Learning search control knowledge for equational deduction

Automated theorem provers for first order logic are increasingly being used in formal mathematics or for verification tasks. For these applications, efficient treatment of the equality relation is particularly important. Due to the special properties of the equality relation, the proof search is particularly hard for problems containing equality, even if state of the art calculi are used. In this thesis we develop techniques to automatically learn good search heuristics to control the proof search of a superposition-based theorem prover for clausal logic with equality. We describe a variant of the superposition calculus and an efficient proof procedure implementing this calculus. An analysis of the choice points of this algorithm shows that the order in which new logical consequences are being processed by the prover is the single most important decision during the proof search. We develop methods to extract information about important good and bad search decisions from existing proof searches. These search decisions are represented by signatureindepended annotated representative clause patterns, which represent all analogous search decision by a single unique term. Annotations carry information about the role of the seach decisions in different proof attempts. To utilize the stored knowledge for a new proof attempt, experiences generated from proof problems similar to the one at hand are extracted from the stored knowledge. The selected proof experiences, represented by a set of patterns with associated evaluations, are used as input for a new, hybrid learning algorithm which generates a term space map, a structure that allows the evaluation of new potential search decisions. We experimentally demonstrate the performance of different variants of the term space mapping algorithm for artificial term classification problems as well as a siginificant gain in performance for the learning theorem prover E/TSM compared to the variant using only conventional search heuristics.

[1]  Wolfgang Reif,et al.  Interactive correctness proofs for software modules using KIV , 1995, COMPASS '95 Proceedings of the Tenth Annual Conference on Computer Assurance Systems Integrity, Software Safety and Process Security'.

[2]  L. Wos,et al.  Paramodulation and Theorem-Proving in First-Order Theories with Equality , 1983 .

[3]  Ian D. Watson,et al.  An Introduction to Case-Based Reasoning , 1995, UK Workshop on Case-Based Reasoning.

[4]  Reinhold Letz,et al.  Strategy Parallelism in Automated Theorem Proving , 1998, FLAIRS.

[5]  Andreas Wolf.  Strategy Selection for Automated Theorem Proving , 1998, AIMSA.

[6]  Jon Whittle,et al.  Analogy as a Control Strategy in Theorem Proving , 1997 .

[7]  Wolfgang Reif,et al.  The KIV System: Systematic Construction of Verified Software , 1992, CADE.

[8]  Thomas Kolbe,et al.  Termination of Theorem Proving by Reuse , 1996, CADE.

[9]  Jörg Denzinger,et al.  Automatic Acquisition of Search Control Knowledge from Multiple Proof Attempts , 2000, Inf. Comput..

[10]  Matthias Fuchs,et al.  High Performance ATP Systems by Combining Several AI Methods , 1997, IJCAI.

[11]  C. Goller,et al.  A Connectionist Control Component for the Theorem Prover SETHEO , 1994 .

[12]  Mathematische Gesellschaft der Ddr.  Journal of information processing and cybernetics : EIK , 1987 .

[13]  Jaime G. Carbonell,et al.  Integrating derivational analogy into a general problem-solving architecture , 1988 .

[14]  Barbara Hammer,et al.  Neural networks can approximate mappings on structured objects , 1997 .

[15]  Luc De Raedt,et al.  Inductive Logic Programming: Theory and Methods , 1994, J. Log. Program..

[16]  Tanel Tammet,et al.  Towards Efficient Subsumption , 1998, CADE.

[17]  Christoph Goller,et al.  Learning task-dependent distributed representations by backpropagation through structure , 1996, Proceedings of International Conference on Neural Networks (ICNN'96).

[18]  Matthias Fuchs.  Learning search heuristics for automated deduction , 1997 .

[19]  Michaël Rusinowitch,et al.  On Word Problems in Equational Theories , 1987, ICALP.

[20]  Donald W. Loveland,et al.  Mechanical Theorem-Proving by Model Elimination , 1968, JACM.

[21]  Matthias Fuchs,et al.  Goal Oriented Equational Theorem Proving Using Team Work , 1994, KI.

