### 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 ≡ _{ε} s indexed by rationals, expressing that “t is approximately equal to s up to an error ε”. 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).

Language | English |
---|---|

Article number | 15 |

Number of pages | 29 |

Journal | Logical Methods in Computer Science |

Volume | 14 |

Issue number | 4 |

DOIs | |

Publication status | Published - 16 Nov 2018 |

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### Keywords

- axiomatization
- behavioral distances
- Markov chains

### Cite this

*Logical Methods in Computer Science*,

*14*(4), [15]. https://doi.org/10.23638/LMCS-14(4:15)2018

}

*Logical Methods in Computer Science*, vol. 14, no. 4, 15. https://doi.org/10.23638/LMCS-14(4:15)2018

**A complete quantitative deduction system for the bisimilarity distance on Markov chains.** / Bacci, Giorgio; Bacci, Giovanni; Larsen, Kim G.; Mardare, Radu.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A complete quantitative deduction system for the bisimilarity distance on Markov chains

AU - Bacci, Giorgio

AU - Bacci, Giovanni

AU - Larsen, Kim G.

AU - Mardare, Radu

PY - 2018/11/16

Y1 - 2018/11/16

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 ≡ ε s indexed by rationals, expressing that “t is approximately equal to s up to an error ε”. 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 ≡ ε s indexed by rationals, expressing that “t is approximately equal to s up to an error ε”. 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).

KW - axiomatization

KW - behavioral distances

KW - Markov chains

UR - http://www.scopus.com/inward/record.url?scp=85060257142&partnerID=8YFLogxK

U2 - 10.23638/LMCS-14(4:15)2018

DO - 10.23638/LMCS-14(4:15)2018

M3 - Article

VL - 14

JO - Logical Methods in Computer Science

T2 - Logical Methods in Computer Science

JF - Logical Methods in Computer Science

SN - 1860-5974

IS - 4

M1 - 15

ER -