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 A (respectively 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 A is minimal tends toward 1/2 whenn 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 3ζ(3)2π2nn′, where ζ is the Riemann “zeta”-function. 4. The average complexity of L(A)∗ is bounded by a constant 5. 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.