Average State Complexity of Operations on Unary Automata (1999)
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@MISC{Nicaud99averagestate,
author = {Cyril Nicaud},
title = {Average State Complexity of Operations on Unary Automata},
year = {1999}
}
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Abstract
. Define the complexity of a rational language as the number of states of its minimal automaton. Let A (respectively A 0 ) be an n-state (resp. n 0 -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 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 0 ) is equivalent to 3i(3) 2 2 nn 0 , where i is the Riemann "zeta"-function. 4. The average complexity of L(A) is bounded by a constant, 5. If n n 0 P (n), for some polynomial P , the average complexity of L(A)L(A 0 ) 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 0 for intersection, (n \Gamma 1) 2 + 1 for star and nn 0 for concatenation product. 1 Introduction This paper addresses a rather natural problem: find the averag...
