A New Technique for Reachability of States in Concatenation Automata

  • Sylvie DaviesEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10952)


We present a new technique for demonstrating the reachability of states in deterministic finite automata representing the concatenation of two languages. Such demonstrations are a necessary step in establishing the state complexity of the concatenation of two languages, and thus in establishing the state complexity of concatenation as an operation. Typically, ad-hoc induction arguments are used to show particular states are reachable in concatenation automata. We prove some results that seem to capture the essence of many of these induction arguments. Using these results, reachability proofs in concatenation automata can often be done more simply and without using induction directly.



I thank Jason Bell, Janusz Brzozowski and the anonymous referees for proofreading and helpful comments. This work was supported by the Natural Sciences and Engineering Research Council of Canada under grant No. OGP0000871.


  1. 1.
    Brzozowski, J.A., Davies, S., Liu, B.Y.V.: Most complex regular ideal languages. Discrete Math. Theoret. Comput. Sci. 18(3) (2016), paper #15Google Scholar
  2. 2.
    Brzozowski, J.A., Jirásková, G., Zou, C.: Quotient complexity of closed languages. Theory Comput. Syst. 54, 277–292 (2014)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brzozowski, J.A.: In search of most complex regular languages. Int. J. Found. Comput. Sci. 24(06), 691–708 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    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, Cham (2016). Scholar
  5. 5.
    Brzozowski, J.A., Davies, S.: Most complex non-returning regular languages. In: Pighizzini, G., Câmpeanu, C. (eds.) DCFS 2017. LNCS, vol. 10316, pp. 89–101. Springer, Cham (2017). Scholar
  6. 6.
    Brzozowski, J.A., Jirásková, G., Li, B.: Quotient complexity of ideal languages. Theoret. Comput. Sci. 470, 36–52 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Brzozowski, J.A., Liu, B.: Quotient complexity of star-free languages. Int. J. Found. Comput. Sci. 23(06), 1261–1276 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Brzozowski, J.A., Sinnamon, C.: Complexity of left-ideal, suffix-closed and suffix-free regular languages. In: Drewes, F., Martín-Vide, C., Truthe, B. (eds.) LATA 2017. LNCS, vol. 10168, pp. 171–182. Springer, Cham (2017). Scholar
  9. 9.
    Brzozowski, J.A., Sinnamon, C.: Complexity of right-ideal, prefix-closed, and prefix-free regular languages. Acta Cybernetica 23(1), 9–41 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Câmpeanu, C., Culik, K., Salomaa, K., Yu, S.: State complexity of basic operations on finite languages. In: Boldt, O., Jürgensen, H. (eds.) WIA 1999. LNCS, vol. 2214, pp. 60–70. Springer, Heidelberg (2001). Scholar
  11. 11.
    Davies, S.: A new technique for reachability of states in concatenation automata (2017).
  12. 12.
    Eom, H.S., Han, Y.S., Jirásková, G.: State complexity of basic operations on non-returning regular languages. Fundam. Inform. 144, 161–182 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Han, Y.S., Salomaa, K., Wood, D.: Operational state complexity of prefix-free regular languages. In: Ésik, Z., Fülöp, Z. (eds.) AFL 2009, pp. 99–115. University of Szeged, Hungary, Institute of Informatics (2009)Google Scholar
  14. 14.
    Han, Y.S., Salomaa, K.: State complexity of basic operations on suffix-free regular languages. Theoret. Comput. Sci. 410(27–29), 2537–2548 (2009)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Jirásková, G., Krausová, M.: Complexity in prefix-free regular languages. In: McQuillan, I., Pighizzini, G., Trost, B. (eds.) DCFS 2010, pp. 236–244. University of Saskatchewan (2010)CrossRefGoogle Scholar
  16. 16.
    Maslov, A.N.: Estimates of the number of states of finite automata. Dokl. Akad. Nauk SSSR 194, 1266–1268 (Russian). English translation: Soviet Math. Dokl. 11(1970), 1373–1375 (1970)Google Scholar
  17. 17.
    Nicaud, C.: Average state complexity of operations on unary automata. In: Kutyłowski, M., Pacholski, L., Wierzbicki, T. (eds.) MFCS 1999. LNCS, vol. 1672, pp. 231–240. Springer, Heidelberg (1999). Scholar
  18. 18.
    Pighizzini, G., Shallit, J.: Unary language operations, state complexity and Jacobsthal’s function. Int. J. Found. Comput. Sci. 13(01), 145–159 (2002)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Yu, S.: State complexity of regular languages. J. Autom. Lang. Comb. 6, 221–234 (2001)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Yu, S., Zhuang, Q., Salomaa, K.: The state complexities of some basic operations on regular languages. Theoret. Comput. Sci. 125(2), 315–328 (1994)MathSciNetCrossRefGoogle Scholar

Copyright information

© IFIP International Federation for Information Processing 2018

Authors and Affiliations

  1. 1.Department of Pure MathematicsUniversity of WaterlooWaterlooCanada

Personalised recommendations