Abstract
The paper determines the number of states in a deterministic finite automaton (DFA) necessary to represent “unambiguous” variants of the union, concatenation, and Kleene star operations on formal languages. For the disjoint union of languages represented by an m-state and an n-state DFA, the state complexity is \(mn-1\); for the unambiguous concatenation, it is known to be \(m2^{n-1} - 2^{n-2}\) (Daley et al. “Orthogonal concatenation: Language equations and state complexity”, J. UCS, 2010), and this paper shows that this number of states is necessary already over a binary alphabet; for the unambiguous star, the state complexity function is determined to be \(\frac{3}{8}2^n+1\). In the case of a unary alphabet, disjoint union requires up to \(\frac{1}{2}mn\) states, unambiguous concatenation has state complexity \(m+n-2\), and unambiguous star requires \(n-2\) states in the worst case.
G. Jirásková—Research supported by VEGA grant 2/0084/15 and grant APVV-15-0091.
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
Similar content being viewed by others
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., Szykuła, M.: Complexity of suffix-free regular languages. J. Comput. Syst. Sci. 89, 270–287 (2017). https://doi.org/10.1016/j.jcss.2017.05.011
Cmorik, R., Jirásková, G.: Basic operations on binary suffix-free languages. In: Kotásek, Z., Bouda, J., Černá, I., Sekanina, L., Vojnar, T., Antoš, D. (eds.) MEMICS 2011. LNCS, vol. 7119, pp. 94–102. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-25929-6_9
Daley, M., Domaratzki, M., Salomaa, K.: Orthogonal concatenation: language equations and state complexity. J. Univers. Comput. Sci. 16(5), 653–675 (2010). https://doi.org/10.3217/jucs-016-05-0653
Han, Y.-S., Salomaa, K.: State complexity of basic operations on suffix-free regular languages. Theoret. Comput. Sci. 410, 2537–2548 (2009). https://doi.org/10.1016/j.tcs.2008.12.054
Han, Y.-S., Salomaa, K.: Nondeterministic state complexity for suffix-free regular languages. In: DCFS 2010, EPTCS, vol. 31, pp. 189–196 (2010). https://doi.org/10.4204/EPTCS.31.21
Han, Y.-S., Salomaa, K., Wood, D.: Operational state complexity of prefix-free regular languages. In: Automata, Formal Languages, and Related Topics, pp. 99–115 (2009)
Han, Y.-S., Salomaa, K., Wood, D.: Nondeterministic state complexity of basic operations for prefix-free regular languages. Fundamenta Informaticae 90(1–2), 93–106 (2009). https://doi.org/10.3233/FI-2009-0008
Holzer, M., Kutrib, M.: Nondeterministic descriptional complexity of regular languages. Int. J. Found. Comput. Sci. 14, 1087–1102 (2003). https://doi.org/10.1142/S0129054103002199
Jirásek, J., Jirásková, G., Szabari, A.: State complexity of concatenation and complementation. Int. J. Found. Comput. Sci. 16(3), 511–529 (2005). https://doi.org/10.1142/S0129054105003133
Jirásek, J., Jirásková, G., Šebej, J.: Operations on unambiguous finite automata. In: Brlek, S., Reutenauer, C. (eds.) DLT 2016. LNCS, vol. 9840, pp. 243–255. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53132-7_20
Jirásková, G., Krausová, M.: Complexity in prefix-free regular languages. In: DCFS 2010, EPTCS, vol. 31, pp. 197–204. https://doi.org/10.4204/EPTCS.31.22
Jirásková, G., Olejár, P.: State complexity of intersection and union of suffix-free languages and descriptional complexity. In: NCMA 2009, books@ocg.at, vol. 256, 151–166 (2009)
Jirásková, G., Okhotin, A.: On the state complexity of operations on two-way finite automata. Inf. Comput. 253(1), 36–63 (2017). https://doi.org/10.1016/j.ic.2016.12.007
Kunc, M., Okhotin, A.: State complexity of union and intersection for two-way nondeterministic finite automata. Fundamenta Informaticae 110(1–4), 231–239 (2011). https://doi.org/10.3233/FI-2011-540
Kunc, M., Okhotin, A.: State complexity of operations on two-way deterministic finite automata over a unary alphabet. Theoret. Comput. Sci. 449, 106–118 (2012). https://doi.org/10.1016/j.tcs.2012.04.010
Maslov, A.N.: Estimates of the number of states of finite automata. Soviet Math. Dokl. 11, 1373–1375 (1970)
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
Rampersad, N., Ravikumar, B., Santean, N., Shallit, J.: State complexity of unique rational operations. Theoret. Comput. Sci. 410, 2431–2441 (2009). https://doi.org/10.1016/j.tcs.2009.02.035
Yu, S., Zhuang, Q., Salomaa, K.: The state complexity of some basic operations on regular languages. Theoret. Comput. Sci. 125, 315–328 (1994). https://doi.org/10.1016/0304-3975(92)00011-F
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 IFIP International Federation for Information Processing
About this paper
Cite this paper
Jirásková, G., Okhotin, A. (2018). State Complexity of Unambiguous Operations on Deterministic Finite Automata. In: Konstantinidis, S., Pighizzini, G. (eds) Descriptional Complexity of Formal Systems. DCFS 2018. Lecture Notes in Computer Science(), vol 10952. Springer, Cham. https://doi.org/10.1007/978-3-319-94631-3_16
Download citation
DOI: https://doi.org/10.1007/978-3-319-94631-3_16
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-94630-6
Online ISBN: 978-3-319-94631-3
eBook Packages: Computer ScienceComputer Science (R0)
