The Complexity of Counting Problems in Equational Matching
Identifieur interne : 001B23 ( Istex/Corpus ); précédent : 001B22; suivant : 001B24The Complexity of Counting Problems in Equational Matching
Auteurs : Miki Hermann ; Phokion G. KolaitisSource :
- Journal of Symbolic Computation [ 0747-7171 ] ; 1995.
Abstract
Abstract: We introduce a class of counting problems that arise naturally in equational matching and investigate their computational complexity. If E is an equational theory, then #E-Matching is the problem of counting the number of most general 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 minimal complete sets of 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.
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DOI: 10.1006/jsco.1995.1054
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<front><div type="abstract" xml:lang="en">Abstract: We introduce a class of counting problems that arise naturally in equational matching and investigate their computational complexity. If E is an equational theory, then #E-Matching is the problem of counting the number of most general 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 minimal complete sets of 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.</div>
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