peter H Schmitt The liberalized 	rule in free variable seman	tic tableaux Journal of Automated Reasoning   Reinhold Letz Klaus Mayr and C Goller Controlled integration of the cut ruleinto connection tableau calculi Journal of Automated Reasoning December  F Oppacher and E Suen HARP A tableau	based theorem prover Journal ofAutomated Reasoning   Steve Reeves Semantic tableaux as a framework for automated theorem	provingIn C S Mellish and J Hallam editors Advances in Articial Intelligence Proceedings of AISB pages  Wiley  Wolfgang Reif The KIV	system Systematic construction of veried softwareIn Proceedings th International Conference on Automated Deduction CADESaratoga Springs NY USA LNCS  pages  Springer  Wolfgang Reif Gerhard Schellhorn and Kurt Stenzel Interactive correctnessproofs for software modules using KIV In Proceedings th Annual Conferenceon Computer Assurance COMPASS Washington USA pages   Peter H Schmitt Computational aspects of three	valued logic In J H Siek	mann editor Proceedings th International Conference on Automated DeductionCADE LNCS pages  Springer  Raymond M Smullyan FirstOrder Logic Dover Publications New York secondcorrected edition  First published  by Springer	Verlag Geo Sutclie Christian Suttner and Theodor Yemenis The TPTP prob	lem library In A Bundy editor Proceedings th International Conferenceon Automated Deduction CADE Nancy France LNCS  pages Springer  Current version available on the World Wide Web at the URLhttpwwwcsjcueduauftpusersGSutcliffeTPTPHTMLThis article was processed using the LATEX macro package with LLNCS style

The tableau-based theorem prover 3TAP. Version 4.0

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The Tableau-based Theorem Prover 3TAP - Version 4.0

The Tableau-based Theorem Prover 3TAP - Version 4.0

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