Abstract
It is shown that theories presented by a set of ground equations with several associative-commutative (AC) symbols always admit a finite canonical system. This result is obtained through the construction of a reduction ordering that is AC-compatible and total on the set of congruence classes generated by the associativity and commutativity axioms. As far as we know, this is the first ordering with such properties when several AC-function symbols and free-function symbols are allowed. Such an ordering is also a fundamental tool for deriving a complete theorem proving strategies with built-in associative commutative unification.
Similar content being viewed by others
References
AnantharmanS. and HsiangJ.: An automated proof of the Moufong identities in alternative rings, J. Automated Reasoning 6 (1990), 79–109.
Bachmair, L. and Dershowitz, N.: Commutation, transformation and termination, in J. Siekmann (ed.), Proc. 8th Conf. Automated Deduction, Lecture Notes in Computer Science 230, Springer-Verlag, 1986, pp. 5–20.
Bachmair, L., Dershowitz, N., and Hsiang, J.: Ordering for equational proofs, in Proc. Symp. Logic in Computer Science (LICS), Boston, MA, 1986, pp. 346–357.
BachmairL. and PlaistedD. A.: Termination orderings for associative-commutative rewriting systems, J. Symbolic Computation 1 (1985), 329–349.
BallantyneA. M. and LankfordD.: New decision algorithms for finitely presented commutative semigroups, Comp. Maths. Appl. 7 (1981), 159–165.
BenCherifaA. and LescanneP.: Termination of rewriting systems by polynomial interpretations and its implementation, Sci. Computer Programming 9(2) (October 1987), 137–160.
DershowitzN.: Termination of rewriting, J. Symbolic Computation 1 & 2 (1987), 69–116.
DershowitzN. and JouannaudJ.-P.: Rewrite systems, in VanLeuven (ed.), Handbook of Theoretical Computer Science, North-Holland, Amsterdam, 1990.
DicksonL.: Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors, Amer. J. Math. 35 (1913), 413–422.
GallierJ., NarendranP., PlaistedD., RaatzS., and SnyderW.: An algorithm for finding canonical sets of ground rewrite rules in polynomial time, JACM 40(1) (Jan. 1993), 1–16.
Hsiang, J. and Rusinowitch, M.: A new method for establishing refutational completeness in theorem proving, in J. Siekmann (ed.), Proc. 8th Conf. Automated Deduction, Lecture Notes in Computer Science 230, Springer-Verlag, 1986, pp. 141–152.
HuetG.: Confluent reductions: abstract properties and applications to term rewriting systems, J. Ass. Computing Machinery 27(4) (October 1980), 797–821.
JouannaudJ.-P. and KirchnerH.: Completion of a set of rules modulo a set of equations, SIAM J. Computing 15(4) (1986), 1155–1194.
KapurD. and NarendranP.: A finite Thue system with decidable word problem and without equivalent finite canonical system, Theoret. Computer Sci. 35 (1985), 337–344.
KapurD. and NarendranP.: Complexity of unification problems with associative-commutative operators, J. Automated Reasoning 9(2) (1992), 261–288.
Kapur, D., Sivakumar, G., and Zhang, H.: A new method for proving termination of AC-rewrite systems, Presented at the Conference on the Foundations of Software Technology and Theoretical Computer Science, New Delhi, India, December 1990.
KnuthD. E. and BendixP. B.: Simple word problems in universal algebras, in J.Leech (ed.), Computational Problems in Abstract Algebra, Pergamon, Oxford, 1970, pp. 263–297.
LankfordD. S.: On Proving Term Rewriting Systems are Noetherian, Technical Report, Louisiana Tech. University, Mathematics Dept., Ruston, LA, 1979.
Lankford, D. S. and Ballantyne, A.: Decision Procedures for Simple Equational Theories with Permutative Axioms: Complete Sets of Permutative Reductions, Technical Report, Univ. of Texas at Austin, Dept. of Mathematics and Computer Science, 1977.
Marché, C.: On ground AC-completion, in R. Book (ed.), Proc. 4th Conf. Rewriting Techniques and Applications, Lecture Notes in Computer Science 488, Springer-Verlag, 1991, pp. 411–422.
Mayr, E. W.: An algorithm for the general Petri net reachability problem, in Proc. STOC, 1981.
Narendran, P. and Rusinowitch, M.: Any ground associative-commutative theory has a finite canonical system, in R. Book (ed.), Proc. 4th Conf. Rewriting Techniques and Applications, Lecture Notes in Computer Science 488, Springer-Verlag, 1991, pp. 423–433.
Narendran, P. and Rusinowitch, M.: The unifiability problem in ground AC theories, Presented at the 8th Ann. Symp. Logic in Computer Science (LICS), Montreal, Canada, June 1993.
PetersonG. and StickelM.: Complete sets of reductions for some equational theories, J. Ass. Computing Machinery 28 (1981), 233–264.
Snyder, W.: Efficient completion: An O(n log n) algorithm for generating reduced sets of ground rewrite rules equivalent to a set of ground equations E, in N. Dershowitz (ed.), Proc. 3rd Conf. Rewriting Techniques and Applications (RTA), Lecture Notes in Computer Science, Springer-Verlag, 1989.
Steinbach, J.: AC-termination of rewrite systems — A modified Knuth-Bendix ordering, in Proc. 2nd Int. Conf. Algebraic and Logic Programming, Nancy (France), Lecture Notes in Computer Science 463, Springer-Verlag, 1990, pp. 372–386.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Narendran, P., Rusinowitch, M. Any ground associative-commutative theory has a finite canonical system. J Autom Reasoning 17, 131–143 (1996). https://doi.org/10.1007/BF00247671
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00247671