Abstract
We reduce an instance of Turing machine acceptance to the problem of detecting whether the Knuth-Bendix completion procedure generates a crossed pair of rules. This resolves an open question posed in [5]. Our proof technique generalizes; using similar reductions, we can show that a number of other questions related to whether the Knuth-Bendix completion procedure generates certain types of rules are all undecidable. We suggest that the techniques illustrated herein may be useful in answering a number of related questions about the Knuth-Bendix completion procedure, and discuss several examples; in particular, we demonstrate how our construction provides a simple proof that the universal matching problem is undecidable for regular canonical theories, a result first proved in [4], and prove that the universal unification problem is undecidable for permutative canonical theories.
this work was done while a student at State University of New York at Albany, Albany, N.Y.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
Bundy, A., The Computer Modelling of Human Reasoning, Academic Press, New York, 1983.
Dershowitz, N., “Termination,” in Rewriting Techniques and Applications, Jean-Pierre Jouannaud, ed., Springer Verlag, Berlin, 1985, pp. 180–224.
Gorn, S., “Explicit Definitions and Linguistic Dominoes,” in Systems and Computer Science (J. Hart and S. Takasu, eds.), U. of Toronto Press, 1967, pp. 77–115.
Heilbrunner, S., and Hölldobler, S., “The Undecidability of the Unification and Matching Problem for Canonical Theories,” Acta Informatica, 24, pp. 157–171, 1987.
Hermann, M., and Privara, I., “On nontermination of the Knuth-Bendix algorithm,” in Proc. 13th EATCS Intl. Colloq. on Automata, Languages, and Programming, L. Kott, ed., Springer Verlag, Berlin, 1986, pp. 146–156.
Hopcroft, J. E., and Ullman, J. D., Introduction to Automata Theory, Languages, and Computation, Addison-Wesley Publishing Company, Reading, MA, 1979.
Huet, G., and Lankford, D.S., “On the Uniform Halting Problem for Term Rewriting Systems,” Rapport Laboria 283, INRIA, Paris, 1978.
Huet, G., and Oppen, D., “Equations and rewrite rules: a survey,” in Formal Languages: Perspectives and Open Problems (R. Book, ed.), Academic Press, New York, 1980.
Kapur, D., Musser, D., Narendran, P., and Stillman, J., “Semi-unification,” in Proc. Conference on Foundations of Software Technology and Theoretical Computer Science (FST & TCS), Springer Verlag, Berlin, 1988.
Kirchner, H., “Schematization of infinite sets of rewrite rules. Application to the divergence of completion processes,” in Rewriting Techniques and Applications, Pierre Lescanne, ed., Springer Verlag, Berlin, 1987, pp. 180–191.
Knuth, D., and Bendix, P., “Simple word problems in universal algebras,” in Computational Problems in Abstract Algebra (J. Leech, ed.), Pergamon Press, Oxford, 1970, pp. 263–297.
Narendran, P., and Stillman, J., “On the Complexity of Homeomorphic Embedding,” presented at the Fifth International Conference on Algebra, Algebraic Algorithms, and Error-Correcting Codes, (AAECC-5), Menorca, Spain, June 1987 (proceedings to appear in Lecture Notes in Computer Science).
Plaisted, D.A., “A simple non-termination test for the Knuth-Bendix method,” in Proc. 8th International Conference on Automated Deduction, Jörg Siekmann, ed., Springer-Verlag, Berlin, 1986, pp. 79–88.
Schmidt-Schauß, M., “Unification in permutative equational theories is undecidable,” Technical Report SR-87-03, Universität Kaiserslautern.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1989 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Narendran, P., Stillman, J. (1989). It is undecidable whether the Knuth-Bendix completion procedure generates a crossed pair. In: Monien, B., Cori, R. (eds) STACS 89. STACS 1989. Lecture Notes in Computer Science, vol 349. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0028998
Download citation
DOI: https://doi.org/10.1007/BFb0028998
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-50840-3
Online ISBN: 978-3-540-46098-5
eBook Packages: Springer Book Archive