RAIRO - Theoretical Informatics and Applications

Research Article

Fixpoints, games and the difference hierarchy

Julian C. Bradfield

LFCS, School of Informatics, University of Edinburgh, Edinburgh, EH9 3JZ, UK; jcb@inf.ed.ac.uk.

Abstract

Drawing on an analogy with temporal fixpoint logic, we relate the arithmetic fixpoint definable sets to the winning positions of certain games, namely games whose winning conditions lie in the difference hierarchy over $\Sigma^0_2$. This both provides a simple characterization of the fixpoint hierarchy, and refines existing results on the power of the game quantifier in descriptive set theory. We raise the problem of transfinite fixpoint hierarchies.

(Online publication November 15 2003)

Key Words:

  • Descriptive set theory;
  • fixpoint;
  • game quantifier;
  • induction.

Mathematics Subject Classification:

  • 03E15;
  • 68Q45
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