State Complexity of Unambiguous Operations on Deterministic Finite Automata

  • Galina Jirásková
  • Alexander Okhotin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10952)


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.


  1. 1.
    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). Scholar
  2. 2.
    Brzozowski, J.A., Szykuła, M.: Complexity of suffix-free regular languages. J. Comput. Syst. Sci. 89, 270–287 (2017). Scholar
  3. 3.
    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). Scholar
  4. 4.
    Daley, M., Domaratzki, M., Salomaa, K.: Orthogonal concatenation: language equations and state complexity. J. Univers. Comput. Sci. 16(5), 653–675 (2010). Scholar
  5. 5.
    Han, Y.-S., Salomaa, K.: State complexity of basic operations on suffix-free regular languages. Theoret. Comput. Sci. 410, 2537–2548 (2009). Scholar
  6. 6.
    Han, Y.-S., Salomaa, K.: Nondeterministic state complexity for suffix-free regular languages. In: DCFS 2010, EPTCS, vol. 31, pp. 189–196 (2010). Scholar
  7. 7.
    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)Google Scholar
  8. 8.
    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). Scholar
  9. 9.
    Holzer, M., Kutrib, M.: Nondeterministic descriptional complexity of regular languages. Int. J. Found. Comput. Sci. 14, 1087–1102 (2003). Scholar
  10. 10.
    Jirásek, J., Jirásková, G., Szabari, A.: State complexity of concatenation and complementation. Int. J. Found. Comput. Sci. 16(3), 511–529 (2005). Scholar
  11. 11.
    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). Scholar
  12. 12.
    Jirásková, G., Krausová, M.: Complexity in prefix-free regular languages. In: DCFS 2010, EPTCS, vol. 31, pp. 197–204. Scholar
  13. 13.
    Jirásková, G., Olejár, P.: State complexity of intersection and union of suffix-free languages and descriptional complexity. In: NCMA 2009,, vol. 256, 151–166 (2009)Google Scholar
  14. 14.
    Jirásková, G., Okhotin, A.: On the state complexity of operations on two-way finite automata. Inf. Comput. 253(1), 36–63 (2017). Scholar
  15. 15.
    Kunc, M., Okhotin, A.: State complexity of union and intersection for two-way nondeterministic finite automata. Fundamenta Informaticae 110(1–4), 231–239 (2011). Scholar
  16. 16.
    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). Scholar
  17. 17.
    Maslov, A.N.: Estimates of the number of states of finite automata. Soviet Math. Dokl. 11, 1373–1375 (1970)MATHGoogle Scholar
  18. 18.
    Okhotin, A.: Unambiguous finite automata over a unary alphabet. Inf. Comput. 212, 15–36 (2012). Scholar
  19. 19.
    Pighizzini, G., Shallit, J.: Unary language operations, state complexity and Jacobsthal’s function. Int. J. Found. Comput. Sci. 13(1), 145–159 (2002). Scholar
  20. 20.
    Rampersad, N., Ravikumar, B., Santean, N., Shallit, J.: State complexity of unique rational operations. Theoret. Comput. Sci. 410, 2431–2441 (2009). Scholar
  21. 21.
    Yu, S., Zhuang, Q., Salomaa, K.: The state complexity of some basic operations on regular languages. Theoret. Comput. Sci. 125, 315–328 (1994). Scholar

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Authors and Affiliations

  1. 1.Mathematical InstituteSlovak Academy of SciencesKošiceSlovak Republic
  2. 2.St. Petersburg State UniversitySaint PetersburgRussia

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