The complexity of counting problems in equational matching
We introduce a class of counting problems that arise naturally in equational matching and study their computational complexity. If E is an equational theory, then #E-Matching is the problem of counting the number of complete minimal E-matchers of two given terms. #E-Matching is a well-defined algorithmic problem for every finitary equational theory. Moreover, it captures more accurately the computational difficulties associated with finding complete sets of minimal E-matchers than the corresponding decision problem for E-matching does.
In 1979, L. Valiant developed a computational model for measuring the complexity of counting problems and demonstrated the existence of #P- complete problems, i.e., counting problems that are complete for counting non-deterministic Turing machines of polynomial-time complexity. Using the theory of #P-completeness, we analyze the computational complexity of #E-matching for several important equational theories E. We establish that if E is one of the equational theories A, C, AC, I, U, ACI, Set, ACU, or ACIU, then #E-Matching is a #P-complete problem. We also show that there are equational theories, such as the restriction of AC-matching to linear terms, for which the underlying decision matching problem is solvable in polynomial time, while the associated counting matching problem is #P-complete.
KeywordsPolynomial Time Turing Machine Equational Theory Conjunctive Normal Form Truth Assignment
Unable to display preview. Download preview PDF.
- [BKN87]D. Benanav, D. Kapur, and P. Narendran. Complexity of matching problems. Journal of Symbolic Computation, 3:203–216, 1987.Google Scholar
- [BS86]R. Book and J. Siekmann. On unification: equational theories are not bounded. Journal of Symbolic Computation, 2:317–324, 1986.Google Scholar
- [CH93]N. Creignou and M. Hermann. On #P-completeness of some counting problems. Research report 93-R-188, Centre de Recherche en Informatique de Nancy, 1993.Google Scholar
- [Der87]N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3(1 & 2):69–116, 1987. Special issue on Rewriting Techniques and Applications.Google Scholar
- [Dom92]E. Domenjoud. Number of minimal unifiers of the equation αx1+⋯+αxb, =ac Βy1+⋯+Βyq. Journal of Automated Reasoning, 8:39–44, 1992.Google Scholar
- [FH86]F. Fages and G. Huet. Complete sets of unifiers and matchers in equational theories. Theoretical Computer Science, 43(1):189–200, 1986.Google Scholar
- [Joh90]D.S. Johnson. A catalog of complexity classes. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity, chapter 2, pages 67–161. North-Holland, Amsterdam, 1990.Google Scholar
- [KN86]D. Kapur and P. Narendran. NP-completeness of the set unification and matching problems. In J.H. Siekmann, editor, Proceedings 8th International Conference on Automated Deduction, Oxford (England), volume 230 of Lecture Notes in Computer Science, pages 489–495. Springer-Verlag, July 1986.Google Scholar
- [KN92a]D. Kapur and P. Narendran. Complexity of unification problems with associative-commutative operators. Journal of Automated Reasoning, 9:261–288, 1992.Google Scholar
- [KN92b]D. Kapur and P. Narendran. Double-exponential complexity of computing a complete set of AC-unifiers. In Proceedings 7th IEEE Symposium on Logic in Computer Science, Santa Cruz (California, USA), pages 11–21, 1992.Google Scholar
- [Koz92]D.C. Kozen. The design and analysis of algorithms, chapter 26: Counting problems and #P, pages 138–143. Springer-Verlag, 1992.Google Scholar
- [MN89]U. Martin and T. Nipkow. Boolean unification — the story so far. Journal of Symbolic Computation, 7(3 & 4):275–294, 1989.Google Scholar
- [Plo72]G. Plotkin. Building-in equational theories. Machine Intelligence, 7:73–90, 1972.Google Scholar
- [Sny93]W. Snyder. On the complexity of recursive path orderings. Information Processing Letters, 46:257–262, 1993.Google Scholar
- [SS82]J. Siekmann and P. Szabó. Universal unification and classification of equational theories. In D.W. Loveland, editor, Proceedings 6th International Conference on Automated Deduction, New York (NY, USA), volume 138 of Lecture Notes in Computer Science, pages 369–389. Springer-Verlag, June 1982.Google Scholar
- [TA87]E. Tidén and S. Arnborg. Unification problems with one-sided distributivity. Journal of Symbolic Computation, 3(1 & 2):183–202, 1987.Google Scholar
- [Val79a]L.G. Valiant. The complexity of computing the permanent. Theoretical Computer Science, 8:189–201, 1979.Google Scholar
- [Val79b]L.G. Valiant. The complexity of enumeration and reliability problems. SIAM Journal on Computing, 8(3):410–421, 1979.Google Scholar