On regularity of context-free languages

Abstract

This paper considers conditions under which a context-free language is regular and conditions which imposed on (productions of) a rewriting system generating a context-free language will guarantee that the generated language is regular. In particular: 1. (1) necessary and sufficient conditions on productions of a unitary grammar are given that guarantee the generated language to be regular (a unitary grammar is a semi-Thue system in which the left-hand of each production is the empty word), and 2. (2) it is proved that commutativity of a linear language implies its regularity. To obtain the former result, we give a generalization of the Myhill–Nerode characterization of the regular languages in terms of well-quasi orders, along with a generalization of Higman's well-quasi order result concerning the subsequence embedding relation on Σ*. In obtaining the latter results, we introduce the class of periodic languages, and demonstrate how they can be used to characterize the commutative regular languages. Here we also utilize the theory of well-quasi orders.