Concise Description of Finite Languages

Abstract This paper is a contribution to the theory of grammatical complexity, in particular to the following basic question: consider some language L and grammars of some type X; what is the smallest number of productions of a type X grammar required to generate L? This complexity measure, the so-called X complexity of L, has been investigated before. We study the more basic case when the languages L considered are finite (a case which has been neglected, so far). We obtain a number of results and insights suggesting that such study is of importance. In addition to a number of ‘expected’ (but not necessarily easy to prove) results that with type X grammars more productions are necessary than with some other type Y grammars (even if types X and Y define the same family of languages) we show that if the limit of certain sequences of finite languages is of type X1 then the type X complexity of each of the finite languages involved must be low.

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