Lower bounds for Las Vegas automata by information theory
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 37 (2003) no. 1, pp. 39-49.

We show that the size of a Las Vegas automaton and the size of a complete, minimal deterministic automaton accepting a regular language are polynomially related. More precisely, we show that if a regular language $L$ is accepted by a Las Vegas automaton having $r$ states such that the probability for a definite answer to occur is at least $p$, then $r\ge {n}^{p}$, where $n$ is the number of the states of the minimal deterministic automaton accepting $L$. Earlier this result has been obtained in [2] by using a reduction to one-way Las Vegas communication protocols, but here we give a direct proof based on information theory.

DOI : https://doi.org/10.1051/ita:2003007
Classification : 68Q19,  68Q10,  94A15
Mots clés : Las Vegas automata, information theory
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Hirvensalo, Mika; Seibert, Sebastian. Lower bounds for Las Vegas automata by information theory. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 37 (2003) no. 1, pp. 39-49. doi : 10.1051/ita:2003007. http://www.numdam.org/articles/10.1051/ita:2003007/

[1] T.M. Cover and J.A. Thomas, Elements of Information Theory. John Wiley & Sons, Inc. (1991). | MR 1122806 | Zbl 0762.94001

[2] P. Ďuris, J. Hromkovič, J.D.P. Rolim and G. Schnitger, Las Vegas Versus Determinism for One-way Communication Complexity, Finite Automata, and Polynomial-time Computations. Springer, Lecture Notes in Comput. Sci. 1200 (1997) 117-128. | MR 1473768

[3] J. Hromkovič, personal communication.

[4] H. Klauck, On quantum and probabilistic communication: Las Vegas and one-way protocols, in Proc. of the ACM Symposium on Theory of Computing (2000) 644-651. | MR 2115303

[5] C.H. Papadimitriou, Computational Complexity. Addison-Wesley (1994). | MR 1251285 | Zbl 0833.68049

[6] S. Yu, Regular Languages, edited by G. Rozenberg and A. Salomaa. Springer, Handb. Formal Languages I (1997).

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