New Generation Computing

, Volume 16, Issue 2, pp 163–200 | Cite as

On semantic resolution with lemmaizing and contraction and a formal treatment of caching

  • Maria Paola Bonacina
  • Jieh Hsiang
Regular Papers

Abstract

Reducing redundancy in search has been a major concern for automated deduction. Subgoal-reduction strategies, such as those based on model elimination and implemented in Prolog technology theorem provers, prevent redundant search by usinglemmaizing andcaching, whereas contraction-based strategies prevent redundant search by usingcontraction rules, such assubsumption. In this work we show that lemmaizing and contraction can coexist in the framework ofsemantic resolution. On the lemmaizing side, we define two meta-level inference rules for lemmaizing in semantic resolution, one producing unit lemmas and one producing non-unit lemmas, and we prove their soundness. Rules for lemmaizing are meta-rules because they use global knowledge about the derivation, e.g. ancestry relations, in order to derive lemmas. Our meta-rules for lemmaizing generalize to semantic resolution the rules for lemmaizing in model elimination. On the contraction side, we give contraction rules for semantic strategies, and we define apurity deletion rule for first-order clauses that preserves completeness. While lemmaizing generalizes success caching of model elimination, purity deletion echoes failure caching. Thus, our approach integrates features of backward and forward reasoning. We also discuss the relevance of our work to logic programming.

