Characterizations of locally testable events

Abstract

Let Σ be a finite alphabet, Σ * the free monoid generated by Σ and χ the length of χ ∈ Σ *. For any integer k0, f k (χ) ( t k (χ)) is χ if χ < k + 1, and it is the prefix (suffix) of χ of length k, othewise. Also let m k+1 (χ) = {νχ = uν w and ν = k+1}. For χ, y ε Σ * define χ ∼ k+1 y iff f k (χ) = f k ( y), t k (χ) = t k ( y) and m k+1 (χ) = m k+1 ( y). The relation ∼ k+1 is a congruence of finite index over Σ *. An event E ⊆ Σ * is ( k+1)-testable iff it is a union of congruence classes of ∼ k+1 . E is locally testable (LT) if it is k+1-testable for some k. (This definition differs from that of [6] but is equivalent.) We show that the family of LT events is a proper sub-family of star-free events of dot-depth 1. LT events and k-testable events are characterized in terms of (a) restricted star-free expressions based on finite and cofinite events; (b) finite automata accepting these events; (c) semigroups; and (d) structural decomposition of such automata. Algorithms are given for deciding whether a regular event is (a) LT and (b) k+1-testable. Generalized definite events are also characterized.