Skip to main content
SpringerLink
Log in

Ordered Semantic Hyper-Linking

  • Published:
Journal of Automated Reasoning Aims and scope Submit manuscript

Abstract

The ordered semantic hyper-linking strategy is complete for first-order logic and accepts a user-specified natural semantics that guides the search for a proof. Any semantics in which the meanings of the function and predicate symbols are computable on ground terms may be used. This instance-based strategy is efficient on near-propositional problems, is goal sensitive, and has an extension to equality and term rewriting. However, it sometimes has difficulty generating large terms. We compare this strategy with some others that use semantic information, and present a proof of soundness and completeness. We also give some theoretical results about the search efficiency of the strategy. Some examples illustrate the performance of the strategy.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ballantyne, A. M. and Bledsoe, W. W.: On generating and using examples in proof discovery, in Michie and Pao (eds), Machine Intelligence, Vol. 10, Ellis Horwood, 1982, pp. 3-39.

  2. Bourely, Ch., Caferra, R. and Peltier, N.: A method for building models automatically: Experiments with an extension of OTTER, in Proceedings of CADE-12, Springer LNAI 814, 1994, pp. 72-86. http://www-leibniz.imag.fr/ATINF/PUBLICATIONS/nico-cade94.ps.gz.

  3. Bachmair, L., Dershowitz, N. and Plaisted, D.: Completion without failure, in H. Aït-Kaci and M. Nivat (eds), Resolution of Equations in Algebraic Structures 2: Rewriting Techniques, Academic Press, New York, 1989, pp. 1-30.

    Google Scholar 

  4. Baumgartner, P., Fröhlich, P., Furbach, U. and Nejdl, W.: Semantically guided theorem proving for diagnosis applications, in 15th International Joint Conference on Artificial Intelligence (IJCAI 97), 1997, pp. 460-465.

  5. Bachmair, L. and Ganzinger, H.: Ordered chaining for total orderings, in A. Bundy (ed.), Proceedings of the 12th International Conference on Automated Deduction, Springer-Verlag, New York, 1994, pp. 435-450.

    Google Scholar 

  6. Bachmair, L. and Ganzinger, H.: Equational reasoning in saturation-based theorem proving, in W. Bibel and P. H. Schmitt (eds), Automated Deduction-A Basis for Applications. Volume I: Foundations-Calculi and Methods, Kluwer Acad. Publ., Dordrecht, 1998, pp. 353-398.

    Google Scholar 

  7. Bachmair, L., Ganzinger, H., Lynch, C. and Snyder, W.: Basic paramodulation, Inform. and Comput. 121(2) (1995), 172-192.

    Google Scholar 

  8. Bachmair, L., Ganzinger, H. and Voronkov, A.: Elimination of equality via transformation with ordering constraints, in C. Kirchner and H. Kirchner (eds), Proceedings of the 15th International Conference on Automated Deduction, Springer-Verlag, New York, 1998, pp. 175-190.

    Google Scholar 

  9. Bibel, W.: Automated Theorem Proving, 2nd edn, Vieweg, Braunschweig/Wiesbaden, 1987.

    Google Scholar 

  10. Bledsoe, W. W.: Using examples to generate instantiations of set variables, in Proceedings of the 8th International Joint Conference on Artificial Intelligence, 1983, pp. 892-901.

  11. Baader, F. and Nipkow, T.: Term Rewriting and All That, Cambridge University Press, Cambridge, 1998.

    Google Scholar 

  12. Brand, D.: Proving theorems with the modification method, SIAM J. Comput. 4 (1975), 412-430.

    Google Scholar 

  13. Chang, C. and Lee, R.: Symbolic Logic and Mechanical Theorem Proving, Academic Press, New York, 1973.

    Google Scholar 

  14. Chu, H. and Plaisted, D.: Semantically guided first-order theorem proving using hyper-linking, in Proceedings of the Twelfth International Conference on Automated Deduction, Lecture Notes in Artif. Intell. 814, 1994, pp. 192-206.

  15. Caferra, R. and Peltier, N.: Extending semantic resolution via automated model building: Applications, in Proceeding of IJCAI'95, Morgan Kaufman, 1995, pp. 328-334. http://wwwleibniz. imag.fr/ATINF/PUBLICATIONS/nico-ijcai95.ps.gz.

