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Unrestricted State Complexity of Binary Operations on Regular Languages

  • Janusz BrzozowskiEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9777)

Abstract

I study the state complexity of binary operations on regular languages over different alphabets. It is well known that if \(L'_m\) and \(L_n\) are languages restricted to be over the same alphabet, with m and n quotients, respectively, the state complexity of any binary boolean operation on \(L'_m\) and \(L_n\) is mn, and that of the product (concatenation) is \((m-1)2^n +2^{n-1}\). In contrast to this, I show that if \(L'_m\) and \(L_n\) are over their own different alphabets, the state complexity of union and symmetric difference is \(mn+m+n+1\), that of intersection is \(mn+1\), that of difference is \(mn+m+1\), and that of the product is \(m2^n+2^{n-1}\).

Keywords

Boolean operation Concatenation Different alphabets Most complex languages Product Quotient complexity Regular language State complexity Stream Unrestricted complexity 

Notes

Acknowledgment

I am very grateful to Sylvie Davies, Bo Yang Victor Liu and Corwin Sinnamon for careful proofreading and constructive comments. I thank Marek Szykuła for contributing the important Remark 1 and Proposition 1.

References

  1. 1.
    Bell, J., Brzozowski, J., Moreira, N., Reis, R.: Symmetric groups and quotient complexity of boolean operations. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014, Part II. LNCS, vol. 8573, pp. 1–12. Springer, Heidelberg (2014)Google Scholar
  2. 2.
    Brzozowski, J.: Quotient complexity of regular languages. J. Autom. Lang. Comb. 15(1/2), 71–89 (2010)Google Scholar
  3. 3.
    Brzozowski, J.: In search of the most complex regular languages. Int. J. Found. Comput. Sci. 24(6), 691–708 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brzozowski, J., Tamm, H.: Complexity of atoms of regular languages. Int. J. Found. Comput. Sci. 24(7), 1009–1027 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brzozowski, J., Tamm, H.: Theory of átomata. Theoret. Comput. Sci. 539, 13–27 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gao, Y., Salomaa, K., Yu, S.: Transition complexity of incomplete DFAs. Fund. Inform. 110, 143–158 (2011)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Iván, S.: Complexity of atoms, combinatorially. Inform. Process. Lett. 116(5), 356–360 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Maia, E., Moreira, N., Reis, R.: Incomplete operational transition complexity of regular languages. Inform. Comput. 244, 1–22 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Maslov, A.N.: Estimates of the number of states of finite automata. Dokl. Akad. Nauk SSSR 194, 1266–1268 (1970) (Russian). English translation: Soviet Math. Dokl. 11, 1373–1375 (1970)Google Scholar
  10. 10.
    Yu, S.: State complexity of regular languages. J. Autom. Lang. Comb. 6, 221–234 (2001)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Yu, S., Zhuang, Q., Salomaa, K.: The state complexities of some basic operations on regular languages. Theoret. Comput. Sci. 125, 315–328 (1994)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© IFIP International Federation for Information Processing 2016

Authors and Affiliations

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

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