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Games, Probability, and the Quantitative μ-Calculus qMμ [chapter]

A. K. McIver, C. C. Morgan
2002 Lecture Notes in Computer Science  
The µ-calculus is a powerful tool for specifying and verifying transition systems, including those with demonic (universal) and angelic (existential) choice; its quantitative generalisation qMµ extends  ...  We show for a finite-state system that the straightforward denotational interpretation of the quantitative µ-calculus is equivalent to an operational interpretation given as a turn-based gambling game  ...  Stirling's game interpretation of the standard µ-calculus, and to show it equivalent to the existing Kozen-style interpretation of qMµ.  ... 
doi:10.1007/3-540-36078-6_20 fatcat:4y7mtcvm6jbn7ledxfwunctoum

A Novel Stochastic Game Via the Quantitative μ-calculus

Annabelle McIver, Carroll Morgan
2006 Electronical Notes in Theoretical Computer Science  
The quantitative μ-calculus qMμ extends the applicability of Kozen's standard μ-calculus [5] to probabilistic systems.  ...  The extension is motivated by examples based on economic games: we propose an extension to qMμ so that they can be specified; we show that the extension can be expressed via a reduction to the original  ...  In particular the quantitative μ-calculus qMμ extends the standard μ-calculus of Kozen [5] , giving access to probabilities by interpreting terms as real-rather than Boolean-valued functions of the state  ... 
doi:10.1016/j.entcs.2005.10.039 fatcat:mab7nnzibnbdpdjlka5fua7epi

Results on the quantitative mu-calculus qMu [article]

Annabelle McIver, Carroll Morgan
2003 arXiv   pre-print
The mu-calculus is a powerful tool for specifying and verifying transition systems, including those with both demonic and angelic choice; its quantitative generalisation qMu extends that to probabilistic  ...  The operational, game- based interpretation aids the intuition and continues in the more general context to provide a surprisingly practical specification tool.  ...  Our Stirling-style quantitative interpretation is operational, and is based on his earlier strategy-based game metaphor for the standard µ-calculus.  ... 
arXiv:cs/0309024v1 fatcat:bnehip2w5ba4pcfcgbdk2gaq44

Quantitative μ-Calculus Analysis of Power Management in Wireless Networks [chapter]

A. K. McIver
2006 Lecture Notes in Computer Science  
Our approach is to use the quantitative modal µ-calculus which allows the specification of a quantitative performance property as a game in which a maximising player's optimal strategy corresponds to the  ...  We extend the standard results on discounted games to infinite state systems, and illustrate our results on a small case study.  ...  Conclusions We have proposed a stochastic parity game to investigate the tradeoff between power savings and message throughput in a wireless network.  ... 
doi:10.1007/11921240_4 fatcat:vftazykqfbde3dwdrqe5ulqz5e

Memoryless Strategies for Stochastic Games via Domain Theory

Carroll Morgan, Annabelle McIver
2005 Electronical Notes in Theoretical Computer Science  
The significance of that extension is that it makes a further link between the program-logical approach and the more general methods for finding control strategies that "solve" games of this kind, in the  ...  The contribution here is to extend our logical approach to maximising games.  ...  In particular, we have shown how two-player turn-based demonic /angelic /probabilistic games correspond to expressions in our quantitative µ-calculus qMµ [14, 15, 10] , which is itself a superset of a  ... 
doi:10.1016/j.entcs.2005.03.003 fatcat:todalkg6mff2lf6yribprwrnrm


P. Höfner
2012 Formal Aspects of Computing  
Höfner and B. Möller. Last but not least we send acknowledgements to the authors for their submissions and the reviewers for their work, the latter often done under great time pressure.  ...  Furthermore, we thank the editors and the publishers of Formal Aspects of Computing for publishing this Festschrift-in particular John Cooke, Jim Woodcock and Sabine Taeger for their enormous help, as  ...  The paper [MM02] (co-authored by Annabelle McIver) gives an operational interpretation of the quantitative μ-calculus qMμ as a turn-based game between two players.  ... 
doi:10.1007/s00165-012-0257-0 fatcat:dbej2gxdqrhyxne7yynjkxo2vi

Quantum mechanics over sets [article]

David Ellerman
2013 arXiv   pre-print
This approach leads to a different treatment of Dirac's brackets than in "modal quantum theory" (MQT), and that gives a full probability calculus (unlike MQT that only has zero-one modalities of impossible  ...  Indeed, QM/sets is proposed as the "logic" of QM in the old-fashioned sense of "logic" as giving the simplified essentials of a theory.  ...  The lifting program and the probability calculus 1 Toy models of QM over finite fields In the tradition of "toy models" for quantum mechanics (QM), Schumacher and Westmoreland [20] and Hanson et al.  ... 
arXiv:1310.8221v1 fatcat:6ocpboidjjf3dcuhwabvw346f4

Quantum Games: Mixed Strategy Nash's Equilibrium Represents Minimum Entropy

Edward Jiménez
2003 Entropy  
, and ii) What do the postulates of Quantum Mechanics indicate in Game Theory and Economics?.  ...  This paper introduces Hermite's polynomials, in the description of quantum games. Hermite's polynomials are associated with gaussian probability density.  ...  Uncertainty Principle of Heisenberg Remember some basic definitions about probability and stochastic calculus.  ... 
doi:10.3390/e5040313 fatcat:aptcff4pz5c6paikpb7oushily

