Two iteration theorems for some families of languages

Abstract

The Abstract Families of Languages, abbreviated full AFL's, were introduced by S. Ginsburg and S. Greibach in [I 1]. It is well known that the family of contextfree languages is a full AFL [9]. It is also a rational cone (according to Eilenberg's terminology) [5]. Let Dn* (respectively, D~*) be the Dyck language (respectively, semi-Dyck language) on 2n letters, ai , 5i , i = 1,..., n, i.e., the class of 1 in the congruence generated by (7i5 i = 5i (7 i ~ 1 (respectively, aiSi = 1) [9-14]. The ChomskySchtitzenberger theorem [9] implies that, for any n >~ 2, D~* (respectively, D~*) is a full generator [10-11] of the AFL as well as of the rational cone of the context-free languages. It seems, then, natural to look at the rational cones and the AFL's generated by the Dyck languages DI* and the semi-Dyck language DI* on two letters. We proved elsewhere [2] that the rational cone W generated by D~* can be characterized by the structure of the pushdown automata recognizing the languages in cg. The main restriction we impose on them is that they should use a single pushdown symbol. Since we can consider the pushdown store as a counter, we call such an a automaton and we call one-counter languages the elements of ~7. It is rather important to notice that this family W is not the family ~ studied by Greibach in [13]. However, and c# are closely related: o~ is the full AFL generated by Di* [13]. Consequently, from a result of [11] restated in [3], o~is the closer of under union, product, and star operation. In this paper, we prove two pumping lemmas (Theorems 3 and 4) which yield corollaries such as: