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Fixpoints, games and the difference hierarchy
2003
RAIRO - Theoretical Informatics and Applications
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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. ...
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 ...
It is then natural to ask whether this is coincidence, or whether there is, in arithmetic, a nice relationship between fixpoint alternation and some hierarchy of games, mediated by the game quantifier. ...
doi:10.1051/ita:2003011
fatcat:4hwj5rljynabbm7vc7af2y4kim
On independence-friendly fixpoint logics
2004
Philosophia Scientiæ
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Abstract: We introduce a fixpoint extension of Hintikka and Sandu's IF (independence-friendly) logic. We obtain some results on its complexity and expressive power. ...
We relate it to parity games of imperfect information, and show its application to defining independence-friendly modal mu-calculi. ...
Observe, however, that the contribution from fixpoint alternation is small compared to the cost of computing independent existential quantifiers. ...
doi:10.4000/philosophiascientiae.568
fatcat:vuezjblzpbcadbu6ktxxiq75l4
Discounting the Future in Systems Theory
[chapter]
2003
Lecture Notes in Computer Science
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Discounting (or inflation) is a key paradigm in economics and has been studied in Markov decision processes as well as game theory. ...
Our theory includes several basic elements: discounted versions of system properties that correspond to the ω-regular properties, fixpoint-based algorithms for checking discounted properties, and a quantitative ...
We thank Radha Jagadeesan for pointing out an error in an earlier formulation of the distance between distributions. ...
doi:10.1007/3-540-45061-0_79
fatcat:wlrgea2abjgexnzcid3of7en6m
The expressive power of stratified logic programs
1991
Information and Computation
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an "unbounded" alternation of quantifiers. ...
For any k > 1 and any I > 1, let C,, be the collection of all first-order formulas over 0 in prenex form with I alternations of quantifiers starting with an existential quantifier and having at most k ...
doi:10.1016/0890-5401(91)90059-b
fatcat:aexvileko5f55b2uvolbkueynu
Page 5152 of Mathematical Reviews Vol. , Issue 2004g
[page]
2004
Mathematical Reviews
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of existential and universal quantifiers: I has a winning strategy in the game G(A) iff SapVa,sarVa3... dya)arda3--- § A. ...
Similar to the Kleene hierarchy, the fixpoint alternation hierarchy (2), ),<,, 1s defined by an enu- meration of the alternation of the fixpoint constructors. ...
On Guarded Transformation In The Modal Mu-Calculus
[article]
2013
arXiv
pre-print
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We also investigate transformations between the μ-calculus, vectorial form and hierarchical equation systems, which are an alternative syntax for alternating parity tree automata. ...
Guarded normal form requires occurrences of fixpoint variables in a μ-calculus-formula to occur under the scope of a modal operator. ...
Otherwise, introduce a vacuous outermost fixpoint quantifier. This can be done without altering alternation depth. ...
arXiv:1305.0648v2
fatcat:wnsjlqfv4vgi5oqht4e6ba7lpy
Proof-theoretic Analysis of Rationality for Strategic Games with Arbitrary Strategy Sets
[article]
2010
arXiv
pre-print
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The other is a modal fixpoint language with expressions for optimality, rationality and belief. ...
In the context of strategic games, we provide an axiomatic proof of the statement Common knowledge of rationality implies that the players will choose only strategies that survive the iterated elimination ...
Then by Tarski's Fixpoint Theorem and Note 1(iii) we can use the following alternative definition of νX.ψ in terms of post-fixpoints: νX.ψ = {E ⊆ Ω | E ⊆ ψ | E }. ...
arXiv:1006.4926v1
fatcat:4hfobhkfmncpxftlxkyq2ilj4y
Graded Alternating-Time Temporal Logic
[chapter]
2010
Lecture Notes in Computer Science
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To this aim, we consider the Alternating-time Temporal Logic (Atl), a derivative of Ctl that is interpreted on games, rather than transition systems. ...
In both these settings, graded modalities enrich the universal and existential quantifiers with the capability to express the concept of at least k or all but k, for a non-negative integer k. ...