[22]  Stephan Schulz.  Term Space Mapping for DISCOUNT , 1998 .

[23]  Harald Ganzinger,et al.  On Restrictions of Ordered Paramodulation with Simplification , 1990, CADE.

[24]  Ingo Dahn,et al.  Integration of Automated and Interactive Theorem Proving in ILP , 1997, CADE.

[25]  Chang Liu,et al.  Term rewriting and all that , 2000, SOEN.

[26]  TU Braunschweig,et al.  SETHEO Goes Software Engineering : Application of ATP to Software Reuse ? , .

[27]  Matthias Fuchs,et al.  DISCOUNT: A SYstem for Distributed Equational Deduction , 1995, RTA.

[28]  Stephan Schulz,et al.  Recording, Analyzing and Presenting Distributed Deduction Processes , 1994 .

[29]  Michael R. Lowry,et al.  Explaining synthesized software , 1998, Proceedings 13th IEEE International Conference on Automated Software Engineering (Cat. No.98EX239).

[30]  Jörg Denzinger,et al.  Learning Domain Knowledge to Improve Theorem Proving , 1996, CADE.

[31]  Alfred V. Aho,et al.  The awk programming language , 1988 .

[32]  Christoph Goller,et al.  A connectionist approach for learning search-control heuristics for automated deduction systems , 1999, DISKI.

[33]  Michael R. Lowry,et al.  Deductive Composition of Astronomical Software from Subroutine Libraries , 1994, CADE.

[34]  Jon Whittle,et al.  Internal Analogy in Theorem Proving , 1996, CADE.

[35]  D. Knuth,et al.  Simple Word Problems in Universal Algebras , 1983 .

[36]  James R. Slagle,et al.  Experiments in automatic learning for a multipurpose hueristic program , 1971, CACM.

[37]  Larry Wos,et al.  Efficiency and Completeness of the Set of Support Strategy in Theorem Proving , 1965, JACM.

[38]  Robert E. Tarjan,et al.  Self-adjusting binary search trees , 1985, JACM.

[39]  Felix Brandt,et al.  Using Term Space Maps to Capture Search Control Knowledge in Equational Theorem Proving , 1999, FLAIRS.

[40]  Andreas Wolf.  p-SETHEO: Strategy Parallelism in Automated Theorem Proving , 1998, TABLEAUX.

[41]  Matthias Fuchs.  Automatic Selection of Search-guiding Heuristics for Theorem Proving , 1998 .

[42]  Gernot Stenz,et al.  Strategy Selection by Genetic Programming , 1999, FLAIRS Conference.

[43]  Jörg Denzinger,et al.  Analysis and Representation of Equational Proofs Generated by a Distributed Completion Based Proof System , 1994 .

[44]  Donald W. Loveland,et al.  Automated theorem proving: a logical basis , 1978, Fundamental studies in computer science.

[45]  Christoph Goller.  Learning search-control heuristics for automated deduction systems with folding architecture networks , 1999, ESANN.

[46]  Jörg Denzinger,et al.  Distributing Equational Theorem Proving , 1993, RTA.

[47]  Thomas Kolbe,et al.  Reusing Proofs , 1994, ECAI.

[48]  Matthias Fuchs,et al.  Experiments in the Heuristic Use of Past Proof Experience , 1996, CADE.

[49]  Christoph Goller,et al.  Inductive Learning in Symbolic Domains Using Structure-Driven Recurrent Neural Networks , 1996, KI.

[50]  Michael Greiner,et al.  Foundations: Indifference, Independence & MaxEnt , 1996 .

[51]  Nachum Dershowitz,et al.  Ordering-Based Strategies for Horn Clauses , 1991, IJCAI.

[52]  I. V. Ramakrishnan,et al.  Term Indexing , 1995, Lecture Notes in Computer Science.

[53]  J. A. Robinson,et al.  A Machine-Oriented Logic Based on the Resolution Principle , 1965, JACM.

[54]  Richard C. T. Lee,et al.  Symbolic logic and mechanical theorem proving , 1973, Computer science classics.

