Automaticity: Properties of a measure of descriptional complexity

Abstract

Let Σ and Δ be nonempty alphabets with Σ finite. Let f be a function mapping Σ* to Δ. We explore the notion of automaticity, which attempts to model how “close” f is to a finite-state function. Formally, the automaticity of f is a function Af(n) which counts the minimum number of states in any deterministic finite automaton that computes f correctly on all strings of length ≤ n (and its behavior on longer strings is not specified). The same or similar notions were examined previously by Trakhtenbrot, Grinberg and Korshunov, Karp, Breitbart, Gabarró, Dwork and Stockmeyer, and Kaneps and Freivalds.