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Quotient Complexity of Closed Languages

  • Janusz Brzozowski
  • Galina Jirásková
  • Chenglong Zou
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6072)

Abstract

A language L is prefix-closed if, whenever a word w is in L, then every prefix of w is also in L. We define suffix-, factor-, and subword-closed languages in an analogous way, where by subword we mean subsequence. We study the quotient complexity (usually called state complexity) of operations on prefix-, suffix-, factor-, and subword-closed languages. We find tight upper bounds on the complexity of the subword-closure of arbitrary languages, and on the complexity of boolean operations, concatenation, star, and reversal in each of the four classes of closed languages. We show that repeated application of positive closure and complement to a closed language results in at most four distinct languages, while Kleene closure and complement gives at most eight.

Keywords

automaton closed factor language prefix quotient regular operation state complexity subword suffix upper bound 

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References

  1. 1.
    Ang, T., Brzozowski, J.: Languages convex with respect to binary relations, and their closure properties. Acta Cybernet 19, 445–464 (2009)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Avgustinovich, S.V., Frid, A.E.: A unique decomposition theorem for factorial languages. Internat. J. Algebra Comput. 15, 149–160 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bassino, F., Giambruno, L., Nicaud, C.: Complexity of operations on cofinite languages. In: López-Ortiz, A. (ed.) LATIN 2010. LNCS, vol. 6034, pp. 222–233. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  4. 4.
    Brzozowski, J.: Derivatives of regular expressions. J. ACM 11, 481–494 (1964)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Brzozowski, J.: Quotient complexity of regular languages. In: Dassow, J., Pighizzini, G., Truthe, B. (eds.) 11th International Workshop on Descriptional Complexity of Formal Systems, DCFS 2009, pp. 25–42. Otto-von-Guericke-Universität, Magdeburg (2009), http://arxiv.org/abs/0907.4547 Google Scholar
  6. 6.
    Brzozowski, J., Grant, E., Shallit, J.: Closures in formal languages and Kuratowski’s theorem. In: Diekert, V., Nowotka, D. (eds.) DLT 2009. LNCS, vol. 5583, pp. 125–144. Springer, Heidelberg (2009)Google Scholar
  7. 7.
    Brzozowski, J., Jirásková, G., Li, B.: Quotient complexity of ideal languages. In: López-Ortiz, A. (ed.) LATIN 2010. LNCS, vol. 6034, pp. 208–221. Springer, Heidelberg (2010) Full paper, http://arxiv.org/abs/0908.2083 CrossRefGoogle Scholar
  8. 8.
    Brzozowski, J., Santean, N.: Predictable semiautomata. Theoret. Comput. Sci. 410, 3236–3249 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Brzozowski, J., Shallit, J., Xu, Z.: Decision procedures for convex languages. In: Dediu, A., Ionescu, A., Martin-Vide, C. (eds.) LATA 2009. LNCS, vol. 5457, pp. 247–258. Springer, Heidelberg (2009)Google Scholar
  10. 10.
    Galil, Z., Simon, J.: A note on multiple-entry finite automata. J. Comput. System Sci. 12, 350–351 (1976)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Gill, A., Kou, L.T.: Multiple-entry finite automata. J. Comput. System Sci. 9, 1–19 (1974)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Haines, L.H.: On free monoids partially ordered by embedding. J. Combin. Theory 6, 94–98 (1969)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Han, Y.-S., Salomaa, K.: State complexity of basic operations on suffix-free regular languages. Theoret. Comput. Sci. 410, 2537–2548 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    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. University of Szeged, Hungary (2009)Google Scholar
  15. 15.
    Holzer, M., Salomaa, K., Yu, S.: On the state complexity of k-entry deterministic finite automata. J. Autom. Lang. Comb. 6, 453–466 (2001)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Jirásek, J., Jirásková, G., Szabari, A.: State complexity of concatenation and complementation. Internat. J. Found. Comput. Sci. 16, 511–529 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Kao, J.-Y., Rampersad, N., Shallit, J.: On NFAs where all states are final, initial, or both. Theoret. Comput. Sci. 410, 5010–5021 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Kuratowski, C.: Sur l’opération \(\overline{A}\) de l’analysis situs. Fund. Math. 3, 182–199 (1922)zbMATHGoogle Scholar
  19. 19.
    de Luca, A., Varricchio, S.: Some combinatorial properties of factorial languages. In: Capocelli, R. (ed.) Sequences, pp. 258–266. Springer, Heidelberg (1990)Google Scholar
  20. 20.
    Maslov, A.N.: Estimates of the number of states of finite automata. Dokl. Akad. Nauk SSSR 194, 1266–1268 (1970) (in Russian); English translation: Soviet Math. Dokl. 11, 1373–1375 (1970)Google Scholar
  21. 21.
    Mirkin, B.G.: On dual automata. Kibernetika (Kiev) 2, 7–10 (1966) (in Russian); English translation: Cybernetics 2, 6–9 (1966)Google Scholar
  22. 22.
    Okhotin, A.: On the state complexity of scattered substrings and superstrings. Turku Centre for Computer Science Technical Report No. 849 (2007)Google Scholar
  23. 23.
    Pighizzini, G., Shallit, J.: Unary language operations, state complexity and Jacobsthal’s function. Internat. J. Found. Comput. Sci. 13, 145–159 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Salomaa, A., Wood, D., Yu, S.: On the state complexity of reversals of regular languages. Theoret. Comput. Sci. 320, 315–329 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Thierrin, G.: Convex languages. In: Nivat, M. (ed.) Automata, Languages and Programming, pp. 481–492. North-Holland, Amsterdam (1973)Google Scholar
  26. 26.
    Veloso, P.A.S., Gill, A.: Some remarks on multiple-entry finite automata. J. Comput. System Sci. 18, 304–306 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Yu., S.: State complexity of regular languages. J. Autom. Lang. Comb. 6, 221–234 (2001)MathSciNetGoogle Scholar
  28. 28.
    Yu, S., Zhuang, Q., Salomaa, K.: The state complexities of some basic operations on regular languages. Theoret. Comput. Sci. 125, 315–328 (1994)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Janusz Brzozowski
    • 1
  • Galina Jirásková
    • 2
  • Chenglong Zou
    • 1
  1. 1.David R. Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.Slovak Academy of SciencesMathematical InstituteKošiceSlovakia

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