Abstract
This article presents a taxonomy of strategies for fully-automated general-purpose first-order theorem proving. It covers forward-reasoning ordering-based strategies and backward-reasoning subgoal-reduction strategies, which do not appear together often. Unlike traditional presentations that emphasize logical inferences, this classification strives to give equal weight to the inference and search components of theorem proving, which are equally important in practice. For this purpose, a formal notion of search plan is given and shown to apply to all classes of strategies. For each class, the form of derivation is specified, and it is shown how inference system and search plan cooperate to generate it.
Supported in part by the National Science Foundation with grant CCR-97-01508.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Siva Anantharaman and Nirina Andrianarivelo. Heuristical criteria in refutational theorem proving. In Alfonso Miola, editor, Proceedings of the 1st DISCO, volume 429 of LNCS, pages 184–193. Springer Verlag, 1990.
Siva Anantharaman and Maria Paola Bonacina. An application of automated equational reasoning to many-valued logic. In Stéphane Kaplan and Mitsuhiro Okada, editors, Proceedings of CTRS-90, volume 516 of LNCS, pages 156–161. Springer Verlag, 1990.
Siva Anantharaman and Jieh Hsiang. Automated proofs of the Moufang identities in alternative rings. Journal of Automated Reasoning, 6(1):76–109, 1990.
Owen L. Astrachan and Don W. Loveland. METEORs: High performance theorem provers using model elimination. Pages 31–60 in [40].
Owen L. Astrachan and Don W. Loveland. The use of lemmas in the model elimination procedure. Journal of Automated Reasoning, 19(1):117–141, 1997.
Owen L. Astrachan and Mark E. Stickel. Caching and lemmaizing in model elimination theorem provers. Pages 224–238 in [77].
Franz Baader and Jörg H. Siekmann. Unification theory. In [62].
Leo Bachmair and Nachum Dershowitz. Inference rules for rewrite-based first-order theorem proving. In Proceedings of LICS-87, pages 331–337. IEEE Computer Society Press, 1987.
Leo Bachmair and Nachum Dershowitz. Critical pair criteria for completion. Journal of Symbolic Computation, 6(1):1–18, 1988.
Leo Bachmair, Nachum Dershowitz, and David A. Plaisted. Completion without failure. In Hassam Aït-Kaci and Maurice Nivat, editors, Resolution of Equations in Algebraic Structures, volume II: Rewriting Techniques, pages 1–30. Academic Press, 1989.
Leo Bachmair and Harald Ganzinger. Non-clausal resolution and superposition with selection and redundancy criteria. In Andrei Voronkov, editor, Proceedings of LPAR-92, volume 624 of LNAI, pages 273–284. Springer Verlag, 1992.
Leo Bachmair and Harald Ganzinger. Rewrite-based equational theorem proving with selection and simplification. Journal of Logic and Computation, 4(3):217–247, 1994.
Leo Bachmair and Harald Ganzinger. A theory of resolution. Technical Report MPI-I-97-2-005, Max Planck Institut für Informatik. 1997. To appear in J. Alan Robinson and Andrei Voronkov, eds., Handbook of Automated Reasoning.
Leo Bachmair, Harald Ganzinger, Christopher Lynch, and Wayne Snyder. Basic paramodulation. Information and Computation, 121(2):172–192, 1995.
Peter Baumgartner. Theory Reasoning in Connection Calculi, volume 1527 of LNAI. Springer, 1998.
Peter Baumgartner and Stefan Brüning. A disjunctive positive refinement of model elimination and its application to subsumption deletion. Journal of Auto mated Reasoning, 19:205–262, 1997.
Peter Baumgartner and Ulrich Furbach. PROTEIN: a PROver with a Theory Extension INterface. Pages 769–773 in [41].
Peter Baumgartner and Ulrich Furbach. Consolution as a framework for comparing calculi. Journal of Symbolic Computation, 16(5), 1993.
Peter Baumgartner and Ulrich Furbach. Model elimination without contrapositives and its application to PTTP. Journal of Automated Reasoning, 13:339–359, 1994.
