Abstract
Austinat, Diekert, Hertrampf, and Petersen [2] 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. [2] 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!.
The research was supported by Grant No. 09.1570 from the Latvian Council of Science and the Invitation Fellowship for Research in Japan S12052 by the Japan Society for the Promotion of Science.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
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)
Austinat, H., Diekert, V., Hertrampf, U., Petersen, H.: Regular frequency computations. Theoretical Computer Science 330(1), 15–21 (2005)
Beigel, R., Gasarch, W., Kinber, E.: Frequency computation and bounded queries. Theoretical Computer Science 163(1-2), 177–192 (1996)
Case, J., Kaufmann, S., Kinber, E.B., Kummer, M.: Learning recursive functions from approximations. J. Comput. Syst. Sci. 55(1), 183–196 (1997)
Degtev, A.N.: On (m,n)-computable sets. In: Moldavanskij, D.I. (ed.) Algebraic Systems, pp. 88–99. Ivanovo Gos. Universitet (1981) (in Russian)
Freivalds, R.: Recognition of languages with high probability of different classes of automata. Doklady Akademii Nauk SSSR 239(1), 60–62 (1978) (in Russian)
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)
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)
Freivalds, R.: Non-constructive methods for finite probabilistic automata. International Journal of Foundations of Computer Science 19(3), 565–580 (2008)
Harizanov, V., Kummer, M., Owings, J.: Frequency computations and the cardinality theorem. The Journal of Symbolic Logic 57(2) (1992)
Hinrichs, M., Wechsung, G.: Time bounded frequency computations. Information and Computation 139(2), 234–257 (1997)
Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages and Computation. Addison–Wesley Publishing Company, Reading (1979)
Kinber, E.B.: Frequency computations in finite automata. Cybernetics and Systems Analysis 12(2), 179–187 (1976)
Kinber, E.B.: Frequency calculations of general recursive predicates and frequency enumerations of sets. Soviet Mathematics Doklady 13, 873–876 (1972)
Kummer, M.: A proof of Beigel’s cardinality conjecture. The Journal of Symbolic Logic 57(2), 677–681 (1992)
McNaughton, R.: The theory of automata, a survey. Advances in Computers 2, 379–421 (1961)
Rabin, M.O., Scott, D.: Finite automata and their decision problems. IBM Journal of Research and Development 3(2), 114–125 (1959)
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 papers
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)
Trakhtenbrot, B.A.: On the frequency computation of functions. Algebra i Logika 2(1), 25–32 (1964) (in Russian)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Freivalds, R., Zeugmann, T., Pogosyan, G.R. (2013). On the Size Complexity of Deterministic Frequency Automata. In: Dediu, AH., Martín-Vide, C., Truthe, B. (eds) Language and Automata Theory and Applications. LATA 2013. Lecture Notes in Computer Science, vol 7810. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37064-9_26
Download citation
DOI: https://doi.org/10.1007/978-3-642-37064-9_26
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-37063-2
Online ISBN: 978-3-642-37064-9
eBook Packages: Computer ScienceComputer Science (R0)