Abstract
Algorithms and data structures form the kernel of any efficient theorem prover. In this abstract we discuss research on algorithms and data structures for efficient theorem proving based on our experience with the theorem prover Vampire. We also briefly overview other works related to algorithms and data structures, and to efficient theorem proving in general.
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References
F. Ajili and E. Contejean. Avoiding slack variables in the solving of linear Diophantine equations and in equations. Theoretical Computer Science, 173(1):183–208, 1997.
J. Avenhaus, J. Denzinger, and M. Fuchs. DISCOUNT: a system for distributed equational deduction. In J. Hsiang, editor, Proceedings of the 6th International Conference on Rewriting Techniques and Applications (RTA-95), volume 914 of Lecture Notes in Computer Science, pages 397–402, Kaiserslautern, 1995.
J. Avenhaus and B. Löchner. Cce: Testing ground join ability. In R. Gore, A. Leitsch, and T. Nipkow, editors, IJCAR 2001, 2001. this volume.
F. Baader and W. Snyder. Unification theory. In A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning, volume I, chapter 8, pages 445–532. Elsevier Science, 2001.
L. Bachmair, T. Chen, and I.V. Ramakrishnan. Associative-commutative discrimination nets. In M.-C. Gaudel and J.-P. Jouannaud, editors, Proceedings of the 4th International Joint Conference on Theory and Practice of Software Development (TAPSOFT), volume 668 of Lecture Notes in Computer Science, pages 61–74, Orsay, France, April 1993. Springer Verlag.
L. Bachmair and H. Ganzinger. Resolution theorem proving. In A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning, volume I, chapter 2, pages 19–99. Elsevier Science, 2001.
L. Bachmair, H. Ganzinger, C. Lynch, and W. Snyder. Basic par modulation and superposition. In D. Kapur, editor, 11th International Conference on Automated Deduction, volume 607 of Lecture Notes in Artificial Intelligence, pages 462–476, Saratoga Springs, NY, USA, June 1992. Springer Verlag.
P. Baumgartner. FDPLL — a first-order Davis-Putnam-Logemann-Loveland procedure. In D. McAllester, editor, 17th International Conference on Automated Deduction (CADE-17), volume 1831 of Lecture Notes in Artificial Intelligence, pages 200–219, Pittsburgh, 2000. Springer Verlag.
L.D. Baxter. The NP-completeness of sub sumption. Unpublished manuscript, 1977.
C. Benzmüller and M. Kohlhase. Extensional higher-order resolution. In C. Kirchner and H. Kirchner, editors, Automated Deduction — CADE-15. 15th International Conference on Automated Deduction, volume 1421 of Lecture Notes in Artificial Intelligence, pages 56–71, Lindau, Germany, 1998. Springer Verlag.
C. Benzmüller and M. Kohlhase. LEO—a higher-order theorem prover. In C. Kirchner and H. Kirchner, editors, Automated Deduction — CADE-15. 15th International Conference on Automated Deduction, volume 1421 of Lecture Notes in Artificial Intelligence, pages 139–144, Lindau, Germany, 1998. Springer Verlag.
A. Bundy. The automation of proof by mathematical induction. In A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning, volume I, chapter 13, pages 845-11. Elsevier Science, 2001.
J. Christian. Flatterms, discrimination nets, and fast term rewriting. Journal of Automated Reasoning, 10(1):95–113, February 1993.
H. Comon. Solving symbolic ordering constraints. International Journal of Foundations of Computer Science, 1(4):387–411, 1990.
H. Comon and R. Treinen. Ordering constraints on trees. In S. Tison, editor, Trees in Algebra and Programming: CAAP’94, volume 787 of Lecture Notes in Computer Science, pages 1–14. Springer Verlag, 1994.
H. de Nivelle. Bliksem 1.10 User’s Manual. MPI für Informatik, Saarbrücken, 2000.
A. Degtyarev. On the forms of inference in calculi with equality and Para modulation. In Yu.V. Kapitonova, editor, Automation of Research in Mathematics, pages 14–26. Institute of Cybernetics, Kiev, Kiev, 1982.
A. Degtyarev and A. Voronkov. Equality elimination for semantic tableaux. UP-MAIL Technical Report 90, Uppsala University, Computing Science Department, December 1994.