Peter Baumgartner, Ulrich Furbach, and Frieder Stolzenburg. Model elimination, logic programming and computing answers. Artificial Intelligence, 90(1–2):135–176, 1997.
Bernhard Beckert. Semantic tableaux with equality. Journal of Logic and Com putation, 7(1):39–58, 1997.
Bernhard Beckert, Reiner Hähnle, P. Oel, and M. Sulzmann. The tableau-based theorem prover 3TAP, version 4.0. Pages 303–307 in [101].
Bernhard Beckert and Joachim Posegga. leanTAP: lean tableau-based theorem proving. Pages 793–797 in [41].
Wolfgang Bibel. Deduction: Automated Logic. Academic Press, 1993.
Wolfgang Bibel, Stefan Brüning, Uwe Egly, and T. Rath. KoMeT. Pages 783–788 in [41].
Wolfgang Bibel and E. Eder. Methods and calculi for deduction. Pages 68–183 in [62].
Wolfgang Bibel and P. H. Schmitt, Eds. Automated Deduction — A Basis for Applications. Kluwer, 1998.
Wolgang Bibel. Automated Theorem Proving. Friedr. Vieweg & Sohn, 2nd edition, 1987.
Carsten Bierwald and Thomas Käufl. Tableau prover Tatzelwurm: hyper-links and UR-resolution. In Maria Paola Bonacina and Ulrich Furbach, editors, Proceedings of the 1st FTP, number 97-50 in Technical Reports of RISC, pages 22–28. Johannes Kepler Universität, 1997.
Roland N. Bol, Krzysztof R. Apt, and J. W. Klop. An analysis of loop checking mechanisms in logic programming. Theoretical Computer Science, 86:35–79, 1991.
Maria Paola Bonacina. A note on the analysis of theorem-proving strategies. AAR Newsletter, No. 36, pages 2–8, April 1997. Full version available as Technical Report, Department of Computer Science, University of Iowa, May 1996.
Maria Paola Bonacina. Analysis of distributed-search contraction-based strategies. In Jürgen Dix, Luis Fariñas del Cerro, and Ulrich Furbach, editors, Proceedings of the 6th JELIA, volume 1489 of LNAI, pages 107–121. Springer, 1998. Full version available as Tech. Rep., Dept. of Comp. Sci., Univ. of Iowa, April 1998.
Maria Paola Bonacina and Jieh Hsiang. On rewrite programs: semantics and relationship with Prolog. Journal of Logic Programming, 14(1 & 2):155–180, 1992.
Maria Paola Bonacina and Jieh Hsiang. On subsumption in distributed derivations. Journal of Automated Reasoning, 12:225–240, 1994.
Maria Paola Bonacina and Jieh Hsiang. Parallelization of deduction strategies: an analytical study. Journal of Automated Reasoning, 13:1–33, 1994.
Maria Paola Bonacina and Jieh Hsiang. Towards a foundation of completion procedures as semidecision procedures. Theoretical Computer Science, 146:199–242, 1995.
Maria Paola Bonacina and Jieh Hsiang. A category-theoretic treatment of automated theorem proving. Journal of Information Science and Engineering, 12(1):101–125, 1996.
Maria Paola Bonacina and Jieh Hsiang. On semantic resolution with lemmaizing and contraction and a formal treatment of caching. New Generation Computing, 16(2):163–200, 1998.
Maria Paola Bonacina and Jieh Hsiang. On the modelling of search in theorem proving — towards a theory of strategy analysis. Information and Computation, 147:171–208, 1998.
Robert S. Boyer, Ed. Automated Reasoning — Essays in Honor of Woody Bledsoe. Kluwer, 1991.
Alan Bundy, Ed. Proceedings of the 12th CADE, volume 814 of LNAI. Springer, 1994.
H.-J. Bürckert. A Resolution Principle for a Logic with Restricted Quantifiers, volume 568 of LNAI. Springer Verlag, 1991.
