A Sheaf-Theoretic Approach to Pattern Matching and Related Problems

Abstract We present a general theory of pattern matching by adopting an extensional, geometric view of patterns. Representing the geometry of the pattern via a Grothendieck topology, the extension of the matching relation for a constant target and varying pattern forms a sheaf. We derive a generalized version of the Knuth-Morris-Pratt string-matching algorithm by gradually converting this extensional description into an intensional description, i.e., an algorithm. The generality of this approach is illustrated by briefly considering other applications: Earley's algorithm for parsing, Waltz filtering for scene analysis, matching modulo commutativity, and the n-queens problem.

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