### Abstract

Semi-Markov chains (SMCs) are continuous-time probabilistic transition systems where the residence time on states is governed by generic distributions on the positive real line. This paper shows the tight relation between the total variation distance on SMCs and their model checking problem over linear real-time specifications. Specifically, we prove that the total variation between two SMCs coincides with the maximal difference w.r.t. the likelihood of satisfying arbitrary MTL formulas or ω-languages recognized by timed automata. Computing this distance (i.e., solving its threshold problem) is NPhard and its decidability is an open problem. Nevertheless, we propose an algorithm for approximating it with arbitrary precision.

Original language | English |
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Title of host publication | Foundations of Software Science and Computation Structures |

Subtitle of host publication | Proceedings 18th International Conference, FOSSACS 2015, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2015 |

Editors | Andrew Pitts |

Place of Publication | Berlin |

Publisher | Springer-Verlag |

Pages | 185-199 |

Number of pages | 15 |

Volume | 9034 |

ISBN (Print) | 9783662466773 |

DOIs | |

Publication status | Published - 1 Jan 2015 |

Event | 18th International Conference on Foundations of Software Science and Computation Structures, FoSSaCS 2015 Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2015 - London, United Kingdom Duration: 11 Apr 2015 → 18 Apr 2015 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 9034 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 18th International Conference on Foundations of Software Science and Computation Structures, FoSSaCS 2015 Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2015 |
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Country | United Kingdom |

City | London |

Period | 11/04/15 → 18/04/15 |

### Keywords

- model check
- measurable space
- coupling structure
- arbitrary precision
- total variation distance

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## Cite this

*Foundations of Software Science and Computation Structures: Proceedings 18th International Conference, FOSSACS 2015, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2015*(Vol. 9034, pp. 185-199). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9034). Berlin: Springer-Verlag. https://doi.org/10.1007/978-3-662-46678-0_12