Ricardo Caferra and Nicolas Peltier. Model building in the cross-roads of con sequence and non-consequence relations. In Maria Paola Bonacina and Ulrich Furbach, editors, Proceedings of the 1st FTP, number 97-50 in Technical Reports of RISC, pages 40–44. Johannes Kepler Universität, 1997.
Ricardo Caferra and N. Zabel. A method for simultaneous search for refutations and models by equational constraint solving. Journal of Symbolic Computation, 13:613–641, 1992.
Chin-Liang Chang and Richard Char-Tung Lee. Symbolic Logic and Mechanical Theorem Proving. Academic Press, 1973.
Jim Christian. Flatterms, discrimination nets and fast term rewriting. Journal of Automated Reasoning, 10:95–113, 1993.
Heng Chu and David A. Plaisted. CLINS-S: a semantically guided first-order theorem prover. In [136].
Heng Chu and David A. Plaisted. Model finding in semantically guided instance-based theorem proving. Fundamenta Informaticae, 21(3):221–235, 1994.
M. D’Agostino, Dov M. Gabbay, Reiner Hähnle, and Joachim Posegga, Eds. Handbook of Tableau Methods. Kluwer, 1998.
Martin Davis and Hilary Putnam. A computing procedure for quantification theory. Journal of the ACM, 7:201–215, 1960.
Jörg Denzinger and M. Fuchs. Goal-oriented equational theorem proving using team-work. In Proceedings of the 18th KI, volume 861 of LNAI, pages 343–354. Springer, 1994.
Nachum Dershowitz. Orderings for term-rewriting systems. Theoretical Computer Science, 17:279–301, 1982.
Nachum Dershowitz. Computing with rewrite systems. Information and Control, 65:122–157, 1985.
Nachum Dershowitz. Canonical sets of Horn clauses. In J. Leach Albert, B. Monien, and Mario Rodriguez Artalejo, editors, Proceedings of the 18th ICALP, volume 510 of LNCS, pages 267–278. Springer Verlag, 1991.
Nachum Dershowitz and Jean-Pierre Jouannaud. Rewrite systems. In Jan van Leeuwen, editor, Handbook of Theoretical Computer Science, volume B, pages 243–320. Elsevier, 1990.
Roland Dietrich. Relating resolution and algebraic completion for Horn logic. Pages 62–78 in [123].
Norbert Eisinger and Hans Jürgen Ohlbach. Deduction systems based on resolution. Pages 184–273 in [62].
Christian Fermüller and Alexander Leitsch. Hyperresolution and automated model-building. Journal of Logic and Computation, 6(2):173–203, 1996.
Christian Fermüller and Alexander Leitsch. Decision procedures and model-building in equational clause logic. Journal of the IGPL, 6(1):17–41, 1998.
Melvin Fitting. First-order Logic and Automated Theorem Proving. Springer, 1990.
S. Fleisig, Don W. Loveland, A. Smiley, and D. Yarmush. An implementation of the model elimination proof procedure. Journal of the ACM, 21:124–139, 1974.
Dov M. Gabbay, Christopher J. Hogger, and J. Alan Robinson, Eds. Handbook of Logic in Artificial Intelligence and Logic Programming (Vol. 1 & 2). Oxford University Press, 1993.
Jean Gallier. Logic for Computer Science — Foundations of Automatic Theorem Proving. Harper & Row, 1986.
Chr. Goller, Reinhold Letz, K. Mayr, and Johann Schumann. SETHEO v3.2: recent developments. Pages 778–782 in [41].
John Harrison. Optimizing proof search in model elimination. Pages 313–327 in [101].
Ryuzo Hasegawa, Miyuki Koshimura, and Hiroshi Fujita. MGTP: a parallel the orem prover based on lazy model generation. Pages 776–780 in [77].
Jieh Hsiang. Refutational theorem proving using term rewriting systems. Artificial Intelligence, 25:255–300, 1985.
Jieh Hsiang. Rewrite method for theorem proving in first order theories with equality. Journal of Symbolic Computation, 3:133–151, 1987.
Jieh Hsiang and Michaël Rusinowitch. On word problems in equational theories. In Th. Ottman, editor, Proceedings of the 14th ICALP, volume 267 of LNCS, pages 54–71. Springer Verlag, 1987.