Therefore, we end up with two alternative semantics for the graded quantifier 1 k . ...
doi:10.1007/978-3-642-17511-4_12
fatcat:sp65ihkmojadlibw4ildrhgk2m
A Higher-order Framework for Decision Problems and Games
[article]
2014
arXiv
pre-print
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Modelling in our framework is compositional as the parts of the game are modular and can be easily exchanged. ...
We provide an extended example where we illustrate concepts and highlight the benefits of our alternative modelling approach. ...
Note that the fixpoint selection function models coordination, and the game in which all selection functions are fixpoints is a coordination game. ...
arXiv:1409.7411v1
fatcat:yxjcxmymbzh2vpbouwavnjfzyy
Page 4608 of Mathematical Reviews Vol. , Issue 2000g
[page]
2000
Mathematical Reviews
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Then G and H are elementarily equivalent if and only if they satisfy the same sentences with two alternations of quantifiers if and only if the direct products Z x G and Z x H are isomorphic, where Z denotes ...
We give a new, self-contained game-theoretical proof of the nonexpressibility results of Matz and Thomas. ...
Higher-Order Decision Theory
[chapter]
2017
Lecture Notes in Computer Science
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By that we can define goals being dependent on the actions and the process in addition to outcomes only. This formulation generalises outcome based preferences to context-dependent goals. ...
We propose to represent goals by higherorder functions or operators that take other functions as arguments where the max and arg max operators are special cases. ...
The fixpoint goals are far more interesting when we consider a game with several agents with different concerns, for instance some with usual preferences and some with fixpoint goals. ...
doi:10.1007/978-3-319-67504-6_17
fatcat:hjokz2f25ndapi5vwytuq2pg4a
Three notes on the complexity of model checking fixpoint logic with chop
2007
RAIRO - Theoretical Informatics and Applications
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This is already the case for its alternation-free fragment. The expression complexity of FLC is trivially P-hard and limited from above by the complexity of solving a parity game, i.e. in UP∩co-UP. ...
This paper analyses the complexity of model checking Fixpoint Logic with Chop -an extension of the modal µ-calculus with a sequential composition operator. ...
We would like to thank Damian Niwiński for questioning a linear upper bound on necessary unfoldings of fixpoint formulas in FLC and various suggestions that have led to considerable improvements in this ...
doi:10.1051/ita:2007011
fatcat:qj7kuj4r6rcslbp6yoewgk5huq
Higher-Order Decision Theory
[article]
2015
arXiv
pre-print
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By that we can define goals being dependent on the actions and the process in addition to outcomes only. This formulation generalises outcome based preferences to context-dependent goals. ...
We propose to represent goals by higher-order functions or operators that take other functions as arguments where the max and argmax operators are special cases. ...
The fixpoint goals are far more interesting when we consider a game with several agents with different concerns, for instance some with usual preferences and some with fixpoint goals. ...
arXiv:1506.01003v2
fatcat:e26qnjbojjdrdd6vkwbyz3l5f4
Higher-Order Game Theory
[article]
2015
arXiv
pre-print
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In applied game theory the motivation of players is a key element. It is encoded in the payoffs of the game form and often based on utility functions. ...
Lastly, this framework serves as a stepping stone to powerful tools from computer science that can be usefully applied to economic game theory in the future such as computational and computability aspects ...
Conclusion In this paper we have introduced an alternative representation of games. Quantifiers and selection functions provide an abstract and general way to describe players' goals. ...
arXiv:1506.01002v2
fatcat:jwi3w264hvaylm5vrqwme7mpmq
The mu-calculus and Model Checking
[chapter]
2018
Handbook of Model Checking
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It gives a quite detailed explanation of the satisfiability algorithm, followed by the results on alternation hierarchy, proof systems, and interpolation. ...
The chapter describes in length the game characterization of the semantics of the mu-calculus. ...
In Section 2 we introduce the logic, and present some basic notions such as: special forms of formulas, vectorial syntax, alternation depth of fixpoints. ...
doi:10.1007/978-3-319-10575-8_26
fatcat:7adhegoggvcjthaa5x5uhfpt24
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