A Complete Quantitative Deduction System for the Bisimilarity Distance on Markov Chains

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Resumé

In this paper we propose a complete axiomatization of the bisimilarity distance of Desharnais et al. for the class of finite labelled Markov chains. Our axiomatization is given in the style of a quantitative extension of equational logic recently proposed by Mardare, Panangaden, and Plotkin (LICS 2016) that uses equality relations t =_epsilon s indexed by rationals, expressing that t is approximately equal to s up to an error epsilon. Notably, our quantitative deduction system extends in a natural way the equational system for probabilistic bisimilarity given by Stark and Smolka by introducing an axiom for dealing with the Kantorovich distance between probability distributions. The axiomatization is then used to propose a metric extension of a Kleene's style representation theorem for finite labelled Markov chains, that was proposed (in a more general coalgebraic fashion) by Silva et al. (Inf. Comput. 2011).
OriginalsprogEngelsk
TidsskriftLogical Methods in Computer Science
ISSN1860-5974
StatusUdgivet - 2017

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Axiomatization
Deduction
Markov processes
Markov chain
Equational Logic
Probability distributions
Approximately equal
Representation Theorem
Axiom
Equality
Probability Distribution
Metric
Style

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title = "A Complete Quantitative Deduction System for the Bisimilarity Distance on Markov Chains",
abstract = "In this paper we propose a complete axiomatization of the bisimilarity distance of Desharnais et al. for the class of finite labelled Markov chains. Our axiomatization is given in the style of a quantitative extension of equational logic recently proposed by Mardare, Panangaden, and Plotkin (LICS 2016) that uses equality relations t =_epsilon s indexed by rationals, expressing that t is approximately equal to s up to an error epsilon. Notably, our quantitative deduction system extends in a natural way the equational system for probabilistic bisimilarity given by Stark and Smolka by introducing an axiom for dealing with the Kantorovich distance between probability distributions. The axiomatization is then used to propose a metric extension of a Kleene's style representation theorem for finite labelled Markov chains, that was proposed (in a more general coalgebraic fashion) by Silva et al. (Inf. Comput. 2011).",
author = "Giovanni Bacci and Giorgio Bacci and Larsen, {Kim Guldstrand} and Radu Mardare",
year = "2017",
language = "English",
journal = "Logical Methods in Computer Science",
issn = "1860-5974",
publisher = "International Federation for Computational Logic",

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TY - JOUR

T1 - A Complete Quantitative Deduction System for the Bisimilarity Distance on Markov Chains

AU - Bacci, Giovanni

AU - Bacci, Giorgio

AU - Larsen, Kim Guldstrand

AU - Mardare, Radu

PY - 2017

Y1 - 2017

N2 - In this paper we propose a complete axiomatization of the bisimilarity distance of Desharnais et al. for the class of finite labelled Markov chains. Our axiomatization is given in the style of a quantitative extension of equational logic recently proposed by Mardare, Panangaden, and Plotkin (LICS 2016) that uses equality relations t =_epsilon s indexed by rationals, expressing that t is approximately equal to s up to an error epsilon. Notably, our quantitative deduction system extends in a natural way the equational system for probabilistic bisimilarity given by Stark and Smolka by introducing an axiom for dealing with the Kantorovich distance between probability distributions. The axiomatization is then used to propose a metric extension of a Kleene's style representation theorem for finite labelled Markov chains, that was proposed (in a more general coalgebraic fashion) by Silva et al. (Inf. Comput. 2011).

AB - In this paper we propose a complete axiomatization of the bisimilarity distance of Desharnais et al. for the class of finite labelled Markov chains. Our axiomatization is given in the style of a quantitative extension of equational logic recently proposed by Mardare, Panangaden, and Plotkin (LICS 2016) that uses equality relations t =_epsilon s indexed by rationals, expressing that t is approximately equal to s up to an error epsilon. Notably, our quantitative deduction system extends in a natural way the equational system for probabilistic bisimilarity given by Stark and Smolka by introducing an axiom for dealing with the Kantorovich distance between probability distributions. The axiomatization is then used to propose a metric extension of a Kleene's style representation theorem for finite labelled Markov chains, that was proposed (in a more general coalgebraic fashion) by Silva et al. (Inf. Comput. 2011).

M3 - Journal article

JO - Logical Methods in Computer Science

JF - Logical Methods in Computer Science

SN - 1860-5974

ER -