Complexity of Left-Ideal, Suffix-Closed and Suffix-Free Regular Languages

  • Janusz A. Brzozowski
  • Corwin SinnamonEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10168)


A language L over an alphabet \(\varSigma \) is suffix-convex if, for any words \(x,y,z\in \varSigma ^*\), whenever z and xyz are in L, then so is yz. Suffix-convex languages include three special cases: left-ideal, suffix-closed, and suffix-free languages. We examine complexity properties of these three special classes of suffix-convex regular languages. In particular, we study the quotient/state complexity of boolean operations, product (concatenation), star, and reversal on these languages, as well as the size of their syntactic semigroups, and the quotient complexity of their atoms.


Different alphabets Left ideal Most complex Quotient/state complexity Regular language Suffix-closed Suffix-convex Suffix-free Syntactic semigroup Transition semigroup Unrestricted complexity 


  1. 1.
    Ang, T., Brzozowski, J.A.: Languages convex with respect to binary relations, and their closure properties. Acta Cybern. 19(2), 445–464 (2009)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Berstel, J., Perrin, D., Reutenauer, C.: Codes and Automata (Encyclopedia of Mathematics and its Applications). Cambridge University Press, Cambridge (2010)zbMATHGoogle Scholar
  3. 3.
    Brzozowski, J.A.: Quotient complexity of regular languages. J. Autom. Lang. Comb. 15(1/2), 71–89 (2010)zbMATHGoogle Scholar
  4. 4.
    Brzozowski, J.A.: In search of the most complex regular languages. Int. J. Found. Comput. Sci. 24(6), 691–708 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brzozowski, J.: Unrestricted state complexity of binary operations on regular languages. In: Câmpeanu, C., Manea, F., Shallit, J. (eds.) DCFS 2016. LNCS, vol. 9777, pp. 60–72. Springer, Heidelberg (2016). doi: 10.1007/978-3-319-41114-9_5 CrossRefGoogle Scholar
  6. 6.
    Brzozowski, J.A., Davies, S.: Quotient complexities of atoms in regular ideal languages. Acta Cybern. 22(2), 293–311 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Brzozowski, J.A., Davies, S., Liu, B.Y.V.: Most complex regular ideal languages. Discrete Math. Theoret. Comput. Sci. 18(3) (2016). Paper #5Google Scholar
  8. 8.
    Brzozowski, J.A., Jirásková, G., Li, B.: Quotient complexity of ideal languages. Theoret. Comput. Sci. 470, 36–52 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Brzozowski, J.A., Jirásková, G., Zou, C.: Quotient complexity of closed languages. Theory Comput. Syst. 54, 277–292 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Brzozowski, J.A., Li, B., Ye, Y.: Syntactic complexity of prefix-, suffix-, bifix-, and factor-free regular languages. Theoret. Comput. Sci. 449, 37–53 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Brzozowski, J.A., Sinnamon, C.: Complexity of left-ideal, suffix-closed, and suffix-free regular languages (2016).
  12. 12.
    Brzozowski, J.A., Sinnamon, C.: Unrestricted state complexity of binary operations on regular and ideal languages (2016).
  13. 13.
    Brzozowski, J., Szykuła, M.: Upper bounds on syntactic complexity of left and two-sided ideals. In: Shur, A.M., Volkov, M.V. (eds.) DLT 2014. LNCS, vol. 8633, pp. 13–24. Springer, Heidelberg (2014). doi: 10.1007/978-3-319-09698-8_2 Google Scholar
  14. 14.
    Brzozowski, J., Szykuła, M.: Complexity of suffix-free regular languages. In: Kosowski, A., Walukiewicz, I. (eds.) FCT 2015. LNCS, vol. 9210, pp. 146–159. Springer, Heidelberg (2015). doi: 10.1007/978-3-319-22177-9_12. Full paper at
  15. 15.
    Brzozowski, J.A., Szykuła, M., Ye, Y.: Syntactic complexity of regular ideals, (September 2015).
  16. 16.
    Brzozowski, J.A., Tamm, H.: Quotient complexities of atoms of regular languages. Int. J. Found. Comput. Sci. 24(7), 1009–1027 (2013)CrossRefzbMATHGoogle Scholar
  17. 17.
    Brzozowski, J.A., Tamm, H.: Theory of átomata. Theoret. Comput. Sci. 539, 13–27 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Brzozowski, J., Ye, Y.: Syntactic complexity of ideal and closed languages. In: Mauri, G., Leporati, A. (eds.) DLT 2011. LNCS, vol. 6795, pp. 117–128. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-22321-1_11 CrossRefGoogle Scholar
  19. 19.
    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). doi: 10.1007/978-3-642-25929-6_9 CrossRefGoogle Scholar
  20. 20.
    Han, Y.S., Salomaa, K.: State complexity of basic operations on suffix-free regular languages. Theoret. Comput. Sci. 410(27–29), 2537–2548 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Holzer, M., König, B.: On deterministic finite automata and syntactic monoid size. Theoret. Comput. Sci. 327(3), 319–347 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Iván, S.: Complexity of atoms, combinatorially. Inform. Process. Lett. 116(5), 356–360 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Jirásková, G., Olejár, P.: State complexity of union and intersection of binary suffix-free languages. In: Bordihn, H., et al. (eds.) NMCA, pp. 151–166. Austrian Computer Society (2009)Google Scholar
  24. 24.
    Krawetz, B., Lawrence, J., Shallit, J.: State complexity and the monoid of transformations of a finite set. In: Domaratzki, M., Okhotin, A., Salomaa, K., Yu, S. (eds.) CIAA 2004. LNCS, vol. 3317, pp. 213–224. Springer, Heidelberg (2005). doi: 10.1007/978-3-540-30500-2_20 CrossRefGoogle Scholar
  25. 25.
    Myhill, J.: Finite automata and representation of events. Wright Air Development Center. Technical report, pp. 57–624 (1957)Google Scholar
  26. 26.
    Pin, J.E.: Syntactic semigroups. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages. vol. 1: Word, Language, Grammar, pp. 679–746. Springer, New York (1997)CrossRefGoogle Scholar
  27. 27.
    Thierrin, G.: Convex languages. In: Nivat, M. (ed.) Automata, Languages and Programming, pp. 481–492. North-Holland (1973)Google Scholar
  28. 28.
    Yu, S.: State complexity of regular languages. J. Autom. Lang. Comb. 6, 221–234 (2001)MathSciNetzbMATHGoogle Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.David R. Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada

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