[55]  Christoph Weidenbach,et al.  SPASS & FLOTTER Version 0.42 , 1996, CADE.

[56]  Jörg Denzinger,et al.  Recording and Analysing Knowledge-Based Distributed Deduction Processes , 1996, J. Symb. Comput..

[57]  William McCune,et al.  OTTER 3.0 Reference Manual and Guide , 1994 .

[58]  Nachum Dershowitz.  Orderings for Term-Rewriting Systems , 1979, FOCS.

[59]  Johann Schumann.  Automatic Verification of Cryptographic Protocols with SETHEO , 1997, CADE.

[60]  Oren Etzioni,et al.  Explanation-Based Learning: A Problem Solving Perspective , 1989, Artif. Intell..

[61]  Matthias Fuchs,et al.  Feature-Based Learning of Search-Guiding Heuristics for Theorem Proving , 1998, AI Commun..

[62]  Albert Rubio,et al.  Theorem Proving with Ordering Constrained Clauses , 1992, CADE.

[63]  P. Schulz.  Some Experiments on the Applicability of Folding Architecture Networks to Guide Theorem Proving , 1997 .

[64]  Larry Wos,et al.  Problems and Experiments for and with Automated Theorem-Proving Programs , 1976, IEEE Transactions on Computers.

[65]  Jaime G. Carbonell,et al.  Derivational analogy: a theory of reconstructive problem solving and expertise acquisition , 1993 .

[66]  Jörg Denzinger.  Knowledge-Based Distributed Search Using Teamwork , 1995, ICMAS.

[67]  Barbara Hammer,et al.  Universal approximation of mappings on structured objects using the folding architecture , 1996 .

[68]  Jörg Denzinger,et al.  Teamwork - eine Methode zum Entwurf verteilter, wissensbasierter Theorembeweiser , 1993 .

[69]  Thomas Hillenbrand,et al.  WALDMEISTER: Development of a High Performance Completion - Based Theorem Prover , 1996 .

[70]  Wolfgang Reif,et al.  The KIV-Approach to Software Verification , 1995, KORSO Book.

[71]  Oren Etzioni,et al.  A Structural Theory of Explanation-Based Learning , 1993, Artif. Intell..

[72]  Stephan Schulz.  System Abstract: E 0.3 , 1999, CADE.

[73]  Wolfgang Ertel,et al.  Automatic Acquisition of Search Guiding Heuristics , 1990, CADE.

[74]  Thierry Boy de la Tour.  An Optimality Result for Clause Form Translation , 1992, J. Symb. Comput..

[75]  Robert Nieuwenhuis,et al.  Saturation of First-Order (Constrained) Clauses with the Saturate System , 1993, RTA.

[76]  Thomas Hillenbrand,et al.  On Gaining Efficiency in Completion-Based Theorem Proving , 1996, RTA.

[77]  Simon Thompson,et al.  Haskell: The Craft of Functional Programming , 1996 .

[78]  Zohar Manna,et al.  Proving termination with multiset orderings , 1979, CACM.

[79]  Jörg Denzinger,et al.  Cooperation of Heterogeneous Provers , 1999, IJCAI.

[80]  Richard S. Bird,et al.  Introduction to functional programming using haskeu , 1998 .

[81]  Andreas Wolf.  Paralleles Theorembeweisen: Leistungssteigerung, Kooperation und Beweistransformation , 1999 .

[82]  Matthias Fuchs,et al.  A Comparison of Equality Reasoning Heuristics , 1998 .

[83]  Jörg Denzinger,et al.  Planning for Distributed Theorem Proving: The Teamwork Approach , 1996, KI.

[84]  Harald Ganzinger,et al.  Rewrite-Based Equational Theorem Proving with Selection and Simplification , 1994, J. Log. Comput..

[85]  Maximilian J. Moser.  Goal-directed reasoning in clausal logic with equality , 1996 .

[86]  Volker Sorge,et al.  ΩMEGA : Towards a mathematical assistant , 1997 .