On the State Complexity of the Shuffle of Regular Languages

  • Janusz Brzozowski
  • Galina Jirásková
  • Bo Liu
  • Aayush Rajasekaran
  • Marek SzykułaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9777)


We investigate the shuffle operation on regular languages represented by complete deterministic finite automata. We prove that \(f(m,n)=2^{mn-1} + 2^{(m-1)(n-1)}(2^{m-1}-1)(2^{n-1}-1)\) is an upper bound on the state complexity of the shuffle of two regular languages having state complexities m and n, respectively. We also state partial results about the tightness of this bound. We show that there exist witness languages meeting the bound if \(2\leqslant m\leqslant 5\) and \(n\geqslant 2\), and also if \(m=n=6\). Moreover, we prove that in the subset automaton of the NFA accepting the shuffle, all \(2^{mn}\) states can be distinguishable, and an alphabet of size three suffices for that. It follows that the bound can be met if all f(mn) states are reachable. We know that an alphabet of size at least mn is required provided that \(m,n \geqslant 2\). The question of reachability, and hence also of the tightness of the bound f(mn) in general, remains open.


Regular language Shuffle State complexity Upper bound 



We would like to thank an anonymous referee for proposing the notions of a uniquely distinguishable state and of a subgraph of unique in-transitions which allow us to simplify the proof of distinguishability. We are also grateful for his comments and suggestions that helped us improve the presentation of the paper.


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Copyright information

© IFIP International Federation for Information Processing 2016

Authors and Affiliations

  • Janusz Brzozowski
    • 1
  • Galina Jirásková
    • 2
  • Bo Liu
    • 1
  • Aayush Rajasekaran
    • 1
  • Marek Szykuła
    • 3
    Email author
  1. 1.David R. Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.Mathematical InstituteSlovak Academy of SciencesKošiceSlovakia
  3. 3.Institute of Computer ScienceUniversity of WrocławWrocławPoland

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