Keywords

Resolution Set-of-Support Lemmaizing Caching Forward and Backward Reasoning Theorem Proving Logic Programming 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1).
    Anantharaman, S. and Bonacina, M. P., “An Application of Automated Equational Reasoning to Many-Valued Logic,” inProceedings of the Second International Workshop on Conditional and Typed Term Rewriting Systems (M. Okada and S. Kaplan, eds.),number 516 in Lecture Notes in Computer Science, Montréal, Canada, June 1990, Springer-Verlag, pp. 156–161, 1990.Google Scholar
  2. 2).
    Anantharaman, S. and Hsiang, J., “Automated Proofs of the Moufang Identities in Alternative Rings,”Journal of Automated Reasoning, 6, 1, pp. 76–109, 1990.CrossRefMathSciNetGoogle Scholar
  3. 3).
    Astrachan, O. L., “Investigations in Theorem Proving Based on Model Elimination,”Ph.D. thesis, Department of Computer Science, Duke University, 1992.Google Scholar
  4. 4).
    Astrachan, O. L. and Loveland, D. W., “METEORs: High Performance Theorem Provers Using Model Elimination,” inAutomated Reasoning: Essays in Honor of Woody Bledsoe (R. S. Boyer, ed.), Automated Reasoning Series, Kluwer Academic Publisher, Dordrecht, Netherlands, pp. 31–60, 1991.Google Scholar
  5. 5).
    Astrachan, O. L. and Stickel, M. E., “Caching and Lemmaizing in Model Elimination Theorem Provers,” inEleventh Conference on Automated Deluction, volume 607 of Lecture Notes in Artificial Intelligence (D. Kapur, ed.), Saratoga Springs, New York, U.S.A., June 1992, Springer-Verlag, pp. 224–238, 1992. Full version available asTechnical Report, 513, SRI International, December 1991.Google Scholar
  6. 6).
    Bachmair, L. and Ganzinger, H., “On Restrictions of Ordered Paramodulation with Simplification,” inTenth Conference on Automated Deduction, volume 449 of Lecture Notes in Artificial Intelligence (M. E. Stickel, ed.), Springer-Verlag, pp. 427–441, 1990.Google Scholar
  7. 7).
    Bonacina, M. P. and Hsiang, J., “On Rewrite Programs: Semantics and Relationship with Prolog,”Journal of Logic Programming, 14, 1 & 2, pp. 155–180, October 1992.MATHCrossRefMathSciNetGoogle Scholar
  8. 8).
    Bonacina, M. P. and Hsiang, J., “Towards a Foundation of Completion Procedures as Semidecision Procedures,”Theoretical Computer Science, 146, pp. 199–242, July 1995.MATHCrossRefMathSciNetGoogle Scholar
  9. 9).
    Bonacina, M. P. and Hsiang, J., “On Semantic Resolution with Lemmaizing and Contraction,” inFourth Pacific Rim International Conference on Artificial Intelligence, volume 1114 of Lecture Notes in Artificial Intelligence (N. Foo and R. Goebel, eds.), Cairns, Australia, August 1996, Springer-Verlag, pp. 372–386, 1996.Google Scholar
  10. 10).
    Chang, C. L. and Lee, R. C.,Symbolic Logic and Mechanical Theorem Proving, Academic Press, New York, U.S.A., 1973.MATHGoogle Scholar
  11. 11).
    Christian, J. D., “Fast Knuth-Bendix Completion: Summary,”, inThird Conference on Rewriting Techniques and Applications, volume 355 of Lecture Notes in Computer Science (N. Dershowitz, ed.), Chapel Hill, North Carolina, U.S.A., April 1989, Springer-Verlag, pp. 551–555, 1989.Google Scholar
  12. 12).
    Davis, M. and Putnam, H., “A Computing Procedure for Quantification Theory,”Journal of the ACM, 7, pp. 201–215, 1960.MATHCrossRefMathSciNetGoogle Scholar
  13. 13).
    Dershowitz, N., “Canonical Sets of Horn Clauses,” inEighteenth International Conference on Automata, Languages and Programming, volume 510 of Lecture Notes in Computer Science (J. Leach Albert, B. Monien, and M. Rodríguez Artalejo, eds.), Madrid, Spain, 1991, Springer-Verlag, pp. 267–278, 1991.Google Scholar
  14. 14).
    Fleisig, S., Loveland, D., Smiley, A., and Yarmush, D., “An Implementation of the Model Elimination Proof Procedure,”Journal of the ACM, 21, pp. 124–139, 1974.MATHCrossRefMathSciNetGoogle Scholar
  15. 15).
    Hsiang, J. and Rusinowitch, M., “Proving Refutational Completeness of Theorem Proving Strategies: The Transfinite Semantic Tree Method,”Journal of the ACM, 38, pp. 559–587, 1991.MATHCrossRefMathSciNetGoogle Scholar
  16. 16).
    Kapur, D. and Zhang, H., “RRL: A Rewrite Rule Laboratory,” inNineth Conference on Automated Deduction, volume 310 of Lecture Notes in Computer Science (E. Lusk and R. Overbeek, eds.), Argonne, Illinois, U.S.A., May 1988, Springer-Verlag, pp. 768–770, 1988.Google Scholar