Dirac's Refined Unification Of Quantum Mechanics And Special Relativity: An Intertheoretic Context

Rinat Magdievich Nugayev
2022 Teorie vědy. Časopis pro mezioborová zkoumání vědy  
The subtle structure of Dirac's scientific research program and first and foremost the odd principles that put up its powerful heuristics is scrutinized with special emphasis on the highly controversial  ...  The peculiar character of Dirac's synthesis of special relativity and quantum mechanics is revealed by comparison with Einstein's sophisticated methodology of the General Relativity contrivance.  ...  Acknowledgements: It is a pleasure to thank the unknown reviewers for helpful critical remarks.  ... 
doi:10.46938/tv.2022.518 fatcat:36zzignv5ncnffunuxtxkagjza

Separable GPL: Decidable Model Checking with More Non-Determinism

Andrey Gorlin, C. R. Ramakrishnan, Michael Wagner
2018 International Conference on Concurrency Theory  
Generalized Probabilistic Logic (GPL) is a temporal logic, based on the modal mu-calculus, for specifying properties of branching probabilistic systems.  ...  probability over all schedulers) of a fuzzy formula.  ...  Other recent extensions of µ-calculus include the Lukasiewicz µ-calculus [22] , µ p -calculus [3], PµTL [17] , and the quantitative µ-calculi, such as qMµ [18] and Qµ [13].  ... 
doi:10.4230/lipics.concur.2018.36 dblp:conf/concur/GorlinR18 fatcat:esa2rsmdgjhytcej7cwciggi5e

XPL: An extended probabilistic logic for probabilistic transition systems [article]

Andrey Gorlin, C. R. Ramakrishnan
2017 arXiv   pre-print
Generalized Probabilistic Logic (GPL) is a temporal logic, based on the modal mu-calculus, for specifying properties of reactive probabilistic systems.  ...  We explore XPL, an extension to GPL allowing the semantics of nondeterminism present in Markov decision processes (MDPs).  ...  Quantitative µ-calculi, such as qMµ [20] and Qµ [14] , are more similar to pLµ, so we do not offer an independent comparison to XPL.  ... 
arXiv:1604.06118v5 fatcat:ifd2omhharexfhms4zvawlywfm

Playing (Almost-)Optimally in Concurrent Büchi and co-Büchi Games [article]

Benjamin Bordais, Patricia Bouyer, Stéphane Le Roux
2022 arXiv   pre-print
We study two-player concurrent stochastic games on finite graphs, with Büchi and co-Büchi objectives. The goal of the first player is to maximize the probability of satisfying the given objective.  ...  These well-behaved game forms are like elementary bricks which, when they behave well in isolation, can be assembled in graph games and ensure the good property for the whole game.  ...  Furthermore, at step n ∈ N, the probability that the event T occurs is equal to ε n . We will denote by P the probability measure of the corresponding measurable sets.  ... 
arXiv:2203.06966v1 fatcat:b3pawokc2vdphdd2bilxgfvvtm

English Translation of Poisson's "Recherches sur la probabilité des jugements en matière criminelle et en matière civile" / "Researches into the Probabilities of Judgements in Criminal and Civil Cases" [article]

S.-D. Poisson
2019 arXiv   pre-print
In spite of its title, the book mostly treats probability theory: the law of large numbers (regarded as a principle); formal definition of a random variable and law of distribution; the misnamed Cauchy  ...  distribution; functions now named after Dirac; difference between objective and subjective probabilities; approximation of the Bernoulli theorem for rare events (but not yet the Poisson distribution);  ...  When dividing the limits of n, corresponding to formula (79.1), by µ, and noting the meaning of U, we will obtain the limits of the ratio n/µ corresponding to the probability R 2 μ μ f pq qm .  ... 
arXiv:1902.02782v1 fatcat:l3rrle2prvezrgwetofe5ygere

Optimal strategies in concurrent reachability games [article]

Benjamin Bordais, Patricia Bouyer, Stéphane Le Roux
2021 arXiv   pre-print
We study two-player reachability games on finite graphs. At each state the interaction between the players is concurrent and there is a stochastic Nature. Players also play stochastically.  ...  Second, we consider three-state games with one main state, one target, and one bin.  ...  The Journal of Symbolic Logic, 63(4):1565-1581, 1998. 14 Annabelle McIver and Carroll Morgan. Games, probability and the quantitative µ-calculus qmµ.  ... 
arXiv:2110.14724v1 fatcat:4ijdp4dfajfadk4w4sfdke7ozq

Type Indeterminacy: A Model for the KT(Kahneman-Tversky)-Man [article]

Ariane Lambert-Mogiliansky, Shmuel Zamir, Herve Zwirn
2007 arXiv   pre-print
They are determined (i.e., realized, and not merely revealed) only when the action takes place.  ...  We explore some implications of our approach in a comparison between classical and type indeterminate rational choice behavior. The potential of the approach is illustrated in two examples.  ...  In particular, in accordance with the calculus of probability we see that the conditional probability formula holds: p AB (i A ∧ j B ) = p A (i A ) p B (j B |i A ).  ... 
arXiv:physics/0604166v2 fatcat:7ll5dlmyf5hf5dcbe5vkkfc6aa
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