Abstract
The simultaneous elementary E-matching problem for an equational theory E is to decide whether there is an E-matcher for a given system of equations in which the only nonconstant function symbols occurring in the terms to be matched are the ones constrained by the equational axioms of E. We study the computational complexity of simultaneous elementary matching problems for the equational theories A of semigroups, AC of commutative semigroups, and ACU of commutative monoids. In each case, we delineate the boundary between NP-completeness and solvability in polynomial time by considering two parameters, the number of equations in the systems and the number of constant symbols in the signature. Moreover, we analyze further the intractable cases of simultaneous elementary AC-matching and ACU-matching by also taking into account the maximum number of occurrences of each variable. Using combinatorial optimization techniques, we show that if each variable is restricted to having at most two occurrences, then several cases of simultaneous elementary AC-matching and ACU-matching can be solved in polynomial time.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Angluin, D.: Finding patterns common to a set of strings, J. Comput. System Sci. 21 (1980), 46–62.
Baader, F. and Siekmann, J. H.: Unification theory, in D. Gabbay, C. Hogger, and J. Robinson (eds.), Handbook of Logic in Artificial Intelligence and Logic Programming, Vol. 2: Deduction Methodologies, Oxford Univ. Press, Oxford (UK), 1994, pp. 41–125.
Benanav, D., Kapur, D. and Narendran, P.: Complexity of matching problems, J. Symbolic Comput. 3 (1987), 203–216.
Berge, C.: Graphs, 2nd revised edn, North-Holland, Amsterdam, 1985.
Berregeb, N., Bouhoula, A. and Rusinowitch, M.: SPIKE-AC: A system for proofs by induction in associative-commutative theories, in H. Ganzinger (ed.), Proceedings 7th Conference on Rewriting Techniques and Applications (RTA'96), New Brunswick (NJ, USA), Lecture Notes in Comput. Sci. 1103, 1996, pp. 428–431.
Bürckert, H.-J., Herold, A., Kapur, D., Siekmann, J. H., Stickel, M. E., Tepp, M. and Zhang, H.: Opening the AC-unification race, J. Automated Reasoning 4(4) (1988), 465–474.
Creignou, N. and Hermann, M.: Complexity of generalized satisfiability counting problems, Inform. and Comput. 125(1) (1996), 1–12.
Edmonds, J. and Johnson, E.: Matching: A well-solved class of integer linear programs, in Combinatorial Structures and Their Applications, Calgary (Canada), Gordon and Breach, 1969, pp. 89–92.
Eker, S.: Improving the efficiency of AC matching and unification, Research report 2104, Institut de Recherche en Informatique et en Automatique, 1993.
Garey, M. and Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman and Co., 1979.
Garland, S. and Guttag, J.: An overview of LP, the Larch Prover, in N. Dershowitz (ed.), Proceedings 3rd Conference on Rewriting Techniques and Applications (RTA'89), Chapel Hill, (NC, USA), Lecture Notes in Comput. Sci. 355, 1989, pp. 137–151.
Hermann, M. and Kolaitis, P.: The complexity of counting problems in equational matching, J. Symbolic Computation 20(3) (1995), 343–362.
Johnson, D.: A catalog of complexity classes, in J. van Leeuwen (ed.), Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity, North-Holland, Amsterdam, Chapt. 2, 1990, pp. 67–161.
Jouannaud, J.-P. and Kirchner, C.: Solving equations in abstract algebras: A rule-based survey of unification, in J.-L. Lassez and G. Plotkin (eds.), Computational Logic. Essays in Honor of Alan Robinson, MIT Press, Cambridge, Chapt. 8, 1991, pp. 257–321.
Kapur, D. and Zhang, H.: An overview of Rewrite Rule Laboratory (RRL), Comput. Math. Appl. 29(2) (1995), 91–114.
Kirchner, C., Kirchner, H. and Vittek, M.: Designing constraint logic programming languages using computational systems, in V. Saraswat and P. V. Hentenryck (eds.), Principles and Practice of Constraint Programming, MIT Press, Chapt. 8, 1995, pp. 133–160.
Lenstra, Jr, H.: Integer programming with a fixed number of variables, Math. Oper. Res. 8(4) (1983), 538–548.
McCune, W.: Solution of the Robbins problem, J. Automated Reasoning 19(3) (1997), 263–276.
Papadimitriou, C.: On the complexity of integer programming, J. Assoc. Comput. Mach. 28(4) (1981), 765–768.
Papadimitriou, C.: Computational Complexity, Addison-Wesley, 1994.
Shiloach, Y.: Another look at the degree constrained subgraph problem, Inform. Proces. Lett. 12(2) (1981), 89–92.
Stickel, M.: A unification algorithm for associative-commutative functions, J. Assoc. Comput. Mach. 28(3) (1981), 423–434.
Toda, S.: On the computational power of PP and ⊕P, in Proceedings 30th IEEE Symposium on Foundations of Computer Science (FOCS'89), Research Triangle Park (NC, USA), 1989, pp. 514–519.
Urquhart, R.: Degree constrained subgraphs of linear graphs, Ph.D. Thesis, University of Michigan, Ann Arbor, 1967.
Valiant, L.: The complexity of computing the permanent, Theoret. Comput. Sci. 8(2) (1979a), 189–201.
Valiant, L.: The complexity of enumeration and reliability problems, SIAM J. Comput. 8(3) (1979b), 410–421.
van Leeuwen, J.: Graph algorithms, in J. van Leeuwen (ed.), Handbook of Theoretical Computer Science A: Algorithms and Complexity, Elsevier, Amsterdam, Chapt. 10, 1990, pp. 525–631.
Verma, R. and Ramakrishnan, I.: Tight complexity bounds for term matching problems, Inform. Comput. 101 (1992), 33–69.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hermann, M., Kolaitis, P.G. Computational Complexity of Simultaneous Elementary Matching Problems. Journal of Automated Reasoning 23, 107–136 (1999). https://doi.org/10.1023/A:1006136609427
Issue Date:
DOI: https://doi.org/10.1023/A:1006136609427