### Abstract

Continuous Markovian Logic (CML) is a multimodal logic that expresses quantitative and qualitative properties of continuous-space and continuous-time labelled Markov processes (CMPs). The modalities of CML approximate the rates of the exponentially distributed random variables that characterize the duration of the labeled transitions. In this paper we present a sound and complete Hilbert-style axiomatization of CML for the CMP-semantics and prove some meta-properties including the small model property. CML characterizes stochastic bisimulation and supports the definition of a quantified extension of satisfiability relation that measures the compatibility of a model and a property. Relying on the small model property, we prove that this measure can be approximated, within a given error, by using a distance between logical formulas.

Language | English |
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Title of host publication | Computer Science Logic 2011 - 25th International Workshop/20th Annual Conference of the EACSL, CSL 2011 |

Pages | 144-158 |

Number of pages | 15 |

Volume | 12 |

DOIs | |

Publication status | Published - 1 Dec 2011 |

Event | 25th International Workshop on Computer Science Logic, CSL 2011/20th Annual Conference of the European Association for Computer Science Logic, EACSL - Bergen, Norway Duration: 12 Sep 2011 → 15 Sep 2011 |

### Conference

Conference | 25th International Workshop on Computer Science Logic, CSL 2011/20th Annual Conference of the European Association for Computer Science Logic, EACSL |
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Country | Norway |

City | Bergen |

Period | 12/09/11 → 15/09/11 |

### Fingerprint

### Keywords

- axiomatization
- Markov processes
- metric semantics
- probabilistic logic

### Cite this

*Computer Science Logic 2011 - 25th International Workshop/20th Annual Conference of the EACSL, CSL 2011*(Vol. 12, pp. 144-158) https://doi.org/10.4230/LIPIcs.CSL.2011.144

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*Computer Science Logic 2011 - 25th International Workshop/20th Annual Conference of the EACSL, CSL 2011.*vol. 12, pp. 144-158, 25th International Workshop on Computer Science Logic, CSL 2011/20th Annual Conference of the European Association for Computer Science Logic, EACSL, Bergen, Norway, 12/09/11. https://doi.org/10.4230/LIPIcs.CSL.2011.144

**Continuous Markovian logic - From complete axiomatization to the metric space of formulas.** / Cardelli, Luca; Larsen, Kim G.; Mardare, Radu.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution book

TY - GEN

T1 - Continuous Markovian logic - From complete axiomatization to the metric space of formulas

AU - Cardelli, Luca

AU - Larsen, Kim G.

AU - Mardare, Radu

PY - 2011/12/1

Y1 - 2011/12/1

N2 - Continuous Markovian Logic (CML) is a multimodal logic that expresses quantitative and qualitative properties of continuous-space and continuous-time labelled Markov processes (CMPs). The modalities of CML approximate the rates of the exponentially distributed random variables that characterize the duration of the labeled transitions. In this paper we present a sound and complete Hilbert-style axiomatization of CML for the CMP-semantics and prove some meta-properties including the small model property. CML characterizes stochastic bisimulation and supports the definition of a quantified extension of satisfiability relation that measures the compatibility of a model and a property. Relying on the small model property, we prove that this measure can be approximated, within a given error, by using a distance between logical formulas.

AB - Continuous Markovian Logic (CML) is a multimodal logic that expresses quantitative and qualitative properties of continuous-space and continuous-time labelled Markov processes (CMPs). The modalities of CML approximate the rates of the exponentially distributed random variables that characterize the duration of the labeled transitions. In this paper we present a sound and complete Hilbert-style axiomatization of CML for the CMP-semantics and prove some meta-properties including the small model property. CML characterizes stochastic bisimulation and supports the definition of a quantified extension of satisfiability relation that measures the compatibility of a model and a property. Relying on the small model property, we prove that this measure can be approximated, within a given error, by using a distance between logical formulas.

KW - axiomatization

KW - Markov processes

KW - metric semantics

KW - probabilistic logic

UR - http://www.scopus.com/inward/record.url?scp=84865047468&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.CSL.2011.144

DO - 10.4230/LIPIcs.CSL.2011.144

M3 - Conference contribution book

SN - 9783939897323

VL - 12

SP - 144

EP - 158

BT - Computer Science Logic 2011 - 25th International Workshop/20th Annual Conference of the EACSL, CSL 2011

ER -