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

TL;DR

This paper investigates the complexity of finite intersections and unions of deterministic context-free languages, introducing new pumping lemmas to better understand their hierarchical structure and non-membership properties.

Contribution

It develops novel pumping lemmas for finite unions of dcf languages, extending the understanding of their intersection hierarchy beyond previous bounded language limitations.

Findings

Established non-membership relations using new pumping lemmas

Extended the hierarchy understanding to a broader class of languages

Linked the results to deterministic limited automata theory

Abstract

We study the computational complexity of finite intersections and finite unions of deterministic context-free (dcf) languages. Earlier, Wotschke [J. Comput. System Sci. 16 (1978) 456--461] demonstrated that intersections of $(d+1)$ dcf languages are in general more powerful than intersections of $d$ dcf languages for any positive integer $d$ based on the separation result of the intersection hierarchy of Liu and Weiner [Math. Systems Theory 7 (1973) 185--192]. The argument of Liu and Weiner, however, works only on bounded languages of particular forms, and therefore Wotschke's result is not directly extendable to other non-bounded languages. To deal with a wide range of languages for the non-membership to the intersection hierarchy, we circumvent the specialization of their proof technics and devise a new and practical technical tool: two pumping lemmas for finite unions of dcf languages. Since the family of dcf languages is closed under complementation and also under intersection with regular languages, these pumping lemmas help us establish the non-membership relation of languages formed by finite intersections of target languages. We also concern ourselves with a relationship to deterministic limited automata of Hibbard [Inf. Control 11 (1967) 196--238] in this regard.