Advertisement

Invited Talk: Rewrite-Based Deduction and Symbolic Constraints

  • Robert Nieuwenhuis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1632)

Abstract

Building a state-of-the-art theorem prover requires the combination of at least three main ingredients: good theory, clever heuristics, and the necessary engineering skills to implement it all in an efficient way. Progress in each of these ingredients interacts in different ways.

On the one hand, new theoretical insights replace heuristics by more precise and effective techniques. For example, the completeness proof of basic paramod- ulation [NR95,BGLS95] shows why no inferences below Skolem functions are needed, as conjectured by McCune in [McC90]. Regarding implementation tech- niques, ad-hoc algorithms for procedures like demodulation or subsumption are replaced by efficient, re-usable, general-purpose indexing data structures for which the time and space requirements are well-known.

Keywords

Theorem Prove Ground Instance Automate Deduction Empty Clause 14th IEEE Symposium 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. BG94.
    Leo Bachmair and Harald Ganzinger. Rewrite-based equational theorem proving with selection and simplification. Journal of Logic and Computation, 4(3):217–247, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  2. BG98.
    Leo Bachmair and Harald Ganzinger. Equational reasoning in saturationbased theorem proving. In W. Bibel and P. Schmitt, editors, Automated Deduction: A Basis for Applications. Kluwer, 1998.Google Scholar
  3. BGLS95.
    L. Bachmair, H. Ganzinger, Chr. Lynch, and W. Snyder. Basic paramodulation. Information and Computation, 121(2):172–192, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  4. BGNR99.
    Miquel Bofill, Guillem Godoy, Robert Nieuwenhuis, and Albert Rubio. Paramodulation with non-monotonic orderings. In 14th IEEE Symposium on Logic in Computer Science (LICS), Trento, Italy, July 2–5, 1999.Google Scholar
  5. BGW93a.
    Leo Bachmair, Harald Ganzinger, and Uwe Waldmann. Set constraints are the monadic class. In Eighth Annual IEEE Symposium on Logic in Computer Science, pages 75–83, Montreal, Canada, June 19-23, 1993. IEEE Computer Society Press.Google Scholar
  6. BGW93b.
    Leo Bachmair, Harald Ganzinger, and Uwe Waldmann. Superposition with simplification as a decision procedure for the monadic class with equality. In 3rd Kurt Gödel Colloquium: Computational Logic and Proof Theory, LNCS 713, pages 83–96. SpringerVerlag, 1993.Google Scholar
  7. BS99.
    Franz Baader and Wayne Snyder. Unification theory. In J.A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning. Elsevier Science Publishers (to appear), 1999.Google Scholar
  8. CNNR98.
    Hubert Comon, Paliath Narendran, Robert Nieuwenhuis, and Michael Rusinowitch. Decision problems in ordered rewriting. In 13th IEEE Symposium on Logic in Computer Science (LICS), pages 410–422, Indianapolis, USA, June 27-30, 1998.Google Scholar
  9. Com90.
    Hubert Comon. Solving symbolic ordering constraints. International Journal of Foundations of Computer Science, 1(4):387–411, 1990.zbMATHCrossRefMathSciNetGoogle Scholar
  10. GdN99.
    Harald Ganzinger and Hans de Nivelle. A superposition decision procedure for the guarded fragment with equality. In 14th IEEE Symposium on Logic in Computer Science (LICS), Trento, Italy, July 2–5, 1999.Google Scholar
  11. GMV99.
    Harald Ganzinger, Christoph Meyer, and Margus Veanes. The two-variable guarded fragment with transitive relations. In 14th IEEE Symposium on Logic in Computer Science (LICS), Trento, Italy, July 2–5, 1999.Google Scholar
  12. GNN95.
    Harald Ganzinger, Robert Nieuwenhuis, and Pilar Nivela. The Saturate System, 1995. Software and documentation available at: http://www.mpi-sb.mpg.de/SATURATE/Saturate.html.
  13. GS92.
