Asymptotic Approximation by Regular Languages

Abstract: This paper investigates a new property of formal languages called REG-measurability where REG is the class of regular languages. Intuitively, a language (L) is REG-measurable if there exists an infinite sequence of regular languages that "converges" to (L). A language without REG-measurability has a complex shape in some sense so that it can not be (asymptotically) approximated by regular languages. We show that several context-free languages are REG-measurable (including languages with transcendental generating function and transcendental density, in particular), while a certain simple deterministic context-free language and the set of primitive words are REG-immeasurable in a strong sense.