Abstract
The paper investigates the state complexity of two operations on regular languages, known as GF(2)-concatenation and GF(2)-inverse (Bakinova et al., “Formal languages over GF(2)”, LATA 2018), in the case of a one-symbol alphabet. The GF(2)-concatenation is a variant of the classical concatenation obtained by replacing Boolean logic in its definition with the GF(2) field; it is proved that GF(2)-concatenation of two unary languages recognized by an m-state and an n-state DFA is recognized by a DFA with 2mn states, and this number of states is necessary in the worst case, as long as m and n are relatively prime. This operation is known to have an inverse, and the state complexity of the GF(2)-inverse operation over a unary alphabet is proved to be exactly \(2^{n-1}+1\).
Supported by Russian Science Foundation, project 18-11-00100.
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References
Bakinova, E., Basharin, A., Batmanov, I., Lyubort, K., Okhotin, A., Sazhneva, E.: Formal languages over GF(2). In: Klein, S.T., Martín-Vide, C., Shapira, D. (eds.) LATA 2018. LNCS, vol. 10792, pp. 68–79. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-77313-1_5
Brzozowski, J.A.: Quotient complexity of regular languages. J. Autom. Lang. Comb. 15(1/2), 71–89 (2010). https://doi.org/10.25596/jalc-2010-071
Chrobak, M.: Finite automata and unary languages. Theor. Comput. Sci. 47(3), 149–158 (1986). https://doi.org/10.1016/0304-3975(86)90142-8
Daley, M., Domaratzki, M., Salomaa, K.: Orthogonal concatenation: language equations and state complexity. J. UCS 16(5), 653–675 (2010). https://doi.org/10.3217/jucs-016-05-0653
Geffert, V., Mereghetti, C., Pighizzini, G.: Converting two-way nondeterministic unary automata into simpler automata. Theor. Comput. Sci. 295, 189–203 (2003). https://doi.org/10.1016/S0304-3975(02)00403-6
Jirásková, G., Okhotin, A.: State complexity of unambiguous operations on deterministic finite automata. In: Konstantinidis, S., Pighizzini, G. (eds.) DCFS 2018. LNCS, vol. 10952, pp. 188–199. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-94631-3_16
Kunc, M., Okhotin, A.: Describing periodicity in two-way deterministic finite automata using transformation semigroups. In: Mauri, G., Leporati, A. (eds.) DLT 2011. LNCS, vol. 6795, pp. 324–336. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22321-1_28
Makarov, V., Okhotin, A.: On the expressive power of GF(2)-grammars. In: Catania, B., Královič, R., Nawrocki, J., Pighizzini, G. (eds.) SOFSEM 2019. LNCS, vol. 11376, pp. 310–323. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-10801-4_25
Maslov, A.N.: Estimates of the number of states of finite automata. Soviet Math. Doklady 11, 1373–1375 (1970)
Mereghetti, C., Pighizzini, G.: Optimal simulations between unary automata. SIAM J. Comput. 30(6), 1976–1992 (2001). https://doi.org/10.1137/S009753979935431X
Okhotin, A.: Unambiguous finite automata over a unary alphabet. Inf. Comput. 212, 15–36 (2012). https://doi.org/10.1016/j.ic.2012.01.003
Pighizzini, G., Shallit, J.: Unary language operations, state complexity and Jacobsthal’s function. Int. J. Found. Comput. Sci. 13(1), 145–159 (2002). https://doi.org/10.1142/S012905410200100X
Yu, S., Zhuang, Q., Salomaa, K.: The state complexities of some basic operations on regular languages. Theor. Comput. Sci. 125(2), 315–328 (1994). https://doi.org/10.1016/0304-3975(92)00011-F
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Okhotin, A., Sazhneva, E. (2019). State Complexity of GF(2)-Concatenation and GF(2)-Inverse on Unary Languages. In: Hospodár, M., Jirásková, G., Konstantinidis, S. (eds) Descriptional Complexity of Formal Systems. DCFS 2019. Lecture Notes in Computer Science(), vol 11612. Springer, Cham. https://doi.org/10.1007/978-3-030-23247-4_19
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DOI: https://doi.org/10.1007/978-3-030-23247-4_19
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