Austinat, Diekert, Hertrampf, and Petersen  proved that every language L that is (m,n)-recognizable by a deterministic frequency automaton such that m > n/2 can be recognized by a deterministic finite automaton as well. First, the size of deterministic frequency automata and of deterministic finite automata recognizing the same language is compared. Then approximations of a language are considered, where a language L′ is called an approximation of a language L if L′ differs from L in only a finite number of strings. We prove that if a deterministic frequency automaton has k states and (m,n)-recognizes a language L, where m > n/2, then there is a language L′ approximating L such that L′ can be recognized by a deterministic finite automaton with no more than k states.
Austinat et al.  also proved that every language L over a single-letter alphabet that is (1,n)-recognizable by a deterministic frequency automaton can be recognized by a deterministic finite automaton. For languages over a single-letter alphabet we show that if a deterministic frequency automaton has k states and (1,n)-recognizes a language L then there is a language L′ approximating L such that L′ can be recognized by a deterministic finite automaton with no more that k states. However, there are approximations such that our bound is much higher, i.e., k!.
Frequency Computation Regular Language Input String Probabilistic Algorithm Chinese Remainder Theorem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.
Ablaev, F.M., Freivalds, R.: Why Sometimes Probabilistic Algorithms Can Be More Effective. In: Wiedermann, J., Gruska, J., Rovan, B. (eds.) MFCS 1986. LNCS, vol. 233, pp. 1–14. Springer, Heidelberg (1986)CrossRefGoogle Scholar
Degtev, A.N.: On (m,n)-computable sets. In: Moldavanskij, D.I. (ed.) Algebraic Systems, pp. 88–99. Ivanovo Gos. Universitet (1981) (in Russian)Google Scholar
Freivalds, R.: Recognition of languages with high probability of different classes of automata. Doklady Akademii Nauk SSSR 239(1), 60–62 (1978) (in Russian)MathSciNetGoogle Scholar
Freivalds, R.: On the growth of the number of states in result of the determinization of probabilistic finite automata. Avtomatika i Vychislitel’naya Tekhnika 3, 39–42 (1982) (in Russian)Google Scholar
Freivalds, R.: Complexity of Probabilistic Versus Deterministic Automata. In: Bārzdiņš, J., Bjørner, D. (eds.) Baltic Computer Science. LNCS, vol. 502, pp. 565–613. Springer, Heidelberg (1991)CrossRefGoogle Scholar
Freivalds, R.: Non-constructive methods for finite probabilistic automata. International Journal of Foundations of Computer Science 19(3), 565–580 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
Harizanov, V., Kummer, M., Owings, J.: Frequency computations and the cardinality theorem. The Journal of Symbolic Logic 57(2) (1992)Google Scholar
Rabin, M.O., Scott, D.: Finite automata and their decision problems. IBM Journal of Research and Development 3(2), 114–125 (1959)MathSciNetCrossRefGoogle Scholar
Rose, G.F.: An extended notion of computability. In: International Congress for Logic, Methodology and Philosophy of Science, Stanford University, Stanford, California, August 24-September 2 (1960); Abstracts of contributed papersGoogle Scholar
Tantau, T.: Towards a Cardinality Theorem for Finite Automata. In: Diks, K., Rytter, W. (eds.) MFCS 2002. LNCS, vol. 2420, pp. 625–636. Springer, Heidelberg (2002)CrossRefGoogle Scholar
Trakhtenbrot, B.A.: On the frequency computation of functions. Algebra i Logika 2(1), 25–32 (1964) (in Russian)Google Scholar