Semantically guided first-order theorem proving using hyper-linking

  • Heng Chu
  • David A. Plaisted
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 814)

Abstract

We present a new procedure, semantic hyper-linking, which uses semantics to guide an instance-based clause-form theorem prover. Semantics for the input clauses is given as input. During the search for the proof, ground instances of the input clauses are generated and new semantic structures are built based on the input semantics and a model of the ground clause set. A proof is found if the ground clause set is unsatisfiable. We give some results in proving hard theorems using semantic hyper-linking; no other special human guidance was given to prove those hard problems. We also show that our method is powerful even with a trivial semantics (that is, even with no guidance in the form of semantic information).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. M. Ballantyne and W. W. Bledsoe. On generating and using examples in proof discovery. Machine Intelligence, 10:3–39, 1982.Google Scholar
  2. 2.
    W. W. Bledsoe. Using examples to generate instantiations of set variables. In Proc. of the 8th IJCAI, pages 892–901, Karlsruhe, FRG, 1983.Google Scholar
  3. 3.
    W. W. Bledsoe. Challenge problems in elementary calculus. J. Automated Reasoning, 6:341–359, 1990.Google Scholar
  4. 4.
    M. Bruschi. The halting problem. AAR Newsletter, March 1991.Google Scholar
  5. 5.
    Ricardo Caferra and Nicolas Zabel. A method for simutaneous search for refutations and models by equational constaint solving. J. Symbolic Computation, 13:613–641, 1992.Google Scholar
  6. 6.
    Heng Chu. Semantically Guided First-Order Theorem Proving Using Hyper-Linking. PhD thesis, University of North Carolina at Chapel Hill, 1994. Expected.Google Scholar
  7. 7.
    Heng Chu and David A. Plaisted. Model finding in semantically guided instanebased theorem proving. Foundamenta Informaticae Journal, 1993. To appear.Google Scholar
  8. 8.
    Heng Chu and David A. Plaisted. Model finding strategies in semantically guided instance-based theorem proving. In Jan Komorowski and Zbigniew W. RaŚ, editors, Proceedings of the 7th International Symposium on Methodologies for Intelligent Systems (ISMIS-93), pages 19–28, 15–18 June 1993.Google Scholar
  9. 9.
    Heng Chu and David A. Plaisted. Rough resolution: A refinement of resolution to remove large literals. In Proceedings of the Eleventh National Conference on Artificial Intelligence (AAAI-93), pages 15–20, 11–15 July 1993.Google Scholar
  10. 10.
    Heng Chu and David A. Plaisted. The use of presburger formulas in semantically guided theorem proving. Presented in The Third International Symposium on Artificial Intelligence and Mathematics, January 1994.Google Scholar
  11. 11.
    M. Davis and H. Putnam. A computing procedure for quantification theory. J. ACM, 7(3):201–215, 1960.Google Scholar
  12. 12.
    P. C. Gilmore. A proof method for quantification theory: its justification and realization. IBM J. Res. Dev., pages 28–35, 1960.Google Scholar
  13. 13.
    J. Herbrand. Researches in the theory of demonstration. In J. van Heijenoort, editor, From Frege to Gödel: a source book in Mathematical Logic, 1879–1931, pages 525–581. Harvard Univ. Press, 1974.Google Scholar
  14. 14.
    Shie-Jue Lee. CLIN: An Automated Reasoning System Using Clause Linking. PhD thesis, University of North Carolina at Chapel Hill, 1990.Google Scholar
  15. 15.
    Shie-Jue Lee and David. A. Plaisted. Eliminating duplication with the hyper-linking strategy. J. Automated Reasoning, 9:25–42, 1992.Google Scholar
  16. 16.
    Rainer Manthey and François Bry. SATCHMO: a theorem prover implemented in Prolog. In E. Lusk and R. Overbeek, editors, Proc. of CADE-9, pages 415–434, Argonne, IL, 1988.Google Scholar
  17. 17.
    William W. McCune. OTTER 2.0 Users Guide. Argonne National Laboratory, Argonne, Illinois, March 1990.Google Scholar
  18. 18.
    Xumin Nie and David A. Plaisted. A complete semantic back chaining proof system. In Mark E. Stickel, editor, Proc. of CADE-10, pages 16–27, Kaiserslautern, Germany, 1990.Google Scholar
  19. 19.
    F. J. Pelletier. Seventy-five problems for testing automatic theorem provers. J. Automated Reasoning, 2:919–216, 1986.Google Scholar
  20. 20.
    D. Plaisted and S. Greenbaum. A structure-preserving clause form translation. J. Symbolic Computation, 2:293–304, 1986.Google Scholar
  21. 21.
    David A. Plaisted. Non-Horn clause logic programming without contrapositives. J. Automated Reasoning, 4:287–325, 1988.Google Scholar
  22. 22.
    David. A. Plaisted, Geoffrey D. Alexander, Heng Chu, and Shie-Jue Lee. Conditional term rewriting and first-order theorem proving. In Proceedings of the Third International Workshop on Conditional Term-Rewriting Systems, Pont-à-Mousson, France, 8–10 July 1992. Invited Talk.Google Scholar
  23. 23.
    John K. Slaney and Ewing L. Lusk. Finding models: Techniques and applications. Tutorial in CADE-11, 1992.Google Scholar
  24. 24.
    H. Wang. Formalization and automatic theorem-proving. In Proc. of IFIP Congress 65, pages 51–58, Washington, D.C., 1965.Google Scholar
  25. 25.
    Tie Cheng Wang. Designing examples for semantically guided hierarchical deduction. In Proc. of the 9th IJCAI, pages 1201–1207, Los Angeles, CA, 1985.Google Scholar
  26. 26.
    Tie-Cheng Wang and W. W. Bledsoe. Hierarchical deduction. J. Automated Reasoning, 3:35–77, 1987.Google Scholar
  27. 27.
    S. Winker. Generation and verification of finite models and counterexamples using an automated theorem prover answering two open questions. J. ACM, 29:273–284, 1982.Google Scholar
  28. 28.
    L. Wos and S. Winker. Open questions solved with the assistance of AURA. In W. Bledsoe and D. Loveland, editors, Automated Theorem Proving: After 25 Years, pages 5–48. American Mathematical Society, Providence, RI, 1984.Google Scholar
  29. 29.
    Larry Wos. Automated Reasoning: 33 Basic Research Problems. Prentice Hall, Englewood Cliffs, NJ, 1988.Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Heng Chu
    • 1
    • 2
    • 3
  • David A. Plaisted
    • 1
    • 2
    • 3
  1. 1.Department of Computer ScienceUniversity of North CarolinaChapel HillUSA
  2. 2.MPI fuer Informatik Im StadtwaldSaarbruecken
  3. 3.Fachbereich InformatikUniversitaet Kaiserslautern675 Kaiserslautern

Personalised recommendations