Intersection and Union Hierarchies of Deterministic Context-Free Languages and Pumping Lemmas (2112.09383v1)

Abstract: We study the computational complexity of finite intersections and 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 intersection hierarchy separation 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 disprove any given language to be written in the form of $d$ intersection of dcf languages. To deal with the non-membership of a wide range of languages, we circumvent the specialization of their proof argument 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 a non-membership relation of languages formed by finite intersections of non-bounded languages as well. We also refer to a relationship to deterministic limited automata of Hibbard [Inf. Control 11 (1967) 196--238] in this regard.