Abstract
We present a completion procedure (called MKB) that works for multiple reduction orderings. Given equations and a set of reduction orderings, the procedure simulates a computation performed by the parallel processes each of which executes the standard completion procedure (KB) with one of the given orderings. To gain efficiency, however, we develop new inference rules working on objects called nodes, which are data structures consisting of a pair s : t of terms associated with the information to show which processes contain the rule s → t (or t → s) and which processes contain the equation s ↔ t. The idea is based on the observation that some inferences made in different processes are often closely related, so we can design inference rules that simulate these inferences all in a single operation. Our experiments show that MKB is significantly more efficient than the naive simulation of parallel execution of KB procedures, when the number of reduction orderings is large enough. We also present an extension of this technique to the unfailing completion for multiple reduction orderings, which is useful in various areas of automated reasoning, including equational theorem proving.
Similar content being viewed by others
References
Avenhaus, J. and Madlener, K.: Term rewriting and equational reasoning, in R. B. Banerji (ed.), Formal Techniques in Artificial Intelligence: A Sourcebook, North-Holland, 1990, pp. 1–44.
Bachmair, L., Dershowitz, N., and Hsiang, J.: Orderings for equational proofs, in Proc. IEEE Symposium on Logic in Computer Science, 1986, pp. 346–357.
Bachmair, L., Dershowitz, N., and Plaisted, D. A.: Completion without failure, in H. Aït-Kaci and M. Nivat (eds.), Resolution of Equations in Algebraic Structures, vol. 2: Rewriting Techniques, Academic Press, 1989, pp. 1–30.
Bachmair, L.: Canonical Equational Proofs, Birkhäuser, 1991.
Dershowitz, N.: Completion and its applications, in H. Aït-Kaci and M. Nivat (eds.), Resolution of Equations in Algebraic Structures, vol. 2: Rewriting Techniques, Academic Press, 1989, pp. 31–85.
Dershowitz, N. and Jouannaud, J.-P.: Rewrite systems, in J. van Leeuwen (ed.), Handbook of Theoretical Computer Science, vol. B, North-Holland, 1990, pp. 243–320.
Detlefs, D. and Forgaad, R.: A procedure for automatically proving the termination of a set of rewrite rules, in Proc. 1st Conf. on Rewriting Techniques and Applications, Lecture Notes in Comput. Sci. 202, 1985, pp. 255–270.
Huet, G. and Oppen, D. C.: Equations and rewrite rules: A survey, in R. Book (ed.), Formal Language Theory: Perspectives and Open Problems, Academic Press, 1980, pp. 349–405.
Huet, G.: A complete proof of correctness of the Knuth and Bendix completion algorithm, J. Comput. System Sci. 23 (1981), 11–21.
Klop, J. W.: Term rewriting systems, in S. Abramsky et al. (eds.), Handbook of Logic in Computer Science, vol. 2, Oxford Univ. Press, 1992, pp. 1–116.
Knuth, D. E. and Bendix, P. B.: Simple word problems in universal algebras, in J. Leech (ed.), Computational Problems in Abstract Algebra, Pargamon Press, 1970, pp. 263–297.
Lescanne, P.: Completion procedures as transition rules + control, in Proc. TAPSOFT, vol. 1, Lecture Notes in Comput. Sci. 351, 1989, pp. 28–41.
Plaisted, D. A.: Equational reasoning and term rewriting systems, in D. M. Gabbay et al. (eds.), Handbook of Logic in Artificial Intelligence and Logic Programming, vol. 1, Oxford Univ. Press, 1993, pp. 274–367.
Steinbach, J. and Kühler, U.: Check your ordering – termination proofs and open problems, SEKI report SR-90-25 (SFB), Fachbereich Informatik, Universität Kaiserslautern, Germany, 1990.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kurihara, M., Kondo, H. Completion for Multiple Reduction Orderings. Journal of Automated Reasoning 23, 25–42 (1999). https://doi.org/10.1023/A:1006129631807
Issue Date:
DOI: https://doi.org/10.1023/A:1006129631807