  16. Caferra, R. and Peltier, N.: Disinference rules, model building and abduction, to appear in E. Or?owska (ed.), Logic at Work: Essays Dedicated to the Memory of Helena Rasiowa, 1997. http://www-leibniz.imag.fr/ATINF/PUBLICATIONS/nico-eva.ps.gz.

  17. Caferra, R. and Zabel, N.: A method for simultaneous search for refutations and models by equational constraint solving, J. Symbolic Comput. 13 (1992), 613-641.

    Google Scholar 

  18. Caferra, R. and Zabel, N.: Building models by using tableaux extended by equational problems, J. Logic Comput. 3 (1993), 3-25.

    Google Scholar 

  19. Dowling, W. and Gallier, J.: Linear-time algorithms for testing the satisfiability of propositional Horn formulae, J. Logic Programming 1 (1984), 267-284.

    Google Scholar 

  20. Dershowitz, N. and Jouannaud, J.-P.: Rewrite systems, in J. van Leeuwen (ed.), Handbook of Theoretical Computer Science, North-Holland, Amsterdam, 1990.

    Google Scholar 

  21. Davis, M., Logemann, G. and Loveland, D.: A machine program for theorem-proving, Communications of the ACM 5 (1962), 394-397.

    Google Scholar 

  22. Davis, M. and Putnam, H.: A computing procedure for quantification theory, J. ACM 7 (1960), 201-215.

    Google Scholar 

  23. Fermüller, C. and Leitsch, A.: Hyperresolution and automated model building, J. Logic Comput. 6(2) (1996), 173-203.

    Google Scholar 

  24. Gelernter, H., Hansen, J. R. and Loveland, D. W.: Empirical explorations of the geometry theorem proving machine, in E. Feigenbaum and J. Feldman (eds), Computers and Thought, McGraw-Hill, New York, 1963, pp. 153-167.

    Google Scholar 

  25. Ganzinger, H.,Meyer, C. and Weidenbach, C.: Soft typing for ordered resolution, in B. McCune (ed.), Proceedings of the Fourteenth Conference on Automated Deduction, 1997, pp. 321-335.

  26. Hasegawa, R., Inoue, K., Ohta, Y. and Koshimura, M.: Non-Horn magic sets to incorporate top-down inference into bottom-up theorem proving, in W. McCune (ed.), Proceedings of the 14th International Conference on Automated Deduction, July 1997, pp. 176-190.

  27. Hsiang, J. and Rusinowitch, M.: Proving refutational completeness of theorem-proving strategies: The transfinite semantic tree method, J. ACM 38(3) (1991), 559-587.

    Google Scholar 

  28. Leitsch, A.: The Resolution Calculus, Springer-Verlag, Berlin, 1997. Texts in Theoretical Computer Science.

    Google Scholar 

  29. Lloyd, J. W.: Foundations of Logic Programming, 2nd edn, Springer-Verlag, Berlin, 1987.

    Google Scholar 

  30. Lassez, J.-L. and Marriott, K. G.: Explicit representation of terms defined by counterexamples, J. Automated Reasoning 3(3) (1987), 1-17.

    Google Scholar 

  31. Lusk, E. and Overbeek, R.: Non-Horn problems, J. Automated Reasoning 1 (1985), 103-114.

    Google Scholar 

  32. Loveland, D.: A simplified format for the model elimination procedure, J. ACM 16 (1969), 349-363.

    Google Scholar 

  33. Loveland, D.: Automated Theorem Proving: A Logical Basis, North-Holland, New York, 1978.

    Google Scholar 

  34. Lee, S.-J. and Plaisted, D.: Eliminating duplication with the hyper-linking strategy, J. Automated Reasoning 9(1) (1992), 25-42.

    Google Scholar 

  35. Lee, S.-J. and Plaisted, D.: Use of replace rules in theorem proving, Methods of Logic in Computer Science 1 (1994), 217-240.

