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The Descriptive Complexity of Modal μ Model-checking Games

Karoliina Lehtinen
2017 Electronic Proceedings in Theoretical Computer Science  
It addresses the question of whether the descriptive complexity of modal mu model-checking games, previously known to depend on the syntactic complexity of a formula, depends in fact on its semantic complexity  ...  Beyond co-B\"uchi, the descriptive complexity of the model-checking parity games of a formula is shown to be an upper bound on its semantic complexity; whether it is also a lower bound remains an open  ...  The descriptive complexity of parity games The last section recalled the classic result that the syntactic complexity of Ψ is an upper bound on the descriptive complexity of its model-checking game.  ... 
doi:10.4204/eptcs.256.6 fatcat:4ryiv3g6fjdj7kh3ili7epckxe

Model-Checking the Higher-Dimensional Modal mu-Calculus

Martin Lange, Etienne Lozes
2012 Electronic Proceedings in Theoretical Computer Science  
We exemplify the latter connection by giving several examples of decision problems which reduce to model checking of the higher-dimensional modal mu-calculus for some fixed formulas.  ...  The higher-dimensional modal mu-calculus is an extension of the mu-calculus in which formulas are interpreted in tuples of states of a labeled transition system.  ...  It is well-known that the model checking problem for the Modal µ-Calculus can be reduced to solving a parity game [Sti95] .  ... 
doi:10.4204/eptcs.77.6 fatcat:uicv3kknffdudnlfd3cd7wkyay

On independence-friendly fixpoint logics

J. C. Bradfield
2004 Philosophia Scientiæ  
We relate it to parity games of imperfect information, and show its application to defining independence-friendly modal mu-calculi.  ...  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.  ...  The IF modal mu-calculus has a model-checking game that is an IF version of the usual parity games for modal logic, as done above for IF-LFP.  ... 
doi:10.4000/philosophiascientiae.568 fatcat:vuezjblzpbcadbu6ktxxiq75l4

Static Analysis of Parity Games: Alternating Reachability Under Parity [chapter]

Michael Huth, Jim Huan-Pu Kuo, Nir Piterman
2015 Lecture Notes in Computer Science  
It is well understood that solving parity games is equivalent, up to polynomial time, to model checking of the modal mu-calculus.  ...  It is a long-standing open problem whether solving parity games (or model checking modal mu-calculus formulas) can be done in polynomial time.  ...  flow logics can capture model checking of the modal mu-calculus.  ... 
doi:10.1007/978-3-319-27810-0_8 fatcat:co2qjkdsunhnrjzc32bklgutcy

Fixpoints, games and the difference hierarchy

Julian C. Bradfield
2003 RAIRO - Theoretical Informatics and Applications  
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  ...  However, the world of Rabin automata and modal mu-calculus provides a suggestion: the complexity of the Rabin or parity condition corresponds to the complexity of the winning plays, and since the modal  ... 
doi:10.1051/ita:2003011 fatcat:4hwj5rljynabbm7vc7af2y4kim

Alternating tree automata, parity games, and modal {$\mu$}-calculus

Thomas Wilke
2001 Bulletin of the Belgian Mathematical Society Simon Stevin  
These problems, in turn, are solved using parity games.  ...  Then these two problems are in turn reduced to the winner problem for parity games.  ...  This also yields the desired upper bound for the complexity of the model checking problem for modal µ-calculus.  ... 
doi:10.36045/bbms/1102714178 fatcat:zjl4ze65jja7vobsbjovkphgoe

Recursive Checkonly QVT-R Transformations with General when and where Clauses via the Modal Mu Calculus [chapter]

Julian Bradfield, Perdita Stevens
2012 Lecture Notes in Computer Science  
In this paper we show how to interpret checkonly QVT-R -or any future model transformation language structured similarly -in the modal mu calculus and use its well-understood model-checking game to lift  ...  We demonstrate and justify a corresponding extension to the rules of the QVT-R game.  ...  The first author is partly supported by UK EPSRC grant EP/G012962/1 'Solving Parity Games and Mu-Calculi'.  ... 
doi:10.1007/978-3-642-28872-2_14 fatcat:xt7ud4mzd5bnfcgdvn6kih5lra

Modal Mu-Calculus [chapter]

Colin Stirling
2001 Texts in Computer Science  
Contextual background The modal mu-calculus comes not from the philosophical tradition of modal logic, but from the application of modal and temporal logics to program verification.  ...  Notation: Lµ means the modal mu-calculus, considered as a logical language (not as a theory).  ...  The model checking game on finite structures can be abstracted into a simple two player graph game, called the parity game.  ... 
doi:10.1007/978-1-4757-3550-5_5 fatcat:mfjlygtmd5e6lbmjdhr73oaisa

Generating and Solving Symbolic Parity Games

Gijs Kant, Jaco van de Pol
2014 Electronic Proceedings in Theoretical Computer Science  
We present a new tool for verification of modal mu-calculus formulae for process specifications, based on symbolic parity games.  ...  We improved the translation from specification to PBES to preserve the structure of the specification in the PBES, we extended LTSmin to instantiate PBESs to symbolic parity games, and implemented the  ...  The combination of these tools allows for high performance model checking of large systems, using MDDs as data structures.  ... 
doi:10.4204/eptcs.159.2 fatcat:oyzkxkthrjdfpdr7yhxlcjkevy

Enforcing QVT-R with mu-Calculus and Games [chapter]

Julian Bradfield, Perdita Stevens
2013 Lecture Notes in Computer Science  
In previous work, we gave a precise game-theoretic semantics for the checkonly semantics of QVT-R transformations, including the recursive invocation of relations which is allowed and used, but not defined  ...  In this paper, we take up the problem of enforce semantics, where the standard attempts formality, but at crucial points lapses into English.  ...  The first author is partly supported by UK EPSRC grant EP/G012962/1 'Solving Parity Games and Mu-Calculi'.  ... 
doi:10.1007/978-3-642-37057-1_21 fatcat:wgnqawchfrgf5clxllmraf6mty

Higher-Order Model Checking: An Overview

Luke Ong
2015 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science  
Higher-order model checking is about the model checking of trees generated by recursion schemes. The past fifteen years or so have seen considerable progress in both theory and practice.  ...  Because the trees generated by recursion schemes are computation trees of higher-order functional programs, higherorder model checking provides a foundation for model checkers of such programming languages  ...  Acknowledgements: The author is grateful to Takeshi Tsukada for helpful comments on the paper.  ... 
doi:10.1109/lics.2015.9 dblp:conf/lics/Ong15 fatcat:55bfxqlkuzhbtgmzc32m46ooku

On Modal μ-Calculus over Finite Graphs with Bounded Strongly Connected Components

Giovanna D'Agostino, Giacomo Lenzi
2010 Electronic Proceedings in Theoretical Computer Science  
We show that for every k, the Modal mu-Calculus fixpoint hierarchy on SCCk collapses to the level Delta2, but not to Comp(Sigma1,Pi1) (compositions of formulas of level Sigma1 and Pi1).  ...  This contrasts with the class of all graphs, where Delta2=Comp(Sigma1,Pi1).  ...  games for the formal verification of complex systems.  ... 
doi:10.4204/eptcs.25.9 fatcat:mfzghgt6inab3is4nar3uv3oue

Transfinite Extension of the Mu-Calculus [chapter]

Julian Bradfield, Jacques Duparc, Sandra Quickert
2005 Lecture Notes in Computer Science  
We extend this approach into the transfinite: in allowing countable disjunctions we show that this kind of extended mu-calculus matches neatly to the transfinite difference hierarchy of Σ 0 2 sets.  ...  The difference hierarchy is intimately related to parity games. When passing to infinitely many priorities, it might not longer be true that there is a positional winning strategy.  ...  Clearly, the rules of the model checking game ensure that the parity game derived from a model checking game of a transfinite mu-calculus formula is max-closed.  ... 
doi:10.1007/11538363_27 fatcat:zeurv63zfngnth6shazzzc7czq

On-The-Fly Solving for Symbolic Parity Games [article]

Maurice Laveaux, Wieger Wesselink, Tim A.C. Willemse
2022 arXiv   pre-print
In practice, tools that use parity games often rely on a specification in a higher-order logic from which the actual game can be obtained by means of an exploration.  ...  Parity games can be used to represent many different kinds of decision problems.  ...  This work was supported by the TOP Grants research programme with project number 612.001.751 (AVVA), which is (partly) financed by the Dutch Research Council (NWO).  ... 
arXiv:2201.09607v1 fatcat:zulfhot6efeypnv6pnsvd7drje

Recursion Schemes, Collapsible Pushdown Automata and Higher-Order Model Checking [chapter]

Luke Ong
2013 Lecture Notes in Computer Science  
We conclude with a brief overview of recent applications to the model checking of higher-order functional programs.  ...  We study and compare the expressive power of the two models and the algorithmic properties of infinite structures such as trees and graphs generated by them.  ...  Acknowledgements Much of my work presented here is based on collaboration. I am grateful to all my collaborators for their help. Thanks are also due to EPSRC for financial support.  ... 
doi:10.1007/978-3-642-37064-9_3 fatcat:vmzogjyskfahbeklhyssyajvuy
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