Closure of families of languages under substitution operators

• Authors: David J. Lewis
• Venue: STOC '70 Proceedings of the second annual ACM symposium on Theory of computing
• Year: 1970

This paper treats the closure of families of formal languages under operators which may be viewed as substitution into a particular language. A language L over alphabet {a1,...,an} induces an n-place operator on languages by substitution of the n arguments Li for the symbols ai. For example, if L is regular, it induces an operator under which any full AFL is closed. In section two we find a large class of full AFL's which are closed under no other such operators than those induced by regular languages. Also, for any full AFL @@@@', let @@@@ be the class of languages which @@@@' is closed under substitution into. Then @@@@ is itself a full AFL and is closed under substitution. Finally we show that any substitution-closed full AFL @@@@ is obtained in this manner from some non-substitution-closed full AFL @@@@ (except when @@@@ is the universal family). The study is based on the concept of a full AFL, and in section one considerable effort is devoted to a novel approach to the subject. A full AFL is taken to be a family of languages closed under finite-state transducer mappings and substitution into regular sets. By replacing the family of regular sets with more general families we obtain some broad results about canonical forms for derivations of languages from other languages using transducers and substitution. These forms yield several known forms as special cases, and provide tools needed for the study of full AFL's in section two.