Decidable approximations of sets of descendants and sets of normal forms

  • Thomas Genet
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1379)


We present here decidable approximations of sets of descendants and sets of normal forms of Term Rewriting Systems, based on specific tree automata techniques. In the context of rewriting logic, a Term Rewriting System is a program, and a normal form is a result of the program. Thus, approximations of sets of descendants and sets of normal forms provide tools for analysing a few properties of programs: we show how to compute a superset of results, to prove the sufficient completeness property, or to find a criterion for proving termination under a specific strategy, the sequential reduction strategy. The main technical contribution of the paper is the construction of an approximation automaton which recognises a superset of the set of normal forms of terms in a set E, w.r.t. a Term Rewriting System R.


  1. 1.
    T. Arts and J. Giesl. Automatically proving termination where simplification orderings fail. In M. Bidoit and M. Dauchet, editors, Proc. 22nd CAAP Conf., Lille (France), volume 1214 of LNCS, pages 261–272. Springer-Verlag, 1997.Google Scholar
  2. 2.
    T. Arts and J. Giesl. Proving innermost termination automatically. In Proc. 7th RTA Conf., Sitges (Spain), volume 1232 of LNCS, pages 157–171. Springer-Verlag, 1997.Google Scholar
  3. 3.
    H. Comon. Sufficient completeness, term rewriting system and anti-unification. In J. Siekmann, editor, Proc. 8th CADE Conf., Oxford (UK), volume 230 of LNCS, pages 128–140. Springer-Verlag, 1986.Google Scholar
  4. 4.
    H. Comon, M. Dauchet, R. Gilleron, D. Lugiez, S. Tison, and Tommasi. Tree automata techniques and applications. Preliminary Version,, 1997.Google Scholar
  5. 5.
    H. Comon and J.-L. Rémy. How to characterize the language of ground normal forms. Technical Report 676, INRIA-Lorraine, 1987.Google Scholar
  6. 6.
    J. Coquidé, M. Dauchet, R. Gilleron, and S. Vagvölgyi. Bottom-up tree pushdown automata and rewrite systems. In R. V. Book, editor, Proc. 4th RTA Conf., Como (Italy), volume 488 of LNCS, pages 287–298. Springer-Verlag, 1991.Google Scholar
  7. 7.
    M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. In Proc. 5th LICS Symp., Philadelphia (Pa., USA), pages 242–248, June 1990.Google Scholar
  8. 8.
    N. Dershowitz and C. Hoot. Natural termination. TCS, 142(2):179–207, May 1995.MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    N. Dershowitz and J.-P. Jouannaud. Handbook of Theoretical Computer Science, volume B, chapter 6: Rewrite Systems, pages 244–320. Elsevier Science Publishers B. V. (North-Holland), 1990. Also as: Research report 478, LRI.Google Scholar
  10. 10.
    J. H. Gallier and R. V. Book. Reductions in tree replacement systems. TCS, 37:123–150, 1985.MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    F. Gécseg and M. Steinby. Tree automata. Akadémiai Kiadó, Budapest, Hungary, 1984.MATHGoogle Scholar
  12. 12.
    T. Genet. Decidable approximations of sets of descendants and sets of normal forms (extended version). Technical Report RR-3325, INRIA, 1997.Google Scholar
  13. 13.
    T. Genet. Proving termination of sequential reduction relation using tree automata. Technical Report 97-R-091, CRIN, 1997.Google Scholar
  14. 14.
    T. Genet and I. Gnaedig. Termination proofs using gpo ordering constraints. In M. Dauchet, editor, Proc. 22nd CAAP Conf., Lille (France), volume 1214 of LNCS, pages 249–260. Springer-Verlag, 1997.Google Scholar
  15. 15.
    R. Gilleron and S. Tison. Regular tree languages and rewrite systems. Fundamenta Informaticae, 24:157–175, 1995.MathSciNetMATHGoogle Scholar
  16. 16.
    F. Jacquemard. Decidable approximations of term rewriting systems. In H. Ganzinger, editor, Proc. 7th RTA Conf., New Brunswick (New Jersey, USA), pages 362–376. Springer-Verlag, 1996.Google Scholar
  17. 17.
    D. Kapur, P. Narendran, and H. Zhang. On sufficient completeness and related properties of term rewriting systems. Acta Informatica, 24:395–415, 1987.MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    C. Kirchner, H. Kirchner, and M. Vittek. Designing constraint logic programming languages using computational systems. In P. Van Hentenryck and V. Saraswat, editors, Principles and Practice of Constraint Programming. The Newport Papers., chapter 8, pages 131–158. The MIT press, 1995.Google Scholar
  19. 19.
    E. Kounalis. Completeness in data type specifications. In B. Buchberger, editor, Proceedings EUROCAL Conference, Linz (Austria), volume 204 of LNCS, pages 348–362. Springer-Verlag, 1985.Google Scholar
  20. 20.
    M. Kurihara and I. Kaji. Modular term rewriting systems and the termination. IPL, 34:1–4, Feb. 1990.MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    M. Kurihara and A. Ohuchi. Modular term rewriting systems with shared constructors. Journal of Information Processing of Japan, 14(3):357–358, 1991.MathSciNetMATHGoogle Scholar
  22. 22.
    S. Limet and P. Réty. E-unification by means of tree tuple synchronized grammars. In M. Dauchet, editor, Proc. 22nd CAAP Conf., Lille (France), volume 1214 of LNCS, pages 429–440. Springer-Verlag, 1997.Google Scholar
  23. 23.
    T. Nipkow and G. Weikum. A decidability result about sufficient completeness of axiomatically specified abstract data types. In 6th GI Conference, volume 145 of LNCS, pages 257–268. Springer-Verlag, 1983.Google Scholar
  24. 24.
    E. Ohlebusch. Modular Properties of Composable Term Rewriting Systems. PhD thesis, Universität Bielefeld, Bielefeld, 1994.Google Scholar
  25. 25.
    K. Salomaa. Deterministic Tree Pushdown Automata and Monadic Tree Rewriting Systems. J. of Computer and System Sciences, 37:367–394, 1988.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag 1998

Authors and Affiliations

  • Thomas Genet
    • 1
  1. 1.INRIA Lorraine & CRIN CNRSVillers-lès-Nancy CedexFrance

Personalised recommendations