### Abstract

We propose a complete axiomatization for the total variation distance of finite labelled Markov chains. Our axiomatization is given in the form of a quantitative deduction system, a framework recently proposed by Mardare, Panangaden, and Plotkin (LICS 2016) to extend classical equational deduction systems by means of inferences of equality relations t≡_{ε}s indexed by rationals, expressing that “t is approximately equal to s up to an error ε”. Notably, the quantitative equational system is obtained by extending our previous axiomatization (CONCUR 2016) for the probabilistic bisimilarity distance with a distributivity axiom for the prefix operator over the probabilistic choice inspired by Rabinovich's (MFPS 1983). Finally, we propose a metric extension to the Kleene-style representation theorem for finite labelled Markov chains w.r.t. trace equivalence due to Silva and Sokolova (MFPS 2011).

Original language | English |
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Pages (from-to) | 27-39 |

Number of pages | 13 |

Journal | Electronic Notes in Theoretical Computer Science |

Volume | 336 |

DOIs | |

Publication status | Published - 16 Apr 2018 |

### Keywords

- axiomatization
- behavioral distances
- Markov chains
- quantitative deductive systems

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## Cite this

*Electronic Notes in Theoretical Computer Science*,

*336*, 27-39. https://doi.org/10.1016/j.entcs.2018.03.014