On the Length of Shortest Strings Accepted by Two-way Finite Automata

Abstract

Given a two-way finite automaton recognizing a non-empty language, consider the length of the shortest string it accepts, and, for each n ≥ 1, let f(n) be the maximum of these lengths over all n-state automata. It is proved that for n-state two-way finite automata, whether deterministic or nondeterministic, this number is at least Ω(10n/5) and less than (2nn+1), with the lower bound reached over an alphabet of size Θ(n). Furthermore, for deterministic automata and for a fixed alphabet of size m ≥ 1, the length of the shortest string is at least e(1+o(1))mn(log n− log m).