Average State Complexity of Operations on Unary Automata

Define the complexity of a regular language as the number of states of its minimal automaton. Let A (respectively A') be an n-state (resp. n'-state) deterministic and connected uneiry automaton. Our main results can be summarized ais follows: 1. The probability that A is minimal tends toweird 1/2 when n tends toward infinity, 2. The average complexity of L(A) is equivalent to n, 3. The average complexity of L(A) ∩ L(A') is equivalent to 3c(3)/2π2nn', where c is the Riemann "zeta"-function. 4. The average complexity of L(A)* is bounded by a constrant, 5. If n ≤ n' ≤ P(n), for some polynomial P, the average complexity of L(A)L(A') 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.