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Fixpoints, games and the difference hierarchy

Published online by Cambridge University Press:  15 November 2003

Julian C. Bradfield
Julian C. Bradfield*
Affiliation:
LFCS, School of Informatics, University of Edinburgh, Edinburgh, EH9 3JZ, UK; jcb@inf.ed.ac.uk.
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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.

Keywords

Descriptive set theoryfixpointgame quantifierinduction.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 37 , Issue 1 , January 2003 , pp. 1 - 15
Copyright
© EDP Sciences, 2003

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References

U. Bosse, An ``Ehrenfeucht-Fraïssé game" for fixpoint logic and stratified fixpoint logic, in Computer science logic. San Miniato, Lecture Notes in Comput. Sci. 702 (1992) 100-114.
Bradfield, J.C., The modal mu-calculus alternation hierarchy is strict. Theoret. Comput. Sci. 195 (1997) 133-153. CrossRef
J.C. Bradfield, Fixpoint alternation and the game quantifier, in Proc. CSL '99. Lecture Notes in Comput. Sci. 1683 (1999) 350-361.
J.R. Büchi, Using determinancy of games to eliminate quantifers, in Proc. FCT '77. Lecture Notes in Comput. Sci. 56 (1977) 367-378.
Burgess, J.P., Classical hierarchies from a modern standpoint. I. C-sets. Fund. Math. 115 (1983) 81-95.
E.A. Emerson and C.S. Jutla, Tree automata, mu-calculus and determinacy, in Proc. FOCS 91 (1991).
Hinman, P.G., The finite levels of the hierarchy of effective R-sets. Fund. Math. 79 (1973) 1-10.
P.G. Hinman, Recursion-Theoretic Hierarchies. Springer, Berlin (1978).
Lubarsky, R.S., µ-definable sets of integers. J. Symb. Logic 58 (1993) 291-313. CrossRef
Y.N. Moschovakis, Descriptive Set Theory. North-Holland, Amsterdam (1980).
Niwinski, D., Fixed point characterization of infinite behavior of finite state systems. Theoret. Comput. Sci. 189 (1997) 1-69. CrossRef
Selivanov, V., Fine hierarchy of regular ω-languages. Theoret. Comput. Sci. 191 (1998) 37-59. CrossRef