Fixpoints, games and the difference hierarchy
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 37 (2003) no. 1, pp. 1-15.

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 Σ 2 0 . 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.

DOI : 10.1051/ita:2003011
Classification : 03E15, 68Q45
Mots-clés : descriptive set theory, fixpoint, game quantifier, induction 
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Bradfield, Julian C. Fixpoints, games and the difference hierarchy. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 37 (2003) no. 1, pp. 1-15. doi : 10.1051/ita:2003011. http://www.numdam.org/articles/10.1051/ita:2003011/

[1] 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. | Zbl

[2] J.C. Bradfield, The modal mu-calculus alternation hierarchy is strict. Theoret. Comput. Sci. 195 (1997) 133-153. | MR | Zbl

[3] J.C. Bradfield, Fixpoint alternation and the game quantifier, in Proc. CSL '99. Lecture Notes in Comput. Sci. 1683 (1999) 350-361. | Zbl

[4] J.R. Büchi, Using determinancy of games to eliminate quantifers, in Proc. FCT '77. Lecture Notes in Comput. Sci. 56 (1977) 367-378. | Zbl

[5] J.P. Burgess, Classical hierarchies from a modern standpoint 115 (1983) 81-95. | MR | Zbl

[6] E.A. Emerson and C.S. Jutla, Tree automata, mu-calculus and determinacy, in Proc. FOCS 91 (1991).

[7] P.G. Hinman, The finite levels of the hierarchy of effective R-sets. Fund. Math. 79 (1973) 1-10. | MR | Zbl

[8] P.G. Hinman, Recursion-Theoretic Hierarchies. Springer, Berlin (1978). | MR | Zbl

[9] R.S. Lubarsky, μ-definable sets of integers. J. Symb. Logic 58 (1993) 291-313. | MR | Zbl

[10] Y.N. Moschovakis, Descriptive Set Theory. North-Holland, Amsterdam (1980). | MR | Zbl

[11] D. Niwiński, Fixed point characterization of infinite behavior of finite state systems. Theoret. Comput. Sci. 189 (1997) 1-69. | MR | Zbl

[12] V. Selivanov, Fine hierarchy of regular ω-languages. Theoret. Comput. Sci. 191 (1998) 37-59. | MR | Zbl

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