Abstract
Size (the number of states) of finite probabilistic automata with an isolated cut-point can be exponentially smaller than the size of any equivalent finite deterministic automaton. The result is presented in two versions. The first version depends on Artin’s Conjecture (1927) in Number Theory. The second version does not depend on conjectures but the numerical estimates are worse. In both versions the method of the proof does not allow an explicit description of the languages used. Since our finite probabilistic automata are reversible, these results imply a similar result for quantum finite automata.
Research supported by Grant No.05.1528 from the Latvian Council of Science and European Commission, contract IST-1999-11234.
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References
Ambainis, A.: The complexity of probabilistic versus deterministic finite automata. In: Nagamochi, H., Suri, S., Igarashi, Y., Miyano, S., Asano, T. (eds.) ISAAC 1996. LNCS, vol. 1178, pp. 233–237. Springer, Heidelberg (1996)
Ambainis, A., Freivalds, R.: 1-way quantum finite automata: strengths, weaknesses and generalizations. In: Proc. IEEE FOCS 1998, pp. 332–341 (1998)
Artin, E.: Beweis des allgemeinen Reziprozitätsgesetzes. Mat. Sem. Univ. Hamburg B.5, 353–363 (1927)
Aschbacher, M.: Finite Group Theory (Cambridge Studies in Advanced Mathematics), 2nd edn. Cambridge University Press, Cambridge (2000)
Freivalds, R.: On the growth of the number of states in result of the determinization of probabilistic finite automata. Avtomatika i Vichislitel’naya Tekhnika (Russian) (3), 39–42 (1982)
Gabbasov, N.Z., Murtazina, T.A.: Improving the estimate of Rabin’s reduction theorem. Algorithms and Automata, Kazan University, pp. 7–10 (Russian) ( 1979)
Garret, P.: The Mathematics of Coding Theory. Pearson Prentice Hall, Upper Saddle River (2004)
Golovkins, M., Kravtsev, M.: Probabilistic Reversible Automata and Quantum Automata. In: Ibarra, O.H., Zhang, L. (eds.) COCOON 2002. LNCS, vol. 2387, pp. 574–583. Springer, Heidelberg (2002)
Hooley, C.: On Artin’s conjecture. J.ReineAngew.Math. 225, 220–229 (1967)
Heath-Brown, D.R.: Artin’s conjecture for primitive roots. Quart. J. Math. Oxford 37, 27–38 (1986)
Kolmogorov, A.N.: Three approaches to the quantitative definition of information. Problems in Information Transmission 1, 1–7 (1965)
Kondacs, A., Watrous, J.: On the power of quantum finite state automata. Proc. IEEE FOCS 1997, pp. 66–75 (1997)
Paz, A.: Some aspects of probabilistic automata. Information and Control 9(1), 26–60 (1966)
Rabin, M.O.: Probabilistic Automata. Information and Control 6(3), 230–245 (1963)
Spencer, J.: Nonconstructive methods in discrete mathematics. In: Rota, G.-C. (ed.) Studies in Combinatorics (MAA Studies in Mathematics), vol. 17, pp. 142–178. (1978)
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Freivalds, R. (2007). Non-constructive Methods for Finite Probabilistic Automata. In: Harju, T., Karhumäki, J., Lepistö, A. (eds) Developments in Language Theory. DLT 2007. Lecture Notes in Computer Science, vol 4588. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73208-2_18
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