Abstract
The limit behaviors of computations have not been fully explored. It is necessary to consider such limit behaviors when we consider the properties of infinite objects in computer science, such as infinite logic programs, the symbolic solutions of infinite polynomial equations. Usually, we can use finite objects to approximate infinite objects, and we should know what kinds of infinite objects are approximable and how to approximate them effectively. A sequence {R k : k ε ω} of term rewriting systems has the well limit behavior if under the condition that the sequence has the set-theoretic limit or the distance-based limit, the sequence {Th(R k ): k ε ω} of corresponding theoretic closures of R k has the set-theoretic or distance-based limit, and lim k→∞ Th(R k ) is equal to the theoretic closure of the limit of {R k : k ε ω}. Two kinds of limits of term rewriting systems are considered: one is based on the set-theoretic limit, the other is on the distance-based limit. It is proved that given a sequence {R k : κ ε ω} of term rewriting systems R k , if there is a well-founded ordering ≺ on terms such that every R k is ≺-well-founded, and the set-theoretic limit of {R k : κ ε ω} exists, then {R k : κ ε ω} has the well limit behavior; and if (1) there is a well-founded ordering ≺ on terms such that every R k is ≺-well-founded, (2) there is a distance d on terms which is closed under substitutions and contexts and (3) {R k : k ε ω} is Cauchy under d then {R κ: κ ε ω} has the well limit behavior. The results are used to approximate the least Herbrand models of infinite Horn logic programs and real Horn logic programs, and the solutions and Gröbner bases of (infinite) sets of real polynomials by sequences of (finite) sets of rational polynomials.
Similar content being viewed by others
References
Sowa J F. Knowledge Representation, Logical, Philosophical and Computational Foundations. Boston, MA: PWS Publishing Company, 1998, 386
Wu W J. On the mechanization of theorem-proving in elementary differential geometry (in Chinese). Sci. Sinica, 1979, 94–102
Wu W J. Mechanical theorem proving in elementary geometry and elementary differential geometry. In: Proceedings of 1980 Beijing DD-Symp., Beijing. Science Press, 1982, 1073–1092.
Wu W J. A constructive theory of differential algebraic geometry. In: Proceedings of 1985 Shanghai DD-Symp., Lecture Notes in Math. 1987, 1255: 173–189
Wang D M. A method for proving theorems in differential geometry and mechanics. J. of Universal Computer Science, 1995, 1: 658–673
Dershowitz N, Kaplan S, Plaisted D A. Rewrite, rewrite, rewrite, rewrite, rewrite,… In: The 16th ACM Symposium on Principles of Programming Languages, 1989, 250–259
Ma S L, Sui Y F, Xu K. The limits of the Horn logic programs. In: Proceedings of the 18-th International Conference on Logic Programming(ICLP2002), Copenhagen, Denmark, LNCS 2401, 2002, 467
Baader F, Nipkow T. Term rewriting and qll that. Cambridge University Press, 1998
Boigelot B, Wolper P. Representing arithmetic constraints with finite automata: an overview. In: Proceedings of the 18-th International Conference on Logic Programming(ICLP2002), Copenhagen, Denmark, LNCS 2401, 1–19
Li W, Ma S L. Limits of theory sequences over algebraically closed fields and applications. Discrete Mathematics and Theoretical Computer Science, 2004, 136(1): 23–43
Nienhuys-Cheng S H. Distance between Herbrand interpretations: a measure for approximations to a target concept. In: Proceedings of the 7th International Workshop on Inductive Programming, LNAI, Springer-Verlag, 1997
Lloyd J W. Foundations of Logic Programming, Springer-Verlag, Berlin, 1987
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ma, S., Sui, Y. & Xu, K. Well limit behaviors of term rewriting systems. Front. Comput. Sc. China 1, 283–296 (2007). https://doi.org/10.1007/s11704-007-0028-x
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11704-007-0028-x