Jieh Hsiang and Michaël Rusinowitch. Proving refutational completeness of theorem proving strategies: the transfinite semantic tree method. Journal of the ACM, 38(3):559–587, 1991.
Jieh Hsiang, Michaël Rusinowitch, and Ko Sakai. Complete inference rules for the cancellation laws. In Proceedings of the 10th IJCAI, pages 990–992, 1987.
Jean-Pierre Jouannaud and Claude Kirchner. Solving equations in abstract algebras: a rule-based survey of unification. Pages 257–321 in [83].
Deepak Kapur, Dave Musser, and Paliath Narendran. Only prime superposition need be considered in the Knuth-Bendix completion procedure. Journal of Symbolic Computation, 6:19–36, 1988.
Deepak Kapur and Paliath Narendran. An equational approach to theorem proving in first order predicate calculus. In Proceedings of the 9th IJCAI, pages 1146–1153, 1985.
Deepak Kapur and Hantao Zhang. A case study of the completion procedure: proving ring commutativity problems. Pages 360–394 in [83].
Deepak Kapur and Hantao Zhang. An overview of RRL: rewrite rule laboratory. In Nachum Dershowitz, editor, Proceedings of the 3rd RTA, volume 355 of LNCS, pages 513–529. Springer Verlag, 1989.
Deepak Kapur, Ed. Proceedings of the 11th CADE, volume 607 of LNAI. Springer, 1992.
Claude Kirchner, Hélène Kirchner, and Michaël Rusinowitch. Deduction with symbolic constraints. Revue Française d’Intelligence Artificielle, 4(3):9–52, 1990.
Donald E. Knuth and Peter B. Bendix. Simple word problems in universal algebras. In J. Leech, editor, Proceedings of the Conf. on Computational Problems in Abstract Algebras, pages 263–298. Pergamon Press, 1970.
R. E. Korf. Depth-first iterative deepening: an optimal admissible tree search. Artificial Intelligence, 27(1):97–109, 1985.
Robert Kowalski. Search strategies for theorem proving. In B. Meltzer and D. Michie, editors, Machine Intelligence, volume 5, pages 181–201. Edinburgh University Press, 1969.
Robert Kowalski and D. Kuehner. Linear resolution with selection function. Artificial Intelligence, 2:227–260, 1971.
Jean-Louis Lassez and Gordon Plotkin, Eds. Computational Logic — Essays in Honor of Alan Robinson. The MIT Press, 1991.
Shie-Jue Lee and David A. Plaisted. Eliminating duplication with the hyperlinking strategy. Journal of Automated Reasoning, 9:25–42, 1992.
Alexander Leitsch. The Resolution Calculus. Springer, 1997.
Reinhold Letz, K. Mayr, and Chr. Goller. Controlled integration of the cut rule into connection tableau calculi. Journal of Automated Reasoning, 13(3):297–338, 1994.
Reinhold Letz, Johann Schumann, S. Bayerl, and Wolfgang Bibel. SETHEO: a high performance theorem prover. Journal of Automated Reasoning, 8(2):183–212, 1992.
John W. Lloyd. Foundations of Logic Programming. Springer Verlag, 2nd edition, 1987.
Don W. Loveland. A simplified format for the model elimination procedure. Journal of the ACM, 16(3):349–363, 1969.
Don W. Loveland. A unifying view of some linear Herbrand procedures. Journal of the ACM, 19(2):366–384, 1972.
Don W. Loveland. Automated Theorem Proving: A Logical Basis. North-Holland, 1978.
Ewing Lusk and Ross Overbeek, Eds. Proceedings of the 9th CADE, volume 310 of LNCS. Springer Verlag, 1988.
Rainer Manthey and François Bry. SATCHMO: a theorem prover implemented in Prolog. Pages 415–434 in [92].
J. McCharen, Ross Overbeek, and Larry Wos. Complexity and related enhancements for automated theorem proving programs. Computers and Mathematics with Applications, 2(1): 1–16, 1976.
