Decision problems for term rewriting systems and recognizable tree languages

  • Rémi Gilleron
Part of the Lecture Notes in Computer Science book series (LNCS, volume 480)


We study the connections between recognizable tree languages and rewrite systems. We investigate some decision problems. Particularly, let us consider the property (P): a rewrite system S is such that, for every recognizable tree language F, the set of S-normal forms of terms in F is recognizable too. We prove that the property (P) is undecidable. We prove that the existential fragment of the theory of ground term algebras modulo a congruence \(\mathop \leftrightarrow \limits^* E\) generated by a set E of equations such that there exists a finite, noetherian, confluent rewrite system S satisfying (P) with \(\mathop \leftrightarrow \limits^* S = \mathop \leftrightarrow \limits^* E\) is undecidable. Nevertheless, we develop a decision procedure for the validity of linear formulas in a fiagment of such a theory.


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  1. [1]
    A.Bockmayr, A note on a canonical theory with undecidable unification and matching problem, Journal of Automated Reasoning, 3, (1987).Google Scholar
  2. [2]
    R.V. Book, Decidable sentences of Church-Rosser congruences, Theoretical Computer Science, 24, (1983).Google Scholar
  3. [3]
    R.V. Book, Thue systems as Rewriting Systems, J. of Symbolic Computation, 3, (1987).Google Scholar
  4. [4]
    R.V. Book and J.H. Gallier, Reductions in Tree Replacement Systems, Theoretical Computer Science, 37, (1985).Google Scholar
  5. [5]
    H. Comon, Equational formulas in order-sorted algebras, in Proc. ICALP 90, (1990).Google Scholar
  6. [6]
    H. Comon, Unification et disunification: théorie et applications, Thése de l' I.N.P. de Grenoble, France, (1988).Google Scholar
  7. [7]
    J.L. Coquidé, M.Dauchet and S.Tison, About connections between syntactical and computational complexity, FCT'89, Lec.Notes.comp.Sci, 380, (1989).Google Scholar
  8. [8]
    J.L.Coquidé and R.Gilleron, Proofs and Reachability problems for Rewrite Systems, IMYCS 90, to appear in Lec.Notes.Comp.Sci, (1990).Google Scholar
  9. [9]
    J.L. Coquidé, M.Dauchet, R.Gilleron and S.Vagvölgyi, Tree pushdown automata and Rewrite Systems, submitted paper, (1990).Google Scholar
  10. [10]
    B. Courcelle,On recognizable sets and tree automata, Resolution of Equations in Algebraic Structures, Academic Press, M.Nivat & H. Ait-Kaci edts, (1989).Google Scholar
  11. [11]
    M.Dauchet, Simulation of Turing machines by a left-linear rewrite-rule, Proc 3rd R.T.A, Lect. Notes Comput. Sci. 355, (1989).Google Scholar
  12. [12]
    M.Dauchet and S. Tison, The theory of Ground Rewrite System is Decidable, Proc. 5th I.E.E.E symp. on Logic in Computer Science, (1990).Google Scholar
  13. [13]
    N.Dershowitz and J.P. Jouannaud, Rewrite systems, Handbook of Theoretical Computer Science, J.V.Leeuwen editor, North-Holland, to appear.(1989).Google Scholar
  14. [14]
    Z. Fülöp and S. Vàgvölgyi, A characterization of irreducible sets modulo left-linear rewriting systems by tree automata, Fundamenta Informaticae XIII, (1990).Google Scholar
  15. [15]
    F. Gesceg and M. Steinby, Tree automata, Akademiai Kiado, (1984).Google Scholar
  16. [16]
    G. Huet and D.C. Oppen, Equations and Rewrite Rules: A survey, in R.V. Book, ed., New York, Academic Press, Formal Language Theory: Perspectives and Open Problems, (1980).Google Scholar
  17. [17]
    M.Jantzen, Confluent string rewriting, EATCS Monographs on Theoretical Computer Science 14, Springer Verlag, (1988).Google Scholar
  18. [18]
    J.P.Jouannaud and H.Kirchner, Completion of a set of rules modulo a set of equations, SIAM J. Comput. 15, (1986).Google Scholar
  19. [19]
    G.Peterson and M.Stickel, Complete sets of reductions for some equational theories, J. Assoc. Comput. Mach. 28, (1981).Google Scholar
  20. [20]
    K. Salomaa, Deterministic Tree Pushdown Automata and Monadic Tree Rewriting Systems, Journal of Comput. and Syst. Sci., 37, (1988).Google Scholar
  21. [21]
    R.Treinen, A new method for undecidability proofs of first order theories, T.R. A-09/90, Universität des Saarladandes, Saarbrücken, (1990).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Rémi Gilleron
    • 1
  1. 1.L.I.F.L, U.R.A 369 C.N.R.S, I.U.T AUniversité Lille-Flandres-ArtoisVilleneuve d'Ascq CedexFrance

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