On the total variation distance of Semi-Markov chains

Giorgio Bacci; Giovanni Bacci; Kim Guldstrand Larsen; Radu Mardare

doi:10.1007/978-3-662-46678-0_12

On the total variation distance of Semi-Markov chains

Giorgio Bacci
, Giovanni Bacci
, Kim