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

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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.

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References

  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. Anantharaman, S. and Hsiang, J., “Automated Proofs of the Moufang Identities in Alternative Rings,”Journal of Automated Reasoning, 6, 1, pp. 76–109, 1990.

    Article  MathSciNet  Google Scholar 

  3. Astrachan, O. L., “Investigations in Theorem Proving Based on Model Elimination,”Ph.D. thesis, Department of Computer Science, Duke University, 1992.

  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. 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. 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.

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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. Chang, C. L. and Lee, R. C.,Symbolic Logic and Mechanical Theorem Proving, Academic Press, New York, U.S.A., 1973.

    MATH  Google Scholar 

  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. Davis, M. and Putnam, H., “A Computing Procedure for Quantification Theory,”Journal of the ACM, 7, pp. 201–215, 1960.

    Article  MATH  MathSciNet  Google Scholar 

  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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. Korf, R. E., “Depth-First Iterative Deepening: An Optimal Admissible Tree Search,”Artificial Intelligence, 27, 1, pp. 97–109, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  20. Loveland, D. W., “A Simplified Format for the Model Elimination Procedure,”Journal of the ACM, 16, 3, pp. 349–363, 1969.

    Article  MATH  MathSciNet  Google Scholar 

  21. Loveland, D. W., “A Unifying View of Some Linear Herbrand Procedures,”Journal of the ACM, 19, 2, pp. 366–384, 1972.

    Article  MATH  MathSciNet  Google Scholar 

  22. McCune, W., “Experiments with Discrimination Tree Indexing and Path Indexing for Term Retrieval,”Journal of Automated Reasoning, 9, 2, pp. 147–167, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  23. McCune, W., “Otter 3.0 Reference Manual and Guide,”Technical Report, 94/6, Mathematics and Computer Science Division, Argonne National Laboratory, 1994.

  24. McCune, W., “Solution of the Robbins Problem,”Pre-print of the Mathematics and Computer Science Division, Argonne National Laboratory, 1997.

  25. Plaisted, D. A., “Non-Horn Clause Logic Programming without Contrapositives,”Journal of Automated Reasoning, 4, 3, pp. 287–325, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  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.

  27. Robinson, J. A., “Automatic Deduction with Hyper-Resolution,”International Journal of Computer Mathematics, 1, pp. 227–234, 1965.

    MATH  Google Scholar 

  28. Rusinowitch, M., “Theorem-Proving with Resolution and Superposition,”Journal of Symbolic Computation, 11, 1 & 2, pp. 21–50, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  29. Shostak, R. E., “Refutation Graphs,”Artificial Intelligence, 7, pp. 51–64, 1976.

    Article  MathSciNet  Google Scholar 

  30. Slagle, J. R., “Automatic Theorem Proving with Renamable and Semantic Resolution,”Journal of the ACM, 14, 4, pp. 687–697, 1967.

    Article  MATH  MathSciNet  Google Scholar 

  31. Stickel, M. E., “A Prolog Technology Theorem Prover: Implementation by an Extended Prolog Compiler,”Journal of Automated Reasoning, 4, pp. 353–380, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  32. Stickel, M. E., “The Path-Indexing Method for Indexing Terms,”Technical Report, 473, SRI International, 1989.

  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. Wallace, K. and Wrightson, G., “Regressive Merging in Model Elimination Tableau-Based Theorem Provers,”Journal of the IGPL, 3, 6, pp. 921–937, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  35. Warren, D. H. D., “An Abstract Prolog Instruction Set,”Technical Report, 309, SRI International, 1983.

  36. Warren, D. S., “Memoing for Logic Programs,”Communications of the ACM, 35, 3, pp. 94–111, 1992.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Iowa, 52242-1419, Iowa City, IA, USA

    Maria Paola Bonacina

  2. Department of Computer Science, National Taiwan University, Taipei, Taiwan

    Jieh Hsiang

Authors
  1. Maria Paola Bonacina
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  2. Jieh Hsiang
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Additional information

Supported in part by the National Science Foundation with grant CCR-94-08667 and CCR-97-01508.

Supported in part by grant NSC 86-2213-E-002-047 and NSC 87-2213-E-002-029 of the National Science Council of the Republic of China.

Maria Paola Bonacina, Ph.D.: She is on the faculty of the Department of Computer Science of the University of Iowa. She received a laurea (1986) and a doctorate in informatics from the Universita degli Studi di Milano, and a Ph.D. in computer science from the State University of New York at Stony Brook (1992). She was awarded fellowships by the Universita degli Studi di Milano, the European Union and the General Electric Foundation. The unifying theme of her research is automated theorem proving. Her research areas include distributed automated deduction, the theory of search and strategy analysis, completion-based theorem proving, category theory for computer science, term rewriting systems, logic programming, and manyvalued logic.

Jieh Hsiang, Ph.D.: He is a Professor of Computer Science at the National Taiwan University and is also the Director of the Center of Excellence for Research in Computer Systems of the National Taiwan University. Professor Hsiang is known for work in term rewriting systems and automated deduction. His other research interests include logic programming, programming logics, computer viruses, and intelligent agents. Recently he has also become interested in the digitization of Taiwanese and Chinese historical records and heritage. Professor Hsiang received a B.S. degree in mathematics from National Taiwan University in 1976 and a Ph.D. degree in computer science from the University of Illinois at Urbana-Champaign in 1982. Before returning to Taiwan in 1993, he was a Professor of Computer Science at the State University of New York st Stony Brook.

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Bonacina, M.P., Hsiang, J. On semantic resolution with lemmaizing and contraction and a formal treatment of caching. NGCO 16, 163–200 (1998). https://doi.org/10.1007/BF03037315

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  • DOI: https://doi.org/10.1007/BF03037315

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