Intersection and Union Hierarchies of Deterministic Context-Free Languages and Pumping Lemmas

Abstract

We study the computational complexity of finite intersections and unions of deterministic context-free languages. Earlier, Wotschke (1978) demonstrated that intersections of (d+1) deterministic context-free languages are in general more powerful than intersections of d deterministic context-free languages for any positive integer d based on the hierarchy separation of Liu and Weiner (1973). The argument of Liu and Weiner, however, works only on bounded languages of particular forms, and therefore Wotschke’s result cannot be extended to disprove any other language to be written in the form of an intersection of d deterministic context-free languages. To deal with the non-membership of a wide range of languages, we circumvent their proof argument and instead devise a new, practical technical tool: a pumping lemma for finite unions of deterministic context-free languages. Since the family of deterministic context-free languages is closed under complementation, this pumping lemma enables us to show a non-membership relation of languages made up with finite intersections of even non-bounded languages as well. We also refer to a relationship to Hibbard’s limited automata.