Abstract
Narrowing is an important method for solving unification problems in equational theories that are presented by confluent term rewriting systems. Because narrowing is a rather complicated operation, several authors studied calculi in which narrowing is replaced by more simple inference rules. This paper is concerned with one such calculus. Contrary to what has been stated in the literature, we show that the calculus lacks strong completeness, so selection functions to cut down the search space are not applicable. We prove completeness of the calculus and we establish an interesting connection between its strong completeness and the completeness of basic narrowing. We also address the eager variable elimination problem. It is known that many redundant derivations can be avoided if the variable elimination rule, one of the inference rules of our calculus, is given precedence over the other inference rules. We prove the completeness of a restricted variant of eager variable elimination in the case of orthogonal term rewriting systems.
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
L. Bachmair, H. Ganzinger, C. Lynch, and W. Snyder, Basic Paramodulation and Superposition, Proc. 11th CADE, LNCS 607, pp. 462–476, 1992.
N. Dershowitz and J.-P. Jouannaud, Rewrite Systems, in: Handbook of Theoretical Computer Science, Vol. B, ed. J. van Leeuwen), North-Holland, pp. 243–320, 1990.
N. Dershowitz and Z. Manna, Proving Termination with Multiset Orderings, Communications of the ACM 22(8), pp. 465–476, 1979.
M. Fay, First-Order Unification in Equational Theories, Proc. 4th CADE, Austin, pp. 161–167, 1979.
J. Gallier and W. Snyder, Complete Sets of Transformations for General E-Unification, TCS 67, pp. 203–260, 1989.
M. Hanus, Efficient Implementation of Narrowing and Rewriting, Proc. PDK-91, LNAI 567, pp. 344–365, 1991.
M. Hanus, The Integration of Functions into Logic Programming: From Theory to Practice, JLP 19 & 20, pp. 583–628, 1994.
S. Hölldobler, A Unification Algorithm for Conflnent Theories, Proc. 14th ICALP, LNCS 267, pp. 31–41, 1987.
S. Hölldobler, Foundations of Equational Logic Programming, LNAI 353, 1989.
G. Huet and J.-J. Lévy, Computations in Orthogonal Rewriting Systems, I and II, in: Computational Logic, Essays in Honor of Alan Robinson (eds. J.-L. Lassez and G. Plotkin), The MIT Press, pp. 396–443, 1991.
J.-M. Hullot, Canonical Forms and Unification, Proc. 5th CADE, LNCS 87, pp. 318–334, 1980.
J.W. Klop, Term Rewriting Systems, in: Handbook of Logic in Computer Science, Vol. II (eds. S. Abramsky, D. Gabbay, and T. Maibaum), Oxford University Press, pp. 1–116, 1992.
A. Martelli, C. Moiso, and G.F. Rossi, Lazy Unification Algorithms for Canonical Rewrite Systems, in: Resolution of Equations in Algebraic Structures, Vol. II (eds. H. Aït-Kaci and M. Nivat), Academic Press, pp. 245–274, 1989.
A. Martelli and U. Montanari, An Efficient Unification Algorithm, ACM TOPLAS 4(2), pp. 258–282, 1982.
A. Martelli, G.F. Rossi, and C. Moiso, An Algortihm for Unification in Equational Theories, Proc. 1986 Symposium on Logic Programming, pp. 180–186, 1986.
A. Middeldorp and E. Hamoen, Completeness Results for Basic Narrowing, AAECC 5, pp. 213–253, 1994.
M. Moser, Improving Transformation Systems for General E-Unification, Proc. 5th RTA, LNCS 690, pp. 92–105, 1993.
S. Okui, A. Middeldorp, and T. Ida, Lazy Narrowing: Strong Completeness and Eager Variable Elimination, Report ISE-TR-94-114, University of Tsukuba, 1994.
J.R. Slagle, Automatic Theorem Proving in Theories with Simplifieras, Commutativity and Associativity, Journal of the ACM 21, pp. 622–642, 1974.
W. Snyder, A Proof Theory for General Unification, Birkhäuser, 1991.
R. Socher-Ambrosius, A Refined Version of General E-Unification, Proc. 12th CADE, LNAI 814, pp. 665–677, 1994.
Y.H. You, Enumerating Outer Narrowing Derivations for Constructor Based Term Rewriting Systems, JSC 7, pp. 319–343, 1989.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1995 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Okui, S., Middeldorp, A., Ida, T. (1995). Lazy narrowing: Strong completeness and eager variable elimination (extended abstract). In: Mosses, P.D., Nielsen, M., Schwartzbach, M.I. (eds) TAPSOFT '95: Theory and Practice of Software Development. CAAP 1995. Lecture Notes in Computer Science, vol 915. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-59293-8_209
Download citation
DOI: https://doi.org/10.1007/3-540-59293-8_209
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-59293-8
Online ISBN: 978-3-540-49233-7
eBook Packages: Springer Book Archive