William W. McCune. Experiments with discrimination tree indexing and path indexing for term retrieval. Journal of Automated Reasoning, 9(2):147–167, 1992.
William W. McCune. A Davis-Putnam program and its application to finite first-order model search: quasigroup existence problems. Unpublished manuscript, May 1994.
William W. McCune. Otter 3.0 reference manual and guide. Technical Report 94/6, MCS Division, Argonne National Laboratory, 1994.
William W. McCune. 33 Basic test problems: a practical evaluation of some paramodulation strategies. In Robert Veroff, editor, Automated Reasoning and its Applications: Essays in Honor of Larry Wos, pages 71–114. MIT Press, 1997.
William W. McCune. Solution of the Robbins problem. Journal of Automated Reasoning, 19(3):263–276, 1997.
William W. McCune, Ed. Proceedings of the 14th CADE, volume 1249 of LNAI. Springer, 1997.
Michael McRobbie and John Slaney, Eds. Proceedings of the 13th CADE, volume 1104 of LNAI. Springer, 1996.
Jürgen Müller and Rolf Socher-Ambrosius. Topics in completion theorem proving. Technical Report SEKI SR-88-13, Fachbereich Informatik, Univ. Kaiserslautern, 1988.
Xumin Nie and David A. Plaisted. A complete semantic back chaining proof system. Pages 16–27 in [135].
Robert Niewenhuis, José Miguel Rivero, and Miguel Angel Vallejo. The Barcelona prover. In [136].
Robert Niewenhuis and A. Rubio. Theorem proving with ordering and equality constrained clauses. Journal of Symbolic Computation, 19(4):321–351, 1995.
Judea Pearl. Heuristics — Intelligent Search Strategies for Computer Problem Solving. Addison Wesley, 1984.
Gerald E. Peterson. A technique for establishing completeness results in theorem proving with equality. SIAM Journal of Computing, 12(1):82–100, 1983.
David A. Plaisted. Equational reasoning and term rewriting systems. Pages 273–364 in [62].
David A. Plaisted. A simplified problem reduction format. Artificial Intelligence, 18:227–261, 1982.
David A. Plaisted. Non-Horn clause logic programming without contrapositives. Journal of Automated Reasoning, 4(3):287–325, 1988.
David A. Plaisted. Mechanical theorem proving. In Ranan B. Banerji, editor, Formal Techniques in Artificial Intelligence. Elsevier, 1990.
David A. Plaisted. A sequent-style model elimination strategy and a positive refinement. Journal of Automated Reasoning, 6(4):389–402, 1990.
David A. Plaisted and Yunshan Zhu. The Efficiency of Theorem Proving Strate gies. Friedr. Vieweg & Sohns, 1997.
David A. Plaisted and Yunshan Zhu. Ordered semantic hyper linking. In Proceedings of AAAI-97, 1997.
Allan Ramsay. Formal Methods in Artificial Intelligence. Cambridge University Press, 1988.
G. Robinson and Larry Wos. Paramodulation and theorem-proving in first-order theories with equality. In D. Michie and R. Meltzer, editors, Machine Intelligence, volume IV, pages 135–150. Edinburgh Univ. Press, 1969.
J. Alan Robinson. Automatic deduction with hyper-resolution. International Journal of Computer Mathematics, 1:227–234, 1965.
J. Alan Robinson. A machine oriented logic based on the resolution principle. Journal of the ACM, 12(1):23–41, 1965.
Michaël Rusinowitch. Theorem-proving with resolution and superposition. Journal of Symbolic Computation, 11(1 & 2):21–50, 1991.
Johann Schumann. Delta: a bottom-up pre-processor for top-down theorem provers. Pages 774–777 in [41].
Robert E. Shostak. Refutation graphs. Artificial Intelligence, 7:51–64, 1976.
Jörg H. Siekmann and Graham Wrightson, Eds. Automation of reasoning-Classical Papers on Computational Logic. Springer Verlag, 1983.
Jörg H. Siekmann, Ed. Proceedings of the 8th CADE, volume 230 of LNCS. Springer, 1986.