    Google Scholar 

  36. Loveland, D., Reed, D. and Wilson, D.: SATCHMORE: SATCHMO with RElevance, J. Automated Reasoning 14 (1995), 325-351.

    Google Scholar 

  37. Manthey, R. and Bry, F.: SATCHMO: A theorem prover implemented in Prolog, in Proceedings of the 9th Conference on Automated Deduction, Argonne, Illinois, May 1988, pp. 415-434.

  38. McCune, W.: Solution of the Robbins problem, J. Automated Reasoning 19(3) (1997), 263-276.

    Google Scholar 

  39. Nie, X. and Plaisted, D.: A complete semantic back chaining proof system, in Proceedings of the 10th International Conference on Automated Deduction, 1990.

  40. Peltier, N.: Increasing the capabilities of model building by constraint solving with terms with integer exponents, J. Symbolic Comput. 24 (1997), 59-101. http://wwwleibniz. imag.fr/ATINF/PUBLICATIONS/nico-jsc97.ps.gz.

    Google Scholar 

  41. Plaisted, D.: A simplified problem reduction format, Artif. Intell. 18 (1982), 227-261.

    Google Scholar 

  42. Plaisted, D.: Non-Horn clause logic programming without contrapositives, J. Automated Reasoning 4 (1988), 287-325.

    Google Scholar 

  43. Plaisted, D.: Ordered semantic hyper-linking, Technical Report MPI-I-94-235, Max-Planck Institut für Informatik, Saarbrücken, Germany, 1994.

    Google Scholar 

  44. Plaisted, D.: The search efficiency of theorem proving strategies: An analytical comparison, Technical Report MPI-I-94-233, Max-Planck Institut für Informatik, Saarbrücken, Germany, 1994.

    Google Scholar 

  45. Plaisted, D. and Zhu, Y.: The Efficiency of Theorem Proving Strategies: A Comparative and Asymptotic Analysis, Vieweg, Wiesbaden, 1997.

    Google Scholar 

  46. Plaisted, D. and Zhu, Y.: Equational reasoning using AC constraints, in Proceedings of the 15th International Joint Conference on Artificial Intelligence, 1997.

  47. Robinson, J.: A machine-oriented logic based on the resolution principle, J. ACM 12 (1965), 23-41.

    Google Scholar 

  48. Reif, W. and Schellhorn, G.: Theorem proving in large theories, in W. Bibel and P. H. Schmitt (eds), Automated Deduction-A Basis for Applications. Volume III: Applications, Kluwer Acad. Publ., Dordrecht, 1998, pp. 225-241.

    Google Scholar 

  49. Robinson, G. and Wos, L.: Paramodulation and theorem-proving in first order theories with equality, in Machine Intelligence 4, Edinburgh University Press, Edinburgh, 1969, pp. 135-150.

    Google Scholar 

  50. Slagle, J. R.: Automatic theorem proving with renameable and semantic resolution, J. ACM 14 (1967), 687-697.

    Google Scholar 

  51. Slaney, J.: SCOTT: A model-guided theorem prover, in R. Bajcsy (ed.), Proceedings of the Thirteenth International Joint Conference on Artificial Intelligence, 1993, pp. 109-14.

  52. Suttner, C. B. and Sutcliffe, G.: The TPTP problem library (TPTP v2.0.0), Technical Report AR-97-01, Institut für Informatik, Technische Universität München, Germany, 1997.

    Google Scholar 

  53. Wos, L., Overbeek, R., Lusk, E. and Boyle, J.: Automated Reasoning: Introduction and Applications, Prentice-Hall, Englewood Cliffs, NJ, 1984.

    Google Scholar 

  54. Wos, L., Robinson, G. and Carson, D.: Efficiency and completeness of the set of support strategy in theorem proving, J. ACM 12 (1965), 536-541.

    Google Scholar 

  55. Zhu, Y. and Plaisted, D.: FOLPLAN: A semantically guided first-order planner, in Proceedings of the 10th International FLAIRS Conference, 1997.

Download references

Authors
  1. David A. Plaisted
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yunshan Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Plaisted, D.A., Zhu, Y. Ordered Semantic Hyper-Linking. Journal of Automated Reasoning 25, 167–217 (2000). https://doi.org/10.1023/A:1006376231563

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1006376231563

Search

Navigation