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International Conference on Automated Deduction

CADE 1994: Automated Deduction — CADE-12 pp 560–574Cite as

The complexity of counting problems in equational matching

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Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 814))

Abstract

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.

Partially supported by Institut National Polytechnique de Lorraine grant 910 0146 R1.

Part of the research reported here was carried out while this author was visiting CRIN & INRIA-Lorraine supported by the University of Nancy 1 and INRIA-Lorraine. Research of this author is also supported by a 1993 John Simon Guggenheim Fellowship and by NSF Grant CCR-9108631.

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Author information

Authors and Affiliations

  1. CRIN (CNRS) and INRIA-Lorraine, BP 239, 54506, Vandceuvre-lès-Nancy, France

    Miki Hermann

  2. Computer and Information Sciences, University of California, 95064, Santa Cruz, CA, USA

    Phokion G. Kolaitis

Authors
  1. Miki Hermann
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  2. Phokion G. Kolaitis
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Alan Bundy

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© 1994 Springer-Verlag Berlin Heidelberg

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Hermann, M., Kolaitis, P.G. (1994). The complexity of counting problems in equational matching. In: Bundy, A. (eds) Automated Deduction — CADE-12. CADE 1994. Lecture Notes in Computer Science, vol 814. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58156-1_41

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  • DOI: https://doi.org/10.1007/3-540-58156-1_41

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-58156-7

  • Online ISBN: 978-3-540-48467-7

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