Fixpoints, games and the difference hierarchy

Julian C. Bradfield

RAIRO - Theoretical Informatics and Applications (2010)

  • Volume: 37, Issue: 1, page 1-15
  • ISSN: 0988-3754

Abstract

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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.

How to cite

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Bradfield, Julian C.. "Fixpoints, games and the difference hierarchy." RAIRO - Theoretical Informatics and Applications 37.1 (2010): 1-15. <http://eudml.org/doc/92711>.

@article{Bradfield2010,
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. },
author = {Bradfield, Julian C.},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Descriptive set theory; fixpoint; game quantifier; induction.; Gale-Steward games; mu-arithmetic; parity games; modal mu-calculus; arithmetic fixpoint definable sets; difference hierarchy; fixpoint hierarchy; descriptive set theory; transfinite fixpoint hierarchies},
language = {eng},
month = {3},
number = {1},
pages = {1-15},
publisher = {EDP Sciences},
title = {Fixpoints, games and the difference hierarchy},
url = {http://eudml.org/doc/92711},
volume = {37},
year = {2010},
}

TY - JOUR
AU - Bradfield, Julian C.
TI - Fixpoints, games and the difference hierarchy
JO - RAIRO - Theoretical Informatics and Applications
DA - 2010/3//
PB - EDP Sciences
VL - 37
IS - 1
SP - 1
EP - 15
AB - 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.
LA - eng
KW - Descriptive set theory; fixpoint; game quantifier; induction.; Gale-Steward games; mu-arithmetic; parity games; modal mu-calculus; arithmetic fixpoint definable sets; difference hierarchy; fixpoint hierarchy; descriptive set theory; transfinite fixpoint hierarchies
UR - http://eudml.org/doc/92711
ER -

References

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  2. J.C. Bradfield, The modal mu-calculus alternation hierarchy is strict. Theoret. Comput. Sci.195 (1997) 133-153.  Zbl0915.03017
  3. J.C. Bradfield, Fixpoint alternation and the game quantifier, in Proc. CSL '99. Lecture Notes in Comput. Sci. 1683 (1999) 350-361.  Zbl0944.03028
  4. J.R. Büchi, Using determinancy of games to eliminate quantifers, in Proc. FCT '77. Lecture Notes in Comput. Sci. 56 (1977) 367-378.  Zbl0367.02005
  5. J.P. Burgess, Classical hierarchies from a modern standpoint. I. C-sets. Fund. Math.115 (1983) 81-95.  Zbl0515.28002
  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.  Zbl0285.02039
  8. P.G. Hinman, Recursion-Theoretic Hierarchies. Springer, Berlin (1978).  Zbl0371.02017
  9. R.S. Lubarsky, µ-definable sets of integers. J. Symb. Logic58 (1993) 291-313.  Zbl0776.03022
  10. Y.N. Moschovakis, Descriptive Set Theory. North-Holland, Amsterdam (1980).  
  11. D. Niwinski, Fixed point characterization of infinite behavior of finite state systems. Theoret. Comput. Sci.189 (1997) 1-69.  Zbl0893.68102
  12. V. Selivanov, Fine hierarchy of regular ω-languages. Theoret. Comput. Sci.191 (1998) 37-59.  Zbl0908.68085

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