Quotient Complexities of Atoms of Regular Languages

  • Janusz Brzozowski
  • Hellis Tamm
Conference paper

DOI: 10.1007/978-3-642-31653-1_6

Part of the Lecture Notes in Computer Science book series (LNCS, volume 7410)
Cite this paper as:
Brzozowski J., Tamm H. (2012) Quotient Complexities of Atoms of Regular Languages. In: Yen HC., Ibarra O.H. (eds) Developments in Language Theory. DLT 2012. Lecture Notes in Computer Science, vol 7410. Springer, Berlin, Heidelberg


An atom of a regular language L with n (left) quotients is a non-empty intersection of uncomplemented or complemented quotients of L, where each of the n quotients appears in a term of the intersection. The quotient complexity of L, which is the same as the state complexity of L, is the number of quotients of L. We prove that, for any language L with quotient complexity n, the quotient complexity of any atom of L with r complemented quotients has an upper bound of 2n − 1 if r = 0 or r = n, and \(1+\sum_{k=1}^{r} \sum_{h=k+1}^{k+n-r} C_{h}^{n} \cdot C_{k}^{h}\) otherwise, where \(C_j^i\) is the binomial coefficient. For each \(n\geqslant 1\), we exhibit a language whose atoms meet these bounds.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Janusz Brzozowski
    • 1
  • Hellis Tamm
    • 2
  1. 1.David R. Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.Institute of CyberneticsTallinn University of TechnologyTallinnEstonia

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