Note on the boolean properties of context free languages

Abstract

A language is considered to be any set of sentences (strings) made up from a finite vocabulary (alphabet). A grammar is a device which enumerates a language, e.g., a Turing machine with any set of input signals, a finite automaton, or a Post system. One often considers the family of languages which have grammars meeting certain specifications or restrictions, e.g., the finite state languages (the languages generatable by finite automata) . A major question of interest is whether such a family is a closed system with respect to the Boolean operations: set-theoretic union, intersection, and difference. The family of recursively enumerable languages is well known to be closed under union and intersection, but not under difference. I t is also well known that the family of finite state languages is closed under all three operations. In this note we answer this question for another family of languages, called type 2 or by Chomsky (1959). For the sake of having a convenient reference we shall present in modified form Chomsky's definitions. I t will be seen that a context free grammar is essentially a special case of a semi-Thue system, cf. Davis (1958). We consider a context free (CF) grammar to be a finite set G of rewriting a --~ ~, where a is a single symbol and ~ is a finite string of symbols from a finite alphabet (vocabulary) V. V contains precisely the symbols appearing in these rules plus the boundary symbol ~, which does not appear in these rules. Rules of the form a --~ a (which