Abstract
We introduce the notion of an associative-commutative congruence closure and show how such closures can be constructed via completion-like transition rules. This method is based on combining completion algorithms for theories over disjoint signatures to produce a convergent rewrite system over an extended signature. This approach can also be used to solve the word problem for ground AC-theories without the need for AC-simplification orderings total on ground terms.
Associative-commutative congruence closure provides a novel way to construct a convergent rewrite system for a ground AC-theory. This is done by transforming an AC-congruence closure, which is described by rewrite rules over an extended signature, to a rewrite system over the original signature. The set of rewrite rules thus obtained is convergent with respect to a new and simpler notion of associative-commutative reduction.
The research described in this paper was supported in part by the National Science Foundation under grants CCR-9902031, CCR-9711386 and EIA-9705998.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bachmair, L.: Canonical Equational Proofs. Birkhäuser, Boston (1991)
Bachmair, L., Dershowitz, N.: Completion for rewriting modulo a congruence. Theoretical Computer Science 67(2 & 3), 173–201 (1989)
Bachmair, L., Ramakrishnan, C., Ramakrishnan, I., Tiwari, A.: Normalization via rewrite closures. In: Narendran, P., Rusinowitch, M. (eds.) RTA 1999. LNCS, vol. 1631, pp. 190–204. Springer, Heidelberg (1999)
Ballantyne, A.M., Lankford, D.S.: New decision algorithms for finitely presented commutative semigroups. Comp. and Maths. with Appls. 7, 159–165 (1981)
Becker, T., Weispfenning, V.: Gröbner bases: a computational approach to commutative algebra. Springer, Berlin (1993)
Dershowitz, N., Jouannaud, J.P.: Rewrite systems. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science. Formal Models and Semantics, vol. B. North-Holland, Amsterdam (1990)
Dershowitz, N., Manna, Z.: Proving termination with multiset orderings. Communications of the ACM 22(8), 465–476 (1979)
Domenjoud, E., Klay, F.: Shallow AC theories. In: Proceedings of the 2nd CCL Workshop, La Escala, Spain (September 1993)
Kapur, D.: Shostak’s congruence closure as completion. In: Comon, H. (ed.) RTA 1997. LNCS, vol. 1232, pp. 23–37. Springer, Heidelberg (1997)
Koppenhagen, U., Mayr, E.W.: An optimal algorithm for constructing the reduced Gröbner basis of binomial ideals. In: Lakshman, Y.D. (ed.) Proceedings of the International Symposium on Symbolic and Algebraic Computation, pp. 55–62 (1996)
Marche, C.: On ground AC-completion. In: Book, R.V. (ed.) RTA 1991. LNCS, vol. 488, pp. 411–422. Springer, Heidelberg (1991)
Mayr, E.W., Meyer, A.R.: The complexity of the word problems for commutative semigroups and polynomial ideals. Advances in Mathematics 46, 305–329 (1982)
Narendran, P., Rusinowitch, M.: Any ground associative-commutative theory has a finite canonical system. In: Book, R.V. (ed.) RTA 1991. LNCS, vol. 488, pp. 423–434. Springer, Heidelberg (1991)
Nelson, G., Oppen, D.: Simplification by cooperating decision procedures. ACM Transactions on Programming Languages and Systems 1(2), 245–257 (1979)
Nelson, G., Oppen, D.: Fast decision procedures based on congruence closure. Journal ofthe Association for Computing Machinery 27(2), 356–364 (1980)
Peterson, G.E., Stickel, M.E.: Complete sets of reductions for some equational theories. J. ACM 28(2), 233–264 (1981)
Ringeissen, C.: Cooperation of decision procedures for the satisfiability problem. In: Baader, F., Schulz, K.U. (eds.) Frontiers of Combining Systems, pp. 121–139. Kluwer Academic Publishers, Dordrecht (1996)
Rubio, A., Nieuwenhuis, R.: A precedence-based total AC-compatible ordering. In: Kirchner, C. (ed.) RTA 1993. LNCS, vol. 690, pp. 374–388. Springer, Heidelberg (1993)
Tinelli, C., Harandi, M.: A new correctness proof of the Nelson-Oppen combination procedure. In: Baader, F., Schulz, K.U. (eds.) Frontiers of Combining Systems, pp. 103–119. Kluwer Academic Publishers, Dordrecht (1996)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bachmair, L., Ramakrishnan, I.V., Tiwari, A., Vigneron, L. (2000). Congruence Closure Modulo Associativity and Commutativity. In: Kirchner, H., Ringeissen, C. (eds) Frontiers of Combining Systems. FroCoS 2000. Lecture Notes in Computer Science(), vol 1794. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10720084_16
Download citation
DOI: https://doi.org/10.1007/10720084_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-67281-4
Online ISBN: 978-3-540-46421-1
eBook Packages: Springer Book Archive