  17. 17).
    Kapur, D. and Zhang, H., “A Case Study of the Completion Procedure: Proving Ring Commutativity Problems,” inComputational Logic: Essays in Honor of Alan Robinson (J.-L. Lassez and G. Plotkin, eds.), MIT Press, Cambridge, Massachusetts, pp. 360–394, 1991.Google Scholar
  18. 18).
    Korf, R. E., “Depth-First Iterative Deepening: An Optimal Admissible Tree Search,”Artificial Intelligence, 27, 1, pp. 97–109, 1985.MATHCrossRefMathSciNetGoogle Scholar
  19. 19).
    Letz, R., Schumann, J., Bayerl, S., and Bibel, W., “SETHEO: A High Performance Theorem Prover,”Journal of Automated Reasoning, 8, 2, pp. 183–212, 1992.MATHCrossRefMathSciNetGoogle Scholar
  20. 20).
    Loveland, D. W., “A Simplified Format for the Model Elimination Procedure,”Journal of the ACM, 16, 3, pp. 349–363, 1969.MATHCrossRefMathSciNetGoogle Scholar
  21. 21).
    Loveland, D. W., “A Unifying View of Some Linear Herbrand Procedures,”Journal of the ACM, 19, 2, pp. 366–384, 1972.MATHCrossRefMathSciNetGoogle Scholar
  22. 22).
    McCune, W., “Experiments with Discrimination Tree Indexing and Path Indexing for Term Retrieval,”Journal of Automated Reasoning, 9, 2, pp. 147–167, 1992.MATHCrossRefMathSciNetGoogle Scholar
  23. 23).
    McCune, W., “Otter 3.0 Reference Manual and Guide,”Technical Report, 94/6, Mathematics and Computer Science Division, Argonne National Laboratory, 1994.Google Scholar
  24. 24).
    McCune, W., “Solution of the Robbins Problem,”Pre-print of the Mathematics and Computer Science Division, Argonne National Laboratory, 1997.Google Scholar
  25. 25).
    Plaisted, D. A., “Non-Horn Clause Logic Programming without Contrapositives,”Journal of Automated Reasoning, 4, 3, pp. 287–325, 1988.MATHCrossRefMathSciNetGoogle Scholar
  26. 26).
    Plaisted, D. A., “The Search Efficiency of Theorem Proving Strategies,” inTwelfth Conference on Automated Deduction, volume 814 of Lecture Notes in Artificial Intelligence (A. Bundy, ed.), Springer-Verlag, pp. 57–71, 1994. Full version available asTechnical Report of the Max Planck Institut für Informatik, MPI-I-94-233.Google Scholar
  27. 27).
    Robinson, J. A., “Automatic Deduction with Hyper-Resolution,”International Journal of Computer Mathematics, 1, pp. 227–234, 1965.MATHGoogle Scholar
  28. 28).
    Rusinowitch, M., “Theorem-Proving with Resolution and Superposition,”Journal of Symbolic Computation, 11, 1 & 2, pp. 21–50, 1991.MATHCrossRefMathSciNetGoogle Scholar
  29. 29).
    Shostak, R. E., “Refutation Graphs,”Artificial Intelligence, 7, pp. 51–64, 1976.CrossRefMathSciNetGoogle Scholar
  30. 30).
    Slagle, J. R., “Automatic Theorem Proving with Renamable and Semantic Resolution,”Journal of the ACM, 14, 4, pp. 687–697, 1967.MATHCrossRefMathSciNetGoogle Scholar
  31. 31).
    Stickel, M. E., “A Prolog Technology Theorem Prover: Implementation by an Extended Prolog Compiler,”Journal of Automated Reasoning, 4, pp. 353–380, 1988.MATHCrossRefMathSciNetGoogle Scholar
  32. 32).
    Stickel, M. E., “The Path-Indexing Method for Indexing Terms,”Technical Report, 473, SRI International, 1989.Google Scholar
  33. 33).
    Stickel, M. E., “PTTP and Linked Inference,” inAutomated Reasoning: Essays in Honor of Woody Bledsoe (R. S. Boyer, ed.), Automated Reasoning Series, Kluwer Academic Publisher, Dordrecht, Netherlands, pp. 283–296, 1991.Google Scholar
  34. 34).
    Wallace, K. and Wrightson, G., “Regressive Merging in Model Elimination Tableau-Based Theorem Provers,”Journal of the IGPL, 3, 6, pp. 921–937, 1995.MATHCrossRefMathSciNetGoogle Scholar
  35. 35).
    Warren, D. H. D., “An Abstract Prolog Instruction Set,”Technical Report, 309, SRI International, 1983.Google Scholar
  36. 36).
    Warren, D. S., “Memoing for Logic Programs,”Communications of the ACM, 35, 3, pp. 94–111, 1992.CrossRefMathSciNetGoogle Scholar
  37. 37).
    Wos, L., Carson, D., and Robinson, G., “Efficiency and Completeness of the Set of Support Strategy in Theorem Proving,”Journal of the ACM, 12, pp. 536–541, 1965.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Ohmsha, Ltd. and Springer 1998

Authors and Affiliations

  • Maria Paola Bonacina
    • 1
  • Jieh Hsiang
    • 2
  1. 1.Department of Computer ScienceUniversity of IowaIowa CityUSA
  2. 2.Department of Computer ScienceNational Taiwan UniversityTaipeiTaiwan

Personalised recommendations