Average State Complexity of Operations on Unary Automata

Abstract

Define the complexity of a regular language as the number of states of its minimal automaton. Let $\mathcal{A}$ (respectively $\mathcal{A}'$) be an n-state (resp. n’-state) deterministic and connected unary automaton. Our main results can be summarized as follows:

• 1. The probability that $\mathcal{A}$ is minimal tends toward 1/2 when n tends toward infinity

• 2. The average complexity of $L{\text{(}}\mathcal{A}{\text{)}}$ is equivalent to n

• 3. The average complexity of $L{\text{(}}\mathcal{A}{\text{)}} \cap L{\text{(}}\mathcal{A}'{\text{)}}$ is equivalent to $\frac{{3\zeta (3)}} {{2\pi ^2 }}nn'$, where ζ is the Riemann “zeta”-function.

• 4. The average complexity of $L{\text{(}}\mathcal{A}{\text{)}}^ *$ is bounded by a constant

• 5. If n ≤ n’ ≤ P(n), for some polynomial P, the average complexity of $L{\text{(}}\mathcal{A}{\text{)}}L{\text{(}}\mathcal{A}'{\text{)}}$ is bounded by a constant (depending on P).

Remark that results 3, 4 and 5 differ perceptibly from the corresponding worst case complexities, which are nn’ for intersection, (n − 1)² + 1 for star and nn’ for concatenation product.