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The mu-calculus and Model Checking [chapter]

Julian Bradfield, Igor Walukiewicz
2018 Handbook of Model Checking  
The chapter describes in length the game characterization of the semantics of the mu-calculus.  ...  Then it develops the notion of modal automaton: an automaton-based model behind the mu-calculus.  ...  Let us remark that the Church synthesis problem [33, 117] for the mu-calculus formulas is an instance of the above for a particular choice of M.  ... 
doi:10.1007/978-3-319-10575-8_26 fatcat:7adhegoggvcjthaa5x5uhfpt24

Data flow analysis is model checking of abstract interpretations

David A. Schmidt
1998 Proceedings of the 25th ACM SIGPLAN-SIGACT symposium on Principles of programming languages - POPL '98  
In particular, the classic %ow equations for bit-vector-based d-Jo. s reformat trivially into modal mu-Cal&us formulas., A surprising consequence is that two of the classical d&a. s are exposed as unsound  ...  In the process of making the above discoveries, we cIarify the relationship between a. i. and d-&a. in terms of the often-misunderstood notion of collecting semantics and we highlight how the research  ...  design space and the modal mu-calculus defines the design space for iterative static analyses.  ... 
doi:10.1145/268946.268950 dblp:conf/popl/Schmidt98 fatcat:2muvgfeo6bhvva6gnqkukobs7e

Page 1665 of Mathematical Reviews Vol. , Issue 91C [page]

1991 Mathematical Reviews  
{For the entire collection see MR 90i:68005.} 91c:68092 68Q60 03B25 03B70 Streett, Robert S.; Emerson, E. Allen (1-TX-C) An automata theoretic decision procedure for the propositional mu-calculus.  ...  Summary: “The propositional mu-calculus is a propositional logic of programs which incorporates a least fixpoint operator and sub- sumes the propositional dynamic logic of Fischer and Ladner, the infinite  ... 

Efficient Patterns for Model Checking Partial State Spaces in CTL ∩ LTL

Adam Antonik, Michael Huth
2006 Electronical Notes in Theoretical Computer Science  
Hence we can compute, with linear blowup only, a semantic minimization in the same temporal logic whose efficient check renders the precise result for the original imprecise pattern.  ...  In this paper we determine for which popular specification patterns, documented at a community-led pattern repository, this under-approximation is precise in that the converse relationship holds as well  ...  Acknowledgments We acknowledge the kind support of this work by the UK Engineering and Physical Sciences Research Council under grant EP/D50595X/1.  ... 
doi:10.1016/j.entcs.2006.04.004 fatcat:2nzj76v4e5canm2d5t2n3w7dfm

Page 7307 of Mathematical Reviews Vol. , Issue 97M [page]

1997 Mathematical Reviews  
Kozen proposed an elegant axiom system for the propositional ~-calculus, the aim of the present paper is to show that this axiomatization is complete.  ...  The propositional u-calculus is an extension of the modal system K with a least fixed point operator; it has proved to be an interesting logic because it subsumes most propositional temporal logics and  ... 

Page 7662 of Mathematical Reviews Vol. , Issue 2001K [page]

2001 Mathematical Reviews  
It is well known that the category of sets is not able to provide a complete standard semantics for lambda-calculus.  ...  Summary: “We consider the minimal unsatisfiability problem MU(k) for propositional formulas in conjunctive normal form over n variables and with n +k clauses, where k is fixed. k is called the difference  ... 

Partial-order reduction in the weak modal mu-calculus [chapter]

Y. S. Ramakrishna, Scott A. Smolka
1997 Lecture Notes in Computer Science  
We present a partial-order reduction technique for local model checking of hierarchical networks of labeled transition systems in the weak modal mu-calculus.  ...  in modal mu-calculus model checking.  ...  We thank Xinxin Liu for valuable comments on this work, and Gerard Holzmann and Doron Peled for providing us with the SPIN benchmarks, and for making available GKPP97].  ... 
doi:10.1007/3-540-63141-0_2 fatcat:l4mwx6qrkvh2veqtvq5onuxqty

Symbolic model checking [chapter]

E. Clarke, K. McMillan, S. Campos, V. Hartonas-Garmhausen
1996 Lecture Notes in Computer Science  
We describe the syntax and semantics of a dialect of the Mu-Calculus, and present a model checking algorithm for Mu-Calculus formulas that uses BDDs to represent relations and formulas.  ...  For Mu-Calculus formulas derived from CTL formulas, it is convenient to encode each state in the domain with the set of atomic propositions that are true for that state.  ... 
doi:10.1007/3-540-61474-5_93 fatcat:luz3iefbijeepg2q5cmhflefee

