Probabilistic and Nondeterministic Unary Automata

Abstract

We investigate unary regular languages and compare deterministic finite automata (DFA’s), nondeterministic finite automata (NFA’s) and probabilistic finite automata (PFA’s) with respect to their size.

Given a unary PFA with n states and an ε-isolated cutpoint, we show that the minimal equivalent DFA has at most $n^{\frac{1}{2\epsilon}}$ states in its cycle. This result is almost optimal, since for any α< 1 a family of PFA’s can be constructed such that every equivalent DFA has at least $n^{\frac{\alpha}{2\epsilon}}$ states. Thus we show that for the model of probabilistic automata with a constant error bound, there is only a polynomial blowup for cyclic languages.

Given a unary NFA with n states, we show that efficiently approximating the size of a minimal equivalent NFA within the factor $\frac{\sqrt{n}}{\ln n}$ is impossible unless P=NP. This result even holds under the promise that the accepted language is cyclic. On the other hand we show that we can approximate a minimal NFA within the factor ln n, if we are given a cyclic unary n-state DFA.