On the Evaluation of Indexing Techniques for Theorem Proving

  • Robert Nieuwenhuis
  • Thomas Hillenbrand
  • Alexandre Riazanov
  • Andrei Voronkov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2083)

Abstract

The problem of term indexing can be formulated abstractly as follows (see [19]). Given a set L of indexed terms, a binary relation R over terms (called the retrieval condition) and a term t (called the query term), identify the subset M of L that consists of the terms l such that R(l; t) holds. Terms in M will be called the candidate terms. Typical retrieval conditions used in first-order theorem proving are matching, generalization, unifiability, and syntactic equality. Such a retrieval of candidate terms in theorem proving is interleaved with insertion of terms to L, and deletion of them from L.

References

  1. 1.
    P. Cheeseman, B. Kanefsky, and W. M. Taylor. Where the really hard problems are. In R. Reiter J. Mylopoulos, editor, IJCAI 1991. Proceedings of the 12th International Joint Conference on Artificial Intelligence, pages 331–340, Sydney, Australia, 1991. Morgan Kaufmann.Google Scholar
  2. 2.
    J. Christian. Flatterms, discrimination nets, and fast term rewriting. Journal of Automated Reasoning, 10(1):95–113, February 1993.MathSciNetCrossRefGoogle Scholar
  3. 3.
    H. Ganzinger, R. Nieuwenhuis, and P. Nivela. Context trees. In IJCAR 2001, Proceedings of the International Joint Conference on Automated Reasoning, Lecture Notes in Artificial Intelligence, Siena, Italy, June 2001. Springer Verlag.Google Scholar
  4. 4.
    P. Graf. Extended path-indexing. In A. Bundy, editor, CADE-12. 12th International Conference on Automated Deduction, volume 814 of Lecture Notes in Artificial Intelligence, pages 514–528, Nancy, France, June/July 1994.Google Scholar
  5. 5.
    P. Graf. Substitution tree indexing. In J. Hsiang, editor, Procs. 6th International Conference on Rewriting Techniques and Applications (RTA-95), volume 914 of Lecture Notes in Computer Science, pages 117–131, Kaiserslautern, 1995.CrossRefGoogle Scholar
  6. 6.
    P. Graf. Term Indexing, volume 1053 of Lecture Notes in Computer Science. Springer Verlag, 1996.Google Scholar
  7. 7.
    J. Gu, P.W. Purdom, J. Franco, and B.W. Wah. Algorithms for the Satisfiability Problems. Cambridge University Press, 2001.Google Scholar
  8. 8.
    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.Google Scholar
  9. 9.
    T. Hillenbrand, A. Buch, R. Vogt, and B. Löchner. Waldmeister: High-performance equational deduction. Journal of Automated Reasoning, 18(2):265–270, 1997.CrossRefGoogle Scholar
  10. 10.
    A. Martelli and U. Montanari. An efficient unification algorithm. ACM Transactions on Programming Languages and Systems, 4(2):258–282, 1982.CrossRefMATHGoogle Scholar
  11. 11.
    W.W. McCune. Experiments with discrimination-tree indexing and path indexing for term retrieval. Journal of Automated Reasoning, 9(2):147–167, 1992.MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    W.W. McCune. OTTER 3.0 reference manual and guide. Technical Report ANL-94/6, Argonne National Laboratory, January 1994.Google Scholar
  13. 13.
    D.G. Mitchell, B. Selman, and H.J. Levesque. Hard and easy distributions of SAT problems. In W.R. Swartout, editor, Procs. 10th National Conference on Artificial Intelligence, pages 459–465, San Jose, CA, January 1992. AAAI Press/MIT Press.Google Scholar
  14. 14.
    R. Nieuwenhuis. Rewrite-based deduction and symbolic constraints. In H. Ganzinger, editor, CADE-16. 16th Int. Conf. on Automated Deduction, Lecture Notes in Artificial Intelligence, pages 302–313, Trento, Italy, July 1999.Google Scholar
  15. 15.
    R. Nieuwenhuis, J.M. Rivero, and M.Á. Vallejo. The Barcelona prover. Journal of Automated Reasoning, 18(2):171–176, 1997.CrossRefGoogle Scholar
  16. 16.
    H.J. Ohlbach. Abstraction tree indexing for terms. In H.-J. Börkert and W. Nutt, editors, Extended Abstracts of the Third International Workshop on Unification, pages 131–135. Universität Kaiserslautern, 1989. SEKI-Report SR 89-17.Google Scholar
  17. 17.
    M. Paterson and M. Wegman. Linear unification. Journal of Computer and System Sciences, 16:158–167, 1978.MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    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.CrossRefGoogle Scholar
  19. 19.
    I.V. Ramakrishnan, R. Sekar, and A. Voronkov. Term indexing. In A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning, pages 1–97. Elsevier Science and MIT Press, 2001. To appear.Google Scholar
  20. 20.
    A. Riazanov and A. Voronkov.Vampire. In H. Ganzinger, editor, CADE-16. 16th International Conference on Automated Deduction, volume 1632 of Lecture Notes in Artificial Intelligence, pages 292–296, Trento, Italy, July 1999.Google Scholar
  21. 21.
    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.Google Scholar
  22. 22.
    J.M.A. Rivero. Data Structures and Algorithms for Automated Deduction with Equality. Phd thesis, Universitat Politèecnica de Catalunya, Barcelona, May 2000.Google Scholar
  23. 23.
    S. Schulz. Learning Search Control Knowledge for Equational Deduction, volume 230 of Dissertationen zur könstliche Intelligenz. Akademische Verlagsgesellschaft Aka GmmH, 2000.Google Scholar
  24. 24.
    M. Stickel. The path indexing method for indexing terms. Technical Report 473, Artificial Intelligence Center, SRI International, Menlo Park, CA, October 1989.Google Scholar
  25. 25.
    G. Sutcliffe and C. Suttner. The TPTP problem library — CNF release v. 1.2.1. Journal of Automated Reasoning, 21(2), 1998.Google Scholar
  26. 26.
    A. Voronkov. The anatomy of Vampire: Implementing bottom-up procedures with code trees. Journal of Automated Reasoning, 15(2):237–265, 1995.MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    A. Voronkov. CASC 16 1/2. Preprint CSPP-4, Department of Computer Science, University of Manchester, February 2000.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Robert Nieuwenhuis
    • 1
  • Thomas Hillenbrand
    • 2
  • Alexandre Riazanov
    • 3
  • Andrei Voronkov
    • 3
  1. 1.Technical University of CataloniaSpain
  2. 2.Universität KaiserslauternGermany
  3. 3.University of ManchesterUK

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