Abstract
We present binary deterministic finite automata of n states that meet the upper bound 2n on the state complexity of reversal. The automata have a single final state and are one-cycle-free-path, thus the witness languages are deterministic union-free. This result allows us to describe a binary language such that the nondeterministic state complexity of the language and of its complement is n and n + 1, respectively, while the state complexity of the language is 2n. We also show that there is no regular language with state complexity 2n such that both the language and its complement have nondeterministic state complexity n.
Research supported by VEGA grant 2/0183/11.
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References
Brzozowski, J.A.: Canonical Regular Expressions and Minimal State Graphs for Definite Events. In: Proceedings of the Symposium on Mathematical Theory of Automata, New York, NY, April 24-26 (1962); Fox, J. (ed.) MRI Symposia Series, vol. 12, pp. 529–561. Polytechnic Press of the Polytechnic Institute of Brooklyn, Brooklyn, NY (1963)
Brzozowski, J., Jirásková, G., Li, B.: Quotient complexity of ideal languages. In: López-Ortiz, A. (ed.) LATIN 2010. LNCS, vol. 6034, pp. 208–221. Springer, Heidelberg (2010)
Brzozowski, J., Jirásková, G., Zou, C.: Quotient complexity of closed languages. In: Ablayev, F., Mayr, E.W. (eds.) CSR 2010. LNCS, vol. 6072, pp. 84–95. Springer, Heidelberg (2010)
Champarnaud, J.-M., Khorsi, A., Paranthoën, T.: Split and join for minimizing: Brzozowski’s algorithm, http://jmc.feydakins.org/ps/c09psc02.ps
Jirásková, G., Masopust, T.: Complexity in union-free regular languages. In: Gao, Y., Lu, H., Seki, S., Yu, S. (eds.) DLT 2010. LNCS, vol. 6224, pp. 255–266. Springer, Heidelberg (2010)
Leiss, E.: Succinct representation of regular languages by Boolean automata. Theoret. Comput. Sci. 13, 323–330 (1981)
Lupanov, U.I.: A comparison of two types of finite automata. Problemy Kibernetiki 9, 321–326 (1963) (in Russian)
Mera, F., Pighizzini, G.: Complementing unary nondeterministic automata. Theor. Comput. Sci. 330, 349–360 (2005)
Mirkin, B.G.: On dual automata. Kibernetika (Kiev) 2, 7–10 (1966) (in Russian); English translation: Cybernetics 2, 6–9 (1966)
Rabin, M., Scott, D.: Finite automata and their decision problems. IBM Res. Develop. 3, 114–129 (1959)
Salomaa, A., Wood, D., Yu, S.: On the state complexity of reversals of regular languages. Theoret. Comput. Sci. 320, 315–329 (2004)
Sipser, M.: Introduction to the theory of computation. PWS Publishing Company, Boston (1997)
Šebej, J.: Reversal of regular languages and state complexity. In: Pardubská, D. (ed.) Proc. 10th ITAT, pp. 47–54. Šafárik University, Košice (2010)
Yu, S.: Chapter 2: Regular languages. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. I, pp. 41–110. Springer, Heidelberg (1997)
Yu, S., Zhuang, Q., Salomaa, K.: The state complexity of some basic operations on regular languages. Theoret. Comput. Sci. 125, 315–328 (1994)
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Jirásková, G., Šebej, J. (2011). Note on Reversal of Binary Regular Languages. In: Holzer, M., Kutrib, M., Pighizzini, G. (eds) Descriptional Complexity of Formal Systems. DCFS 2011. Lecture Notes in Computer Science, vol 6808. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22600-7_17
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DOI: https://doi.org/10.1007/978-3-642-22600-7_17
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