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Convolution is a broadly useful operation with applications including signalprocessing, machine learning, probability, optics, polynomial multiplication,and efficient parsing. Usually, however, this operation is understood andimplemented in more specialized forms, hiding commonalities and limitingusefulness. This paper formulates convolution in the common algebraic frameworkof semirings and semimodules and populates that framework with variousrepresentation types. One of those types is the grand abstract template anditself generalizes to the free semimodule monad. Other representations servevaried uses and performance trade-offs, with implementations calculated fromsimple and regular specifications.  Of particular interest is Brzozowski's method for regular expressionmatching. Uncovering the method's essence frees it from syntacticmanipulations, while generalizing from boolean to weighted membership (such asmultisets and probability distributions) and from sets to n-ary relations. Theclassic trie data structure then provides an elegant and efficient alternativeto syntax.  Pleasantly, polynomial arithmetic requires no additional implementationeffort, works correctly with a variety of representations, and handlesmultivariate polynomials and power series with ease. Image convolution alsofalls out as a special case.

Generalized Convolution and Efficient Language Recognition