On the Average Complexity of Brzozowski’s Algorithm for Deterministic Automata with a Small Number of Final States

Abstract

Abstract. We analyze the average complexity of Brzozowski's minimization algorithm for distributions of deterministic automata with a small number of final states. We show that, as in the case of the uniform distribution, the average complexity is super-polynomial even if we consider random deterministic automata with only one final state. We therefore go beyond the previous study where the number of final states was linear in the number of states. Our result holds for alphabets with at least 3 letters.

Average Case Analysis of Brzozowski's Algorithm

“…The results presented in this article therefore covers the uniform case. However, they are weaker than those established in [9], which is dedicated to such distributions.…”

“…The result may hold for binary alphabets, see Section 6 for a discussion on that point. Notice also that we slightly improved over [9] by generalizing the possible distributions for the set of final states.…”

“…Section 5 is devoted to the proof of our main theorem. Some further directions are finally proposed in Section 6. This article is the full version of the extended abstract [9], where all the missing proofs have been included.…”

Random Deterministic Automata

“…Unfortunately, as stated in Remark 20, most results presented in this article cannot be adapted to automata with, say, one final state. The only exception is the recent analysis of Brzozowski's algorithm for that kind of distributions [20]. One of the interesting open questions here is to confirm experimental studies that indicates that a random automaton with one final state should be minimal with non-negligible probability.…”

“…Using the same technique as for sink states, this "proves" that an accessible automaton is co-accessible with high probability. By controlling the error terms, this informal argument can be turned into a full proof [20].…”