Symbolic model checking: 1020 States and beyond

J.R. Burch, E.M. Clarke, K.L. McMillan, D.L. Dill, L.J. Hwang
1992 Information and Computation  
We describe the syntax and semantics of a dialect of the Mu-Calculus, and present a model checking algorithm for Mu-Calculus formulas that uses BDDs to represent relations and formulas.  ...  For Mu-Calculus formulas derived from CTL formulas, it is convenient to encode each state in the domain with the set of atomic propositions that are true for that state.  ... 
doi:10.1016/0890-5401(92)90017-a fatcat:dtdxdp4uurhxrhmt3zrgbqjsbm

Faster model checking for the modal Mu-Calculus [chapter]

Ranee Cleaveland, Marion Klein, Bernhard Steffen
1993 Lecture Notes in Computer Science  
Syntax and Semantics of the Mu-Calculus This section first provides a brief overview of labeled transition 8ystents, which are used as models for the mu-calcnlus.  ...  In thla paper, we develop an algorithm for model checking that handlu the full modal mucalculus including alternating fixpoints.  ...  Syntax and Semantics of Formulas The syntax of the modal mu-calculus is parameterized with respect to a (countable) set Vat of variables, a set .4 of atomic propositions, and a set Act of actions.  ... 
doi:10.1007/3-540-56496-9_32 fatcat:4rrsehc4zfgyrc7zxsagv5xy34

Symmetry Reduction for the Local Mu-Calculus [chapter]

Kedar S. Namjoshi, Richard J. Trefler
2018 Lecture Notes in Computer Science  
Considerably generalizing earlier results on invariance, it is shown that all local mu-calculus properties, including safety and liveness properties, are preserved by neighborhood symmetries.  ...  This defines an interleaving semantics, with unchanged(W ) denoting that the values of variables in W are unchanged. A global state is a function mapping variables in V to values in their domain.  ...  Allen Emerson for inspiring discussions on the topic.  ... 
doi:10.1007/978-3-319-89963-3_22 fatcat:xhuau3tcpjayxnvq273r7jngxi

Symmetry Reduction for the Local Mu-Calculus [article]

Kedar S. Namjoshi, Richard J. Trefler
2019 arXiv   pre-print
Considerably generalizing earlier results on invariance, it is shown that all local mu-calculus properties, including safety and liveness properties, are preserved by neighborhood symmetries.  ...  In general, local verification approximates verification over the global state space; however, if process interactions are outward-facing, the relationship is shown to be exact.  ...  Allen Emerson for inspiring discussions on the topic.  ... 
arXiv:1903.10405v1 fatcat:wrwngfzwvrem3lcuiiiqgsttty

Comparing CTL and PCTL on Labeled Markov Chains [chapter]

M. R. A. Huth, M. Z. Kwiatkowska
1998 Programming Concepts and Methods PROCOMET '98  
We interpret modal mu-calculus formulas in a non-standard way as expectations over labeled Markov chains. In this semantics fixed-points are computed in an iterative incremental/ decremental fashion.  ...  Finally, we show soundness of our non-standard semantics of CTL + with respect to the one for PCTL in the following sense: if our semantics computes a positive evidence for ¢ at state s then s satisfies  ...  ACKNOWLEDGMENTS We would like to thank all the anonymous referees for their constructive criticism.  ... 
doi:10.1007/978-0-387-35358-6_18 fatcat:acdkyetax5h2lmwtjg3wtpqcdy

Constraint-Based Abstract Semantics for Temporal Logic: A Direct Approach to Design and Implementation [chapter]

Gourinath Banda, John P. Gallagher
2010 Lecture Notes in Computer Science  
Using the modal µ-calculus to implement CTL, the abstract semantics gives an over-approximation of the set of states in which an arbitrary CTL formula holds.  ...  We apply the framework of abstract interpretation to derive an abstract semantic function for the modal µ-calculus, which is the basis for abstract model checking.  ...  We gratefully acknowledge discussions with Dennis Dams, César Sánchez, Kim Guldstrand Larsen and suggestions by the LPAR-16 referees.  ... 
doi:10.1007/978-3-642-17511-4_3 fatcat:fqim7etzrnaf7gbljd2x7quobe

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.  ...  Nous la relions aux jeux de paritéà information imparfaite, et nous montrons une applicationà la définition d'un mu-calcul modal fait pour l'indépendance.  ...  [Kozen 1983 ] modal mu-calculus, a popular and interesting temporal logic -see [Bradfield & Stirling 2001] for an introductory survey of modal mu-calculus.  ... 
doi:10.4000/philosophiascientiae.568 fatcat:vuezjblzpbcadbu6ktxxiq75l4
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