A. Degtyarev and A. Voronkov. Equality elimination for the tableau method. In J. Calmet and C. Limongelli, editors, Design and Implementation of Symbolic Computation Systems. International Symposium, DISCO’96, volume 1128 of Lecture Notes in Computer Science, pages 46–60, Karlsruhe, wGermany, September 1996.
A. Degtyarev and A. Voronkov. Stratified resolution. In D. McAllester, editor, 17th International Conference on Automated Deduction (CADE-17), volume 1831 of Lecture Notes in Artificial Intelligence, pages 365–384, Pittsburgh, 2000. Springer Verlag.
A. Degtyarev and A. Voronkov. Equality reasoning in sequent-based calculi. In A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning, volume I, chapter 10, pages 609–704. Elsevier Science, 2001.
J. Denzinger and D. Fuchs. Cooperation of heterogeneous provers. In T. Dean, editor, Proc. of the Sixteenth International Joint Conference on Artificial Intelligence (IJCAI-99), volume 1, pages 10–15, Stockholm, 1999.
A. Franke, S.M. Hess, C.G. Jung, M. Kohlhase, and V. Sorge. Agent-oriented integration of distributed mathematical services. Journal of Universal Computer Science, 5(3):156–187, 1999.
A. Franke and M. Kohlhase. Mathweb, an agent-based communication layer for distributed automated theorem proving. In H. Ganzinger, editor, Automated Deduction—CADE-16. 16th International Conference on Automated Deduction, volume 1632 of Lecture Notes in Artificial Intelligence, pages 217–221, Trento, Italy, July 1999.
A. Franke and M. Kohlhase. Mbase, an open mathematical knowledge base. In D. McAllester, editor, 17th International Conference on Automated Deduction (CADE-17), volume 1831 of Lecture Notes in Artificial Intelligence, pages 455–459, Pittsburgh, 2000. Springer Verlag.
H. Ganzinger, R. Nieuwenhuis, and P. Nivela. Context trees. In R. Gore, A. Leitsch, and T. Nipkow, editors, IJCAR 2001, 2001. this volume.
M.R. Garey and D.S. Johnson. Computers and Intractability. Freeman, San Francisco, 1979.
M. Gordon and T. Melham. Introduction to HOL. Cambridge University Press, 1993.
G. Gottlob and A. Leitsch. On the efficiency of subsumption algorithms. Journal of the Association for Computing Machinery, 32(2):280–295, April 1987.
P. Graf. Extended path-indexing. In A. Bundy, editor, Automated Deduction — CADE-12. 12th International Conference on Automated Deduction, volume 814 of Lecture Notes in Artificial Intelligence, pages 514–528, Nancy, France, June/July 1994.
P. Graf. Substitution tree indexing. In J. Hsiang, editor, Proceedings of the 6th International Conference on Rewriting Techniques and Applications (RTA-95), volume 914 of Lecture Notes in Computer Science, pages 117–131, Kaiserslautern, 1995.
P. Graf. Term Indexing, volume 1053 of Lecture Notes in Computer Science. Springer Verlag, 1996.
C. Hewitt. Description and theoretical analysis of Planner: a language for proving theorems and manipulating models in a robot. PhD thesis, Department of Mathematics, MIT, Cambridge, Mass., January 1971.
T. Hillenbrand, A. Buch, R. Vogt, and B. Löchner. Waldmeister: High-performance equational deduction. Journal of Automated Reasoning, 18(2):265–270, 1997.
J.-P. Jouannaud and M. Okada. Satisfiability of systems of ordinal notations with the sub term property is decidable. In J.L. Albert, B. Monien, and M. Rodríguez-Artalejo, editors, Automata, Languages and Programming, 18th International Colloquium, ICALP’91, volume 510 of Lecture Notes in Computer Science, pages 455–468, Madrid, Spain, 1991. Springer Verlag.
D. Kapur and P. Narendran. Double-exponential complexity of computing a complete set of AC-unifiers. In Proc. IEEE Conference on Logic in Computer Science (LICS). IEEE Computer Society Press, 1992.
C. Kirchner, H. Kirchner, and M. Rusinowitch. Deduction with symbolic constraints. Revue Francaise d’Intelligence Artificielle, 4(3):9–52, 1990. Special issue on automated deduction.
K. Korovin and A. Voronkov. A decision procedure for the existential theory of term algebras with the Knuth-Bendix ordering. In Proc. 15th Annual IEEE Symp. on Logic in Computer Science, pages 291–302, Santa Barbara, California, June 2000.
