Closures which preserve finiteness in families of languages*

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Abstract

Each set of operations selected from union, intersection, complement, star, quotients, derivatives, word-reversal, and homomorphisms is investigated with respect to its closure of an arbitrary family of word-sets, as well as to its closure of an arbitrary family of regular languages. Certain sets are shown to produce a finite closure for every finite family of word-sets; others, to produce a finite closure only for every finite family of regular languages; in either case, the closure for a given family of regular languages can be calculated by algorithm. For a third class-of sets, the closure is not necessarily finite even for finite families of regular languages.

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*

Research sponsored in part by the Air Force Cambridge Research laboratories, Office of Aerospace Research, USAF, under contract F1962867C0008, and by the Air Force Office of Scientific Research, Office of Aerospace Research, USAF, under AFOSR Grant No. AFAFOSR-1203-67.