Normalization properties for Shallow TRS and Innermost Rewriting

  • Guillem Godoy


Computation with a term rewrite system (TRS) consists in the application of rules from a given starting term until a normal form is reached. Two natural questions arise from this definition: whether all terms can reach at least one normal form (weak normalization property), and whether all terms can reach at most one normal form (unique normalization property). Innermost rewriting is one of the most studied and used strategies for rewriting, since it corresponds to the “call by value” computation of programming languages. Henceforth, it is meaningful to study whether weak and unique normalization are indeed decidable for a significant class of TRS with the use of the innermost strategy. In this paper we study weak and unique normalization for shallow TRS and innermost rewriting. We show decidability of unique normalization in polynomial time, and decidability of weak normalization in doubly exponential time. We obtain an exptime-hardness proof for the latter case. Nevertheless, when a TRS satisfies the unique normalization property, we conclude the decidability of the weak normalization property in polynomial time, too. Hence, whether all terms reach one and only one normal form is decidable in polynomial time. All of these results are obtained assuming that the maximum arity of the signature is fixed for the problem, which is a usual assumption.


Term rewriting Syntactic restrictions Innermost rewriting 

Mathematics Subject Classification (2000)

16S15 68Q42 


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  1. 1.
    Baader F., Nipkow T.: Term Rewriting and All That. Cambridge University Press, New York (1998)Google Scholar
  2. 2.
    Bogaert, B., Tison, S.: Equality and disequality constraints on direct subterms in tree automata. In: International Symposium on Theoretical Aspects of Computeter Science (STACS), pp. 161–171 (1992)Google Scholar
  3. 3.
    Comon, H., Dauchet, M., Gilleron, R., Löding, C., Jacquemard, F., Lugiez, D., Tison, S., Tommasi, M.: Tree automata techniques and applications. Available on: (2007). Release 12 Oct 2007
  4. 4.
    Coquidé J., Dauchet M., Gilleron R., Vágvölgyi S.: Bottom-up tree pushdown automata: classification and connection with rewrite systems. Theor Comput Sci 127, 69–98 (1994)zbMATHCrossRefGoogle Scholar
  5. 5.
    Durand, I., Sénizergues, G.: Bottom-up rewriting is inverse recognizability preserving. In: International Conference on Rewriting Techniques and Applications (RTA), pp. 107–121 (2007)Google Scholar
  6. 6.
    Ganzinger H., Jacquemard F., Veanes M.: Rigid reachability: the non-symmetric form of rigid E-unification. Intl. J. Found. Comput. Sci. 11(1), 3–27 (2000)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Gascon A., Godoy G., Jacquemard F.: Closure of tree automata languages under innermost rewriting. Electron. Notes Theor. Comput Sci. 237, 23–38 (2009)CrossRefGoogle Scholar
  8. 8.
    Godoy G., Hernández H.: Undecidable properties of flat term rewrite systems. Appl. Algebra Eng. Commun. Comput. 20(2), 187–205 (2009)zbMATHCrossRefGoogle Scholar
  9. 9.
    Godoy, G., Huntingford, E.: Innermost-reachability and innermost-joinability are decidable for shallow term rewrite systems. In: International Conference on Rewriting Techniques and Applications (RTA), pp. 184–199 (2007)Google Scholar
  10. 10.
    Godoy, G., Tison, S.: On the normalization and unique normalization properties of term rewrite systems. In: F. Pfenning (ed.) International Conference on Automated Deduction (CADE-21), pp. 247–262 (2007)Google Scholar
  11. 11.
    Godoy, G., Tiwari, A., Verma, R.: On the confluence of linear shallow term rewrite systems. In: Alt, H., Habib, M. (eds.) International Symposium on Theoretical Aspects of Computer Science (STACS). Lecture Notes in Computer Science, vol. 2607, pp. 85–96. Springer (2003)Google Scholar
  12. 12.
    Godoy G., Tiwari A., Verma R.: Deciding confluence of certain term rewriting systems in polynomial time. Ann. Pure Appl. Log. 130(1–3), 33–59 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Gyenizse P., Vágvölgyi S.: Linear generalized semi-monadic rewrite systems effectively preserve recognizability. Theor. Comput. Sci. 194, 87–122 (1998)zbMATHCrossRefGoogle Scholar
  14. 14.
    Nagaya T., Toyama Y.: Decidability for left-linear growing term rewriting systems. Inf. Comput. 178(2), 499–514 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Salomaa K.: Deterministic tree pushdown automata and monadic tree rewriting systems. J. Comput. Syst. Sci. 37, 367–394 (1998)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Takai, T., Kaji, Y., Seki, H.: Right-linear finite path overlapping term rewriting systems effectively preserve recognizability. In: International Conference on Rewriting Techniques and Applications (RTA), pp. 246–260 (2000)Google Scholar
  17. 17.
    Verma R., Hayrapetyan A.: A new decidability technique for ground term rewriting systems. ACM Trans. Comput. Log. 6(1), 102–123 (2005)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Universitat Politèctina de CatalunyaBarcelonaSpain

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