    Harald Ganzinger and Jürgen Stuber. Inductive theorem proving by consistency for first-order clauses (extended abstract). In M[ichaël] Rusinowitch and J[ean-]L[uc] Rémy, editors, The Third InternationalWorkshop on Conditional Term Rewriting Systems, Extended Abstracts, pages 130–135, Pontá-Mousson, France, July 8-10, 1992. Centre de Recherche en Informatique de Nancy and INRIA Lorraine.Google Scholar
  14. HR91.
    J. Hsiang and M Rusinowitch. Proving refutational completeness of theorem proving strategies: the transfinite semantic tree method. Journal of the ACM, 38(3):559–587, jul 1991.zbMATHCrossRefMathSciNetGoogle Scholar
  15. JMW98.
    Florent Jacquemard, Christoph Meyer, and Christoph Weidenbach. Unification in extensions of shallow equational theories. In Proceedings of the 9th International Conference on Rewriting Techniques and Applications, RTA-9, volume to appear, Tsukuba, Japan, 1998. Springer.Google Scholar
  16. LS98.
    C. Lynch and C. Scharff. Basic completion with E-cycle simplification. In Artificial Intelligence and Symbolic Computation, lncs 1476, pages 121–121, 1998.Google Scholar
  17. McC90.
    William McCune. Skolem functions and equality in automated deduction. In Tom Dietterich and William Swartout, editors, Proceedings of the 8th National Conference on Artificial Intelligence, pages 246–251, Hynes Convention Centre?, July 29-August 3 1990. MIT Press.Google Scholar
  18. McC97.
    William McCune. Solution of the Robbins problem. Journal of Automated Reasoning, 19(3):263–276, December 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  19. Nie98.
    Robert Nieuwenhuis. Decidability and complexity analysis by basic paramodulation. Information and Computation, 147:1–21, 1998. Extended abstract in IEEE LICS’96.zbMATHCrossRefMathSciNetGoogle Scholar
  20. NR95.
    Robert Nieuwenhuis and Albert Rubio. Theorem Proving with Ordering and Equality Constrained Clauses. Journal of Symbolic Computation, 19(4):321–351, April 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  21. NR97.
    Robert Nieuwenhuis and Albert Rubio. Paramodulation with Builtin ACTheories and Symbolic Constraints. Journal of Symbolic Computation, 23(1):1–21, May 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  22. NR99a.
    Robert Nieuwenhuis and Jos*#x00E9; Miguel Rivero. Solved forms for path ordering constraints. In P. Narendran and M. Rusinowitch, editors, Tenth International Conference on Rewriting Techniques and Applications (RTA), LNCS, Trento, Italy, July 2-4, 1999. Springer-Verlag.Google Scholar
  23. NR99b.
    Robert Nieuwenhuis and Albert Rubio. Paramodulation-based theorem proving. In J.A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning. Elsevier Science Publishers (to appear), 1999.Google Scholar
  24. PS81.
    G.E. Peterson and M.E. Stickel. Complete sets of reductions for some equational theories. Journal Assoc. Comput. Mach., 28(2):233–264, 1981.zbMATHMathSciNetGoogle Scholar
  25. RV95.
    Michael Rusinowitch and Laurent Vigneron. Automated deduction with associative commutative operators. J. of Applicable Algebra in Engineering, Communication and Computation, 6(1):23–56, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  26. RW69.
    G. A. Robinson and L. T. Wos. Paramodulation and theorem-proving in first order theories with equality. Machine Intelligence, 4:135–150, 1969.MathSciNetzbMATHGoogle Scholar
  27. Vig94.
    Laurent Vigneron. Associative Commutative Deduction with constraints. In Alan Bundy, editor, 12th International Conference on Automated Deduction, LNAI 814, pages 530–544, Nancy, France, June 1994. Springer-Verlag.Google Scholar
  28. Wei97.
    Christoph Weidenbach. SPASS version 0.49. Journal of Automated Reasoning, 18(2):247–252, April 1997.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Robert Nieuwenhuis
    • 1
  1. 1.Dept. LSI Jordi Girona 1Technical University of CataloniaBarcelonaSpain

Personalised recommendations