Automaticity I: Properties of a measure of descriptional complexity

Abstract

Let Σand 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). We define AL(n) for languages L to be AXL(n), where XL is the characteristic function of L. The same or similar notions were examined previously by Trakhtenbrot, Grinberg and Korshunov, Karp, Breitbart, Gabarró, Dwork and Stockmeyer, and Kaneps and Freivalds. Karp proved that if L ⊆ Σ* is not regular, then AL(n) ≥ (n + 3)/2 infinitely often. We prove that the lower bound is best possible. We obtain results on the growth rate of Af(n). If |Σ| - k ≥ 2 and |Δ| - l 0 we have Af(n) > (1 - ∈) Ckn+1/n for all sufficiently large n. We also obtain bounds on NL(n), the non-deterministic automaticity function. This is similar to Af(n), except that it counts the number of states in the minimal NFA, and it is defined for languages L ⊆ Σ*. For |Σ| = k ≥ 2, we have NL(n) = O(kn/2). Also, for almost all languages L and every ∈ > 0 we have NL(n) > (1 - ∈) kn/2/ √k - 1 for all sufficiently large n. We prove some incomparability results between the automaticity measure and those defined earlier by Gabarró and others. Finally, we examine the notion of automaticity as applied to sequences. © 1996 Academic Press, Inc.