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

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

*qM*

*µ*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

*qM*

*µ*. ...

##
###
A Novel Stochastic Game Via the Quantitative μ-calculus

2006
*
Electronical Notes in Theoretical Computer Science
*

*The*

*quantitative*

*μ*-

*calculus*

*qM*

*μ*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

*qM*

*μ*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*

*qM*

*μ*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 ...

##
###
Results on the quantitative mu-calculus qMu
[article]

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*. ...

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

2006
*
Lecture Notes in Computer Science
*

Our approach is to use

doi:10.1007/11921240_4
fatcat:vftazykqfbde3dwdrqe5ulqz5e
*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. ...##
###
Memoryless Strategies for Stochastic Games via Domain Theory

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*

*qM*

*µ*[14, 15, 10] , which is itself a superset of a ...

##
###
Preface

2012
*
Formal Aspects of Computing
*

Höfner

doi:10.1007/s00165-012-0257-0
fatcat:dbej2gxdqrhyxne7yynjkxo2vi
*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**qM**μ*as a turn-based*game*between two players. ...##
###
Quantum mechanics over sets
[article]

2013
*
arXiv
*
pre-print

This approach leads to a different treatment of Dirac's brackets than in "modal quantum theory" (MQT),

arXiv:1310.8221v1
fatcat:6ocpboidjjf3dcuhwabvw346f4
*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. ...##
###
Quantum Games: Mixed Strategy Nash's Equilibrium Represents Minimum Entropy

2003
*
Entropy
*

,

doi:10.3390/e5040313
fatcat:aptcff4pz5c6paikpb7oushily
*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*. ...##
###
Dirac's Refined Unification Of Quantum Mechanics And Special Relativity: An Intertheoretic Context

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. ...

##
###
Separable GPL: Decidable Model Checking with More Non-Determinism

2018
*
International Conference on Concurrency Theory
*

Generalized Probabilistic Logic (GPL) is a temporal logic, based on

doi:10.4230/lipics.concur.2018.36
dblp:conf/concur/GorlinR18
fatcat:esa2rsmdgjhytcej7cwciggi5e
*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], P*µ*TL [17] ,*and**the**quantitative**µ*-calculi, such as*qM**µ*[18]*and*Q*µ*[13]. ...##
###
XPL: An extended probabilistic logic for probabilistic transition systems
[article]

2017
*
arXiv
*
pre-print

Generalized Probabilistic Logic (GPL) is a temporal logic, based on

arXiv:1604.06118v5
fatcat:ifd2omhharexfhms4zvawlywfm
*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*qM**µ*[20]*and*Q*µ*[14] , are more similar to pL*µ*, so we do not offer an independent comparison to XPL. ...##
###
Playing (Almost-)Optimally in Concurrent Büchi and co-Büchi Games
[article]

2022
*
arXiv
*
pre-print

We study two-player concurrent stochastic

arXiv:2203.06966v1
fatcat:b3pawokc2vdphdd2bilxgfvvtm
*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. ...##
###
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]

2019
*
arXiv
*
pre-print

In spite of its title,

arXiv:1902.02782v1
fatcat:l3rrle2prvezrgwetofe5ygere
*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*. ...##
###
Optimal strategies in concurrent reachability games
[article]

2021
*
arXiv
*
pre-print

We study two-player reachability

arXiv:2110.14724v1
fatcat:4ijdp4dfajfadk4w4sfdke7ozq
*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**qm**µ*. ...##
###
Type Indeterminacy: A Model for the KT(Kahneman-Tversky)-Man
[article]

2007
*
arXiv
*
pre-print

They are determined (i.e., realized,

arXiv:physics/0604166v2
fatcat:7ll5dlmyf5hf5dcbe5vkkfc6aa
*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 ). ...
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