Skip to main content
SpringerLink
Log in

Constraint Solving for Term Orderings Compatible with Abelian Semigroups, Monoids and Groups

  • Published:
Constraints Aims and scope Submit manuscript

Abstract

It is crucial for the performance of ordered resolution or paramodulation-based deduction systems that they incorporate specialized techniques to work efficiently with standard algebraic theories E. Essential ingredients for this purpose are term orderings that are E-compatible, for the given E, and algorithms deciding constraint satisfiability for such orderings.

Here we introduce a uniform technique providing the first such algorithms for some orderings for abelian semigroups, abelian monoids and abelian groups, which we believe will lead to reasonably efficient techniques for practice.

Our algorithms are in NP, and hence optimal, since in addition we show that, for any well-founded E-compatible ordering for these E, the constraint satisfiability problem is NP-hard even for conjunctions of inequations.

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. Bachmair, L., & Plaisted, D. A. (1985). Termination orderings for associative-commutative rewriting systems. Journal of Symbolic Computation, 1: 329–349.

    Google Scholar 

  2. Comon, H. (1990). Solving symbolic ordering constraints. International Journal of Foundations of Computer Science, 1(4): 387–411.

    Google Scholar 

  3. Comon, H., Nieuwenhuis, R., & Rubio, A. (1995). Orderings, AC-theories and symbolic constraint solving. In 10th IEEE Symposium on Logic in Computer Science (LICS), pages 375–385. San Diego, USA.

  4. Dershowitz, N. (1982). Orderings for term-rewriting systems. Theoretical Computer Science, 17(3): 279–301.

    Article  Google Scholar 

  5. Dershowitz, N., & Jouannaud, J.-P. (1990). Rewrite Systems. In van Leeuwen, J., ed., Handbook of Theoretical Computer Science, Vol. B: Formal Models and Semantics, Chapt. 6, pages 244–320. Amsterdam, New York, Oxford, Tokyo: Elsevier Science Publishers B.V.

    Google Scholar 

  6. Godoy, G., & Nieuwenhuis, R. (2000). Paramodulation with built-in abelian groups. In 15th IEEE Symposium on Logic in Computer Science (LICS), pages 413–424. Santa Barbara, USA: IEEE Computer Society Press.

    Google Scholar 

  7. Godoy, G., & Nieuwenhuis, R. (2001). Ordering constraints for deduction with built-in Abelian semigroups, Monoids and groups. In 16th IEEE Symposium on Logic in Computer Science (LICS), pages 38–47. Boston, USA: IEEE Computer Society Press.

    Google Scholar 

  8. Jouannaud, J.-P., & Okada, M. (1991). Satisfiability of systems of ordinal notations with the subterm property is decidable. In 18th International Colloquium Automata, Languages and Programming (ICALP), pages 455–468. Madrid, Spain: Springer-Verlag.

    Google Scholar 

  9. Kirchner, C., Kirchner, H., & Rusinowitch, M. (1990). Deduction with symbolic constraints. Revue Française d'Intelligence Artificielle, 4(3): 9–52.

    Google Scholar 

  10. _ Korovin, K., & Voronkov, A. (2000). A decision procedure for the existential theory of term algebras with the Knuth-Bendix ordering. In 15th IEEE Symposium on Logic in Computer Science (LICS), pages 291–302.

  11. _ Korovin, K., & Voronkov, A. (2001). Knuth-Bendix constraint solving is NP-complete. In Automata, Languages and Programming, International Colloquium, 28th Int. colloquium, (ICALP). Crete, Greece: Springer-Verlag.

    Google Scholar 

  12. Marché, C. (1996). Normalized rewriting: An alternative to rewriting modulo a set of equations. Journal of Symbolic Computation, 21(3): 253–288.

    Article  Google Scholar 

  13. McCune, W. (1997). Solution of the Robbins problem. Journal of Automated Reasoning, 19(3): 263–276.

    Article  Google Scholar 

  14. Narendran, P., Rusinowitch, M., & Verma, R. (1999). RPO constraint solving is in NP. In Gottlob, G., Grandjean, E., & Seyr, K., eds., 12th Int. Conference of the European Association of Computer Science Logic (CSL'98), pages 385–398. Brno, Czech Republic: Springer-Verlag.

    Google Scholar 

  15. Nieuwenhuis, R. (1993). Simple LPO constraint solving methods. Information Processing Letters, 47: 65–69.

    Article  Google Scholar 

  16. Nieuwenhuis, R., & Rubio, A. (1995). Theorem proving with ordering and equality constrained clauses. Journal of Symbolic Computation, 19(4): 321–351.

    Article  Google Scholar 

  17. Nieuwenhuis, R., & Rubio, A. (1997). Paramodulation with built-in AC-theories and symbolic constraints. Journal of Symbolic Computation, 23(1): 1–21.

    Article  Google Scholar 

  18. Nieuwenhuis, R., & Rubio, A. (2000). Paramodulation-based theorem proving. In Robinson, J., & Voronkov, A., eds., Handbook of Automated Reasoning. Elsevier Science Publishers (to appear).

  19. Nieuwenhuis, R., & Rubio, A. (2001). Paramodulation-based theorem proving. In Robinson, J., & Voronkov, A., eds., Handbook of Automated Reasoning, pages 371–444. Elsevier Science and MIT Press.

  20. Rubio, A., & Nieuwenhuis, R. (1995). A total AC-compatible ordering based on RPO. Theoretical Computer Science, 142(2): 209–227.

    Article  Google Scholar 

  21. Schrijver, A. (1987). Theory of Linear and Integer Programming. NewYork: John Wiley and Sons.

    Google Scholar 

  22. Vigneron, L. (1994), Associative commutative deduction with constraints. In Bundy, A., ed., 12th International Conference on Automated Deduction (CADE), pages 530–544. Nancy, France: Springer-Verlag.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. LSI, Technical University of Catalonia, Jordi Girona 1, 08034, Barcelona, Spain

    Guillem Godoy

  2. Dept. LSI, Technical University of Catalonia, Jordi Girona 1, 08034, Barcelona, Spain

    Robert Nieuwenhuis

Authors
  1. Guillem Godoy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Robert Nieuwenhuis
    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

Godoy, G., Nieuwenhuis, R. Constraint Solving for Term Orderings Compatible with Abelian Semigroups, Monoids and Groups. Constraints 9, 167–192 (2004). https://doi.org/10.1023/B:CONS.0000036021.31386.cc

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:CONS.0000036021.31386.cc

Search

Navigation