Abstract
We investigate the state complexity of the square operation on languages represented by deterministic, alternating, and Boolean automata. For each k such that \(1 \le k \le n-2\), we describe a binary language accepted by an n-state DFA with k final states meeting the upper bound \(n2^n - k2^{n-1}\) on the state complexity of its square. We show that in the case of \(k=n-1\), the corresponding upper bound cannot be met. Using the DFA witness for square with \(2^n\) states where half of them are final, we get the tight upper bounds on the complexity of the square operation on alternating and Boolean automata.
Research supported by grant VEGA 2/0084/15 and grant APVV-15-0091. This work was conducted as a part of PhD study of the first author at Comenius University in Bratislava.
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References
Brzozowski, J.A., Leiss, E.L.: On equations for regular languages, finite automata, and sequential networks. Theor. Comput. Sci. 10, 19–35 (1980). http://dx.doi.org/10.1016/0304-3975(80)90069-9
Čevorová, K., Jirásková, G., Krajňáková, I.: On the square of regular languages. In: Holzer, M., Kutrib, M. (eds.) CIAA 2014. LNCS, vol. 8587, pp. 136–147. Springer, Cham (2014). http://dx.doi.org/10.1007/978-3-319-08846-4_10
Fellah, A., Jürgensen, H., Yu, S.: Constructions for alternating finite automata. Int. J. Comput. Math. 35(1–4), 117–132 (1990). http://dx.doi.org/10.1080/00207169008803893
Hospodár, M., Jirásková, G.: Concatenation on deterministic and alternating automata. In: Bordihn, H., Freund, R., Nagy, B., Vaszil, G. (eds.) NCMA 2016, vol. 321, pp. 179–194. Österreichische Computer Gesellschaft, books@ocg.at (2016)
Jirásková, G.: Descriptional complexity of operations on alternating and boolean automata. In: Hirsch, E.A., Karhumäki, J., Lepistö, A., Prilutskii, M. (eds.) CSR 2012. LNCS, vol. 7353, pp. 196–204. Springer, Heidelberg (2012). doi:10.1007/978-3-642-30642-6_19
Leiss, E.L.: Succint representation of regular languages by boolean automata. Theor. Comput. Sci. 13, 323–330 (1981)
Maslov, A.N.: Estimates of the number of states of finite automata. Soviet Math. Doklady 11, 1373–1375 (1970)
Rampersad, N.: The state complexity of \({L}^2\) and \({L}^k\). Inf. Process. Lett. 98(6), 231–234 (2006). http://dx.doi.org/10.1016/j.ipl.2005.06.011
Sipser, M.: Introduction to the Theory of Computation. Cengage Learning, Boston (2012)
Yu, S., Zhuang, Q., Salomaa, K.: The state complexities of some basic operations on regular languages. Theor. Comput. Sci. 125(2), 315–328 (1994). http://dx.doi.org/10.1016/0304-3975(92)00011-F
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Krajňáková, I., Jirásková, G. (2017). Square on Deterministic, Alternating, and Boolean Finite Automata. In: Pighizzini, G., Câmpeanu, C. (eds) Descriptional Complexity of Formal Systems. DCFS 2017. Lecture Notes in Computer Science(), vol 10316. Springer, Cham. https://doi.org/10.1007/978-3-319-60252-3_17
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DOI: https://doi.org/10.1007/978-3-319-60252-3_17
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