article

Complexity of matching problems

Authors:

Dan Benanav,

Deepak Kapur,

Paliath NarendranAuthors Info & Claims

Journal of Symbolic Computation, Volume 3, Issue 1-2

Pages 203 - 216

https://doi.org/10.1016/S0747-7171(87)80027-5

Published: 01 February 1987 Publication History

Abstract

The associative-commutative matching problem is shown to be NP-complete; more precisely, the matching problem for terms in which some function symbols are uninterpreted and others are both associative and commutative, is NP-complete. It turns out that the similar problems of associative-matching and commutative-matching are also NP-complete. However, if every variable appears at most once in a term being matched, then the associative-commutative matching problem is shown to have an upper-bound of O ( | s | ^* | t |^3), where | s | and | t | are respectively the sizes of the pattern 8 and the subject t.

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Reviews

Reviewer: John A. Campbell

The matching of expressions to determine equality or equivalence is a routine operation in symbolic mathematical computations (e.g., computer algebra, program transformation, functional programming). Matching is also a component of unification algorithms, such as those used in logic programming. The general problem is NP-complete, although in practice many programs that use matching do so under special conditions where additional information is available to keep the size of their computations under control. Although most actual users of matching are not worried by NP-completeness, it is of theoretical interest to understand the natures of specialized instances of the general problem. This paper represents a technical contribution in that direction. By using 3SAT (the satisfiability problem for a set of clauses which each have three literals) as a reference problem, the authors show NP-completeness for three specialized examples of matching: when the set F of operators or function symbols contains at least one member that is both associative and commutative and at least three that are not known to be either associative or commutative, when F contains at least one associative member and at least two that are not known to be associative, and when F contains at least one commutative member and at least five that are not known to be commutative. Also, in the case where every variable involved appears no more than once in a term being matched, the first problem (associative-commutative matching) is shown to have a complexity bounded above by O( st 3), where s and t are the sizes of the pattern being matched and the structure against which it is being matched, respectively.

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cover image Journal of Symbolic Computation

Journal of Symbolic Computation Volume 3, Issue 1-2

Feb./April 1987

213 pages

ISSN:0747-7171

Editor:
Bruno Buchberger
Johannes Kepler Univ., Linz, Austria

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Academic Press, Inc.

United States

Publication History

Published: 01 February 1987

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Thomas SBornholt JTsafrir DMUSUVATHI MGupta RAbu-Ghazaleh N(2024)Automatic Generation of Vectorizing Compilers for Customizable Digital Signal ProcessorsProceedings of the 29th ACM International Conference on Architectural Support for Programming Languages and Operating Systems, Volume 110.1145/3617232.3624873(19-34)Online publication date: 27-Apr-2024
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On the computational complexity of 2-interval pattern matching problems

Complexity of Nilpotent Unification and Matching Problems

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