Termination Modulo Combinations of Equational Theories

  • Francisco Durán
  • Salvador Lucas
  • José Meseguer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5749)

Abstract

Rewriting with rules R modulo axioms E is a widely used technique in both rule-based programming languages and in automated deduction. Termination methods for rewriting systems modulo specific axioms E (e.g., associativity-commutativity) are known. However, much less seems to be known about termination methods that can be modular in the set E of axioms. In fact, current termination tools and proof methods cannot be applied to commonly occurring combinations of axioms that fall outside their scope. This work proposes a modular termination proof method based on semantics- and termination-preserving transformations that can reduce the proof of termination of rules R modulo E to an equivalent proof of termination of the transformed rules modulo a typically much simpler set B of axioms. Our method is based on the notion of variants of a term recently proposed by Comon and Delaune. We illustrate its practical usefulness by considering the very common case in which E is an arbitrary combination of associativity, commutativity, left- and right-identity axioms for various function symbols.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Baader, F., Schulz, K.U.: Unification Theory. In: Automated Deduction. Applied Logic Series, vol. I, 8, pp. 225–263. Kluwer, Dordrecht (1998)Google Scholar
  2. 2.
    Bachmair, L., Dershowitz, N.: Completion for rewriting modulo a congruence. Theor. Comput. Sci. 67(2,3), 173–201 (1989)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Cherifa, A.B., Lescanne, P.: Termination of rewriting systems by polynomial interpretations and its implementation. Sci. Comput. Program. 9(2), 137–159 (1987)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Clavel, M., Durán, F., Eker, S., Lincoln, P., Martí-Oliet, N., Meseguer, J., Talcott, C.: All About Maude - A High-Performance Logical Framework. LNCS, vol. 4350. Springer, Heidelberg (2007)MATHGoogle Scholar
  5. 5.
    Comon, H., Delaune, S.: The Finite Variant Property: How to Get Rid of Some Algebraic Properties. In: Giesl, J. (ed.) RTA 2005. LNCS, vol. 3467, pp. 294–307. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  6. 6.
    Durán, F., Lucas, S., Meseguer, J.: MTT: The Maude Termination Tool (System Description). In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 313–319. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  7. 7.
    Durán, F., Lucas, S., Meseguer, J.: Termination Modulo Combinations of Equational Theories (Long Version). University of Illinois Tech. Rep. (June 2009), http://hdl.handle.net/2142/12311
  8. 8.
    Durán, F., Lucas, S., Meseguer, J., Marché, C., Urbain, X.: Proving operational termination of membership equational programs. Higher-Order and Symbolic Computation 21(1-2), 59–88 (2008)CrossRefMATHGoogle Scholar
  9. 9.
    Escobar, S., Meseguer, J., Sasse, R.: Effectively Checking the Finite Variant Property. In: Voronkov, A. (ed.) RTA 2008. LNCS, vol. 5117, pp. 79–93. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  10. 10.
    Escobar, S., Meseguer, J., Sasse, R.: Variant Narrowing and Equational Unification. In: Proc. of WRLA 2008. ENTCS (2008) (to appear, 2009)Google Scholar
  11. 11.
    Escobar, S., Meseguer, J., Sasse, R.: Variant Narrowing and Extreme Termination. University of Illinois Tech. Rep. UIUCDCS-R-2009-3049 (March 2009)Google Scholar
  12. 12.
    Ferreira, M.C.F.: Dummy elimination in equational rewriting. In: Ganzinger, H. (ed.) RTA 1996. LNCS, vol. 1103. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  13. 13.
    Giesl, J., Kapur, D.: Dependency Pairs for Equational Rewriting. In: Middeldorp, A. (ed.) RTA 2001. LNCS, vol. 2051, pp. 93–108. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  14. 14.
    Giesl, J., Schneider-Kamp, P., Thiemann, R.: AProVE 1.2: Automatic Termination Proofs in the Dependency Pair Framework. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 281–286. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  15. 15.
    Goguen, J., Meseguer, J.: Order-sorted algebra I: Equational deduction for multiple inheritance, overloading, exceptions and partial operations. Theor. Comput. Sci. 105, 217–273 (1992)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Huet, G.: Confluent reductions: Abstract properties and applications to term rewriting systems. Journal of the ACM 27, 797–821 (1980)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Jouannaud, J.-P., Kirchner, H.: Completion of a Set of Rules Modulo a Set of Equations. SIAM Journal of Computing 15(4), 1155–1194 (1986)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Jouannaud, J.-P., Marché, C.: Termination and completion modulo associativity, commutativity and identity. Theor. Comput. Sci. 104(1), 29–51 (1992)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kirchner, C., Kirchner, H., Meseguer, J.: Operational Semantics of OBJ3. In: Lepistö, T., Salomaa, A. (eds.) ICALP 1988. LNCS, vol. 317, pp. 287–301. Springer, Heidelberg (1988)CrossRefGoogle Scholar
  20. 20.
    Lucas, S., Meseguer, J.: Operational Termination of Membership Equational Programs: the Order-Sorted Way. In: Proc. of WRLA 2008. ENTCS (2008) (to appear, 2009)Google Scholar
  21. 21.
    Marché, C.: Normalised rewriting and normalised completion. In: Proc. LICS 1994, pp. 394–403. IEEE, Los Alamitos (1994)Google Scholar
  22. 22.
    Marché, C., Urbain, X.: Termination of associative-commutative rewriting by dependency pairs. In: Nipkow, T. (ed.) RTA 1998, vol. 1379, pp. 241–255. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  23. 23.
    Nipkow, T.: Combining Matching Algorithms: The Regular Case. Journal of Symbolic Computation 12, 633–653 (1991)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Ohlebusch, E.: Advanced Topics in Term Rewriting. Springer, Heidelberg (2002)CrossRefMATHGoogle Scholar
  25. 25.
    Ohsaki, H., Middeldorp, A., Giesl, J.: Equational termination by semantic labelling. In: Clote, P.G., Schwichtenberg, H. (eds.) CSL 2000. LNCS, vol. 1862, pp. 457–471. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  26. 26.
    Peterson, G.E., Stickel, M.E.: Complete Sets of Reductions for Some Equational Theories. Journal of the ACM 28(2), 233–264 (1981)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Ringeissen, C.: Combination of matching algorithms. In: Enjalbert, P., Mayr, E.W., Wagner, K.W. (eds.) STACS 1994. LNCS, vol. 775, pp. 187–198. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  28. 28.
    Rubio, A., Nieuwenhuis, R.: A total AC-compatible ordering based on RPO. Theor. Comput. Sci. 142(2), 209–227 (1995)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Viry, P.: Equational rules for rewriting logic. Theor. Comp. Sci. 285, 487–517 (2002)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Francisco Durán
    • 1
  • Salvador Lucas
    • 2
  • José Meseguer
    • 3
  1. 1.LCCUniversidad de MálagaSpain
  2. 2.DSICUniversidad Politécnica de ValenciaSpain
  3. 3.CS Dept.University of Illinois at Urbana-ChampaignUSA

Personalised recommendations