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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
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)