# 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 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 (3ζ(3)/2π²)nn', where ζ is the Riemann "zeta"-function. 4. The average complexity of L(A) is bounded by a constant, 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.
Document type :
Conference papers

https://hal-upec-upem.archives-ouvertes.fr/hal-00620109
Contributor : Cyril Nicaud <>
Submitted on : Wednesday, September 7, 2011 - 1:24:15 PM
Last modification on : Saturday, March 28, 2020 - 2:15:04 AM

### Citation

Cyril Nicaud. Average State Complexity of Operations on Unary Automata. MFCS 1999, Sep 1999, Szklarska Poreba, Poland, Poland. pp.231-240, ⟨10.1007/3-540-48340-3_21⟩. ⟨hal-00620109⟩

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