Transfinite Extension of the Mu-Calculus

  • Julian Bradfield
  • Jacques Duparc
  • Sandra Quickert
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3634)


In [1] Bradfield found a link between finite differences formed by Σ\(^{\rm 0}_{\rm 2}\) sets and the mu-arithmetic introduced by Lubarski [7]. We extend this approach into the transfinite: in allowing countable disjunctions we show that this kind of extended mu-calculus matches neatly to the transfinite difference hierarchy of Σ\(^{\rm 0}_{\rm 2}\) sets. The difference hierarchy is intimately related to parity games. When passing to infinitely many priorities, it might not longer be true that there is a positional winning strategy. However, if such games are derived from the difference hierarchy, this property still holds true.


Winning Strategy Game Tree Positional Strategy Winning Region Parity Game 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Julian Bradfield
    • 1
  • Jacques Duparc
    • 2
  • Sandra Quickert
    • 1
  1. 1.Laboratory for Foundations of Computer ScienceUniversity of Edinburgh 
  2. 2.Ecole des HECUniversité de Lausanne 

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