Language Equations with Complementation

  • Alexander Okhotin
  • Oksana Yakimova
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4036)


Systems of language equations of the form X i =ϕ i (X 1, ..., X n ) (\(1 \leqslant i \leqslant n\)) are studied, in which every ϕ i may contain the operations of concatenation and complementation. The properties of having solutions and of having a unique solution are given mathematical characterizations. As decision problems, the former is NP-complete, while the latter is in co-RE and its decidability remains, in general, open. Uniqueness becomes decidable for a unary alphabet, where it is US-complete, and in the case of linear concatenation, where it is L-complete. The position of the languages defined by these equations in the hierarchy of language families is established.


Unique Solution Boolean Operation Regular Language Language Family Solution Existence 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Alexander Okhotin
    • 1
  • Oksana Yakimova
    • 2
  1. 1.Department of MathematicsUniversity of TurkuFinland
  2. 2.Max-Planck-Institut für MathematikBonnGermany

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