James R. Slagle. Automatic theorem proving with renamable and semantic resolution. Journal of the ACM, 14(4):687–697, 1967.
John Slaney. FINDER: finite domain enumerator. Pages 798–801 in [41].
Raymond M. Smullyan. First-Order Logic. Dover, 1995. (Republication of the work first published as “Band 43” Series Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer Verlag, 1968).
Rolf Socher-Ambrosius. How to avoid the derivation of redundant clauses in reasoning systems. Journal of Automated Reasoning, 9(1):77–98, 1992.
Rolf Socher-Ambrosius and Patricia Johann. Deduction systems. Springer, 1997.
Mark E. Stickel. PTTP and linked inference. Pages 283–296 in [40].
Mark E. Stickel. A Prolog technology theorem prover. New Generation Computing, 2(4):371–383, 1984.
Mark E. Stickel. Automated deduction by theory resolution. Journal of Automated Reasoning, 1:333–355, 1985.
Mark E. Stickel. A Prolog technology theorem prover: implementation by an extended Prolog compiler. Journal of Automated Reasoning, 4:353–380, 1988.
Mark E. Stickel. The path-indexing method for indexing terms. Technical Report 473, SRI International, 1989.
Mark E. Stickel, Richard Waldinger, Michael Lowry, Thomas Pressburger, and Ian Underwood. Deductive composition of astronomical software from subroutine libraries. Pages 341–355 in [41].
Mark E. Stickel, Ed. Proceedings of the 10th CADE, volume 449 of LNAI. Springer, 1990.
Geoff Sutcliffe and Christian Suttner, Eds. The CADE-13 ATP system competition. Journal of Automated Reasoning, 18(2), 1997.
Tanel Tammet. Gandalf. Pages 199–204 in [136].
Laurent Vigneron. Automated deduction techniques for studying rough algebras. Fundamenta Informaticae, 33:85–103, 1998.
Kevin Wallace and Graham Wrightson. Regressive merging in model elimination tableau-based theorem provers. Journal of the IGPL, 3(6):921–937, 1995.
David H. D. Warren. An abstract Prolog instruction set. Technical Report 309, SRI International, 1983.
David S. Warren. Memoing for logic programs. Communications of the ACM, 35(3):94–111, 1992.
Christoph Weidenbach, B. Gaede, and G. Rock. SPASS & FLOTTER, version 0.42. Pages 141–145 in [101].
Larry Wos, D. Carson, and G. Robinson. Efficiency and completeness of the set of support strategy in theorem proving. Journal of the ACM, 12:536–541, 1965.
Larry Wos, G. Robinson, D. Carson, and L. Shalla. The concept of demodulation in theorem proving. Journal of the ACM, 14(4):698–709, 1967.
Lary Wos, Ross Overbeek, Ewing Lusk, and J. Boyle. Automated Reasoning: Introduction and Applications. McGraw-Hill, 2nd edition, 1992.
Hantao Zhang. SATO: an efficient propositional prover. Pages 272–275 in [100].
Hantao Zhang. A new method for the boolean ring based theorem proving. Journal of Symbolic Computation, 17(2):189–211, 1994.
Hantao Zhang, Maria Paola Bonacina, and Jieh Hsiang. PSATO: a distributed propositional prover and its application to quasigroup problems. Journal of Symbolic Computation, 21:543–560, 1996.
Hantao Zhang and Mark E. Stickel. Implementing the Davis-Putnam algorithm by tries. Technical Report 94-12, Department of Computer Science, University of Iowa, 1994.
Jian Zhang and Hantao Zhang. Generating models by SEM. Pages 308–312 in [101].
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Bonacina, M.P. (1999). A Taxonomy of Theorem-Proving Strategies. In: Wooldridge, M.J., Veloso, M. (eds) Artificial Intelligence Today. Lecture Notes in Computer Science(), vol 1600. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48317-9_3
Download citation
DOI: https://doi.org/10.1007/3-540-48317-9_3
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-66428-4
Online ISBN: 978-3-540-48317-5
eBook Packages: Springer Book Archive