K. Korovin and A. Voronkov. Knuth-Bendix constraint solving is NP-complete. Preprint CSPP-8, Department of Computer Science, University of Manchester, November 2000. to appear in ICALP 2001.
R. Letz, J. Schumann, S. Bayerl, and W. Bibel. SETHEO: A high-performance theorem prover. Journal of Automated Reasoning, 8(2):183–212, 1992.
D.W. Loveland. Mechanical theorem proving by model elimination. Journal of the Association for Computing Machinery, 15:236–251, 1968.
E.L. Lusk. Controlling redundancy in large search spaces: Argonne-style theorem proving through the years. In A. Voronkov, editor, Logic Programming and Automated Reasoning. International Conference LPAR’92., volume 624 of Lecture Notes in Artificial Intelligence, pages 96–106, St.Petersburg, Russia, July 1992.
E.L. Lusk, W. McCune, and R.A. Overbeek. Logic machine architecture: Inference mechanisms. In D.W. Loveland, editor, Proceedings 6th Conference on Automated Deduction (CADE’ 82), number 138 in Lecture Notes in Computer Science, pages 85–108, New York, 1982. Springer Verlag.
E.L. Lusk, W. McCune, and R.A. Overbeek. Logic machine architecture: Kernel functions. In D.W. Loveland, editor, Proceedings 6th Conference on Automated Deduction (CADE’ 82), number 138 in Lecture Notes in Computer Science, pages 70–84, New York, 1982. Springer Verlag.
E.W. Lusk, W.W. McCune, and J. Slaney. ROO: a parallel theorem prover. In D. Kapur, editor, 11th International Conference on Automated Deduction (CADE), volume 607 of Lecture Notes in Artificial Intelligence, pages 731–734, Saratoga Springs, NY, USA, June 1992. Springer Verlag.
A. Martelli and U. Montanari. An efficient unification algorithm. ACM Transactions on Programming Languages and Systems, 4(2):258–282, 1982.
W. McCune. Solution of the robbins problem. Journal of Automated Reasoning, 19(3):263–276, 1997.
W. McCune. Automatic proofs and counterexamples for some ortholattice identities. Information Processing Letters, 65(6):285–291, 1998.
W.W. McCune. Experiments with discrimination-tree indexing and path indexing for term retrieval. Journal of Automated Reasoning, 9(2):147–167, 1992.
W.W. McCune. Single axioms for the left group and the right group calculi. Notre Dame J. of Formal Logic, 34(1):132–139, 1993.
W.W. McCune. OTTER 3.0 reference manual and guide. Technical Report ANL-94/6, Argonne National Laboratory, January 1994.
M. Moser, C. Lynch, and J. Steinbach. Model elimination with basic ordered paramodulation. Technical Report AR-95-11, Fakultät für Informatik, Technische Universität München, München, 1995.
W.W. McCune M.P. Bonacina. Distributed theorem proving by peers. In A. Bundy, editor, Automated Deduction — CADE-12. 12th International Conference on Automated Deduction, volume 814 of Lecture Notes in Artificial Intelligence, pages 841–845, Nancy, France, June/July 1994.
P. Narendran, M. Rusinowitch, and R. Verma. RPO constraint solving is in NP. In G. Gottlob, E. Grandjean, and K. Seyr, editors, Computer Science Logic, 12th International Workshop, CSL’98, volume 1584 of Lecture Notes in Computer Science, pages 385–398. Springer Verlag, 1999.
R. Nieuwenhuis. Simple LPO constraint solving methods. Information Processing Letters, 47:65–69, 1993.
R. Nieuwenhuis. Rewrite-based deduction and symbolic constraints. In H. Ganzinger, editor, Automated Deduction—CADE-16. 16th International Conference on Automated Deduction, Lecture Notes in Artificial Intelligence, pages 302–313, Trento, Italy, July 1999.
R. Nieuwenhuis, T. Hillenbrand, A. Riazanov, and A. Voronkov. On the evaluation of indexing techniques for theorem proving. In R. Gore, A. Leitsch, and T. Nipkow, editors, IJCAR 2001, 2001. this volume.
R. Nieuwenhuis and J.M. Rivero. Solved forms for path ordering constraints. In Proc. 10th International Conference on Rewriting Techniques and Applications (RTA), volume 1631 of Lecture Notes in Computer Science, pages 1–15, Trento, Italy, 1999.
R. Nieuwenhuis, J.M. Rivero, and M.Á. Vallejo. The Barcelona prover. Journal of Automated Reasoning, 18(2):171–176, 1997.
R. Nieuwenhuis and A. Rubio. Basic superposition is complete. In ESOP’92, volume 582 of Lecture Notes in Computer Science, pages 371–389. Springer Verlag, 1992.
R. Nieuwenhuis and A. Rubio. AC-superposition with constraints: no AC-unifiers needed. In A. Bundy, editor, Automated Deduction—CADE-12. 12th International Conference on Automated Deduction, volume 814 of Lecture Notes in Artificial Intelligence, pages 545–559, Nancy, France, June/July 1994.
R. Nieuwenhuis and A. Rubio. Theorem proving with ordering and equality constrained clauses. Journal of Symbolic Computations, 19:321–351, 1995.
R. Nieuwenhuis and A. Rubio. Paramodulation-based theorem proving. In A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning, volume I, chapter 7, pages 371–443. Elsevier Science, 2001.
H.J. Ohlbach. Abstraction tree indexing for terms. In H.-J. Bürkert and W. Nutt, editors, Extended Abstracts of the Third International Workshop on Unification, pages 131–135. Fachbereich Informatik, Universität Kaiserslautern, 1989. SEKI-Report SR 89-17.
M. Paterson and M. Wegman. Linear unification. Journal of Computer and System Sciences, 16:158–167, 1978.
L.C. Paulson. Isabelle: A Generic Theorem Prover. Springer-Verlag LNCS 828, 1994.
D. Plaisted and Y. Zhu. Replacement rules with definition detection. In R. Caferra and G. Salzer, editors, Automated Deduction in Classical and Non-Classical Logics, volume 1761 of Lecture Notes in Artificial Intelligence, pages 80–94. Springer Verlag, 1999.
G. Plotkin. Building-in equational theories. In B. Meltzer and D. Michie, editors, Machine Intelligence, volume 7, pages 73–90. Edinburgh University Press, 1972.
P.W. Purdom and C.A. Brown. Fast many-to-one matching algorithms. In J.-P. Jouannaud, editor, Rewriting Techniques and Applications, First International Conference, RTA-85, volume 202 of Lecture Notes in Computer Science, pages 407–416, Dijon, France, 1985. Springer Verlag.
A. Riazanov and A. Voronkov. Vampire. In H. Ganzinger, editor, Automated Deduction—CADE-16. 16th International Conference on Automated Deduction, volume 1632 of Lecture Notes in Artificial Intelligence, pages 292–296, Trento, Italy, July 1999.
A. Riazanov and A. Voronkov. Limited resource strategy in resolution theorem proving. Preprint CSPP-7, Department of Computer Science, University of Manchester, October 2000.
A. Riazanov and A. Voronkov. Partially adaptive code trees. In M. Ojeda-Aciego, I.P. de Guzmán, G. Brewka, and L.M. Pereira, editors, Logics in Artificial Intelligence. European Workshop, JELIA 2000, volume 1919 of Lecture Notes in Artificial Intelligence, pages 209–223, Málaga, Spain, 2000. Springer Verlag.
A. Riazanov and A. Voronkov. An efficient algorithm for backward sub sumption using path indexing and database joins. Preprint, Department of Computer Science, University of Manchester, 2001. To appear.
A. Riazanov and A. Voronkov. Splitting without backtracking. Preprint CSPP-10, Department of Computer Science, University of Manchester, January 2001. to appear at IJCAI’2001.
A. Riazanov and A. Voronkov. Vampire 1.1. (system description). In R. Gore, A. Leitsch, and T. Nipkow, editors, IJCAR 2001, 2001. this volume.
J.M.A. Rivero. Data Structures and Algorithms for Automated Deduction with Equality. Phd thesis, Universitat Politècnica de Catalunya, Barcelona, May 2000.
J.A. Robinson. A machine-oriented logic based on the resolution principle. Journal of the Association for Computing Machinery, 12(1):23–41, 1965.
S. Schulz. System abstract: E 0.3. In H. Ganzinger, editor, Automated Deduction— CADE-16. 16th International Conference on Automated Deduction, Lecture Notes in Artificial Intelligence, pages 297–301, Trento, Italy, July 1999.
S. Schulz. Learning Search Control Knowledge for Equational Deduction, volume 230 of Dissertationen zur künstliche Intelligenz. Akademische Verlagsgesellschaft Aka GmmH, 2000.
J. Schumann and R. Letz. PARTHEO: a high performance parallel theorem prover. In M.E. Stickel, editor, Proc. 10th Int. Conf. on Automated Deduction, volume 449 of Lecture Notes in Artificial Intelligence, pages 40–56, 1990.
R. Sekar, I.V. Ramakrishnan, and A. Voronkov. Term indexing. In A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning, volume II, chapter 26, pages 1853–1964. Elsevier Science, 2001.
J.K. Slaney, E.L. Lusk, and W. McCune. SCOTT: Semantically Constrained Otter (system description). In A. Bundy, editor, Automated Deduction—CADE-12. 12th International Conference on Automated Deduction, volume 814 of Lecture Notes in Artificial Intelligence, pages 764–768, Nancy, France, 1994.
Stickel. Automated deduction by theory resolution. Journal of Automated Reasoning, 1:333–355, 1985.
M. Stickel. The path indexing method for indexing terms. Technical Report 473, Artificial Intelligence Center, SRI International, Menlo Park, CA, October 1989.
M.E. Stickel. A complete unification algorithm for associative-commutative functions. In Advance Papers of the Fourth International Joint Conference on Artificial Intelligence, pages 71–76, Tbilisi, USSR, 1975.
M.E. Stickel. A complete unification algorithm for associative-commutative functions. Journal of the Association for Computing Machinery, 28(3):423–434, 1981.
M.E. Stickel. Schubert’s Steamroller problem: Formulation and solutions. Journal of Automated Reasoning, 2(1):89–101, 1986.
G. Sutcliffe. Systemon tptp. In D. McAllester, editor, 17th International Conference on Automated Deduction (CADE-17), volume 1831 of Lecture Notes in Artificial Intelligence, pages 406–410, Pittsburgh, 2000. Springer Verlag.
T. Tammet. Gandalf. Journal of Automated Reasoning, 18(2):199–204, 1997.
T. Tammet. Towards efficient subsumption. In C. Kirchner and H. Kirchner, editors, Automated Deduction — CADE-15. 15th International Conference on Automated Deduction, volume 1421 of Lecture Notes in Artificial Intelligence, pages 427–441, Lindau, Germany, 1998. Springer Verlag.
V. Turchin. Refal-5, Programming Guide and Reference Manual. New England Publishing Co., 1989.
A. Voronkov. LISS — the logic inference search system. In M. Stickel, editor, Proc. 10th Int. Conf. on Automated Deduction, volume 449 of Lecture Notes in Computer Science, pages 677–678, Kaiserslautern, Germany, 1990. Springer Verlag.
A. Voronkov. The anatomy of Vampire: Implementing bottom-up procedures with code trees. Journal of Automated Reasoning, 15(2):237–265, 1995.
A. Voronkov. Formulae over reduction orderings: some solved and yet unsolved problems. Invited talk at RTA’2000, unpublished, 2000.
C. Weidenbach. Combining superposition, sorts and splitting. In A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning, volume II, chapter 27, pages 1965–2013. Elsevier Science, 2001.
C. Weidenbach, B. Afshordel, U. Brahm, C. Cohrs, T. Engel, E. Keen, C. Theobalt, and D. Topic. System description: Spass version 1.0.0. In H. Ganzinger, editor, Automated Deduction—CADE-16. 16th International Conference on Automated Deduction, volume 1632 of Lecture Notes in Artificial Intelligence, pages 378–382, Trento, Italy, July 1999.
L. Wos, R. Overbeek, and E. Lusk. Subsumption, a sometimes undervalued procedure. In J.-L. Lassez and G. Plotkin, editors, Computational Logic. Essays in Honor of Alan Robinson, pages 3–40. The MIT Press, Cambridge, MA, 1991.
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Voronkov, A. (2001). Algorithms, Datastructures, and other Issues in Efficient Automated Deduction. In: Goré, R., Leitsch, A., Nipkow, T. (eds) Automated Reasoning. IJCAR 2001. Lecture Notes in Computer Science, vol 2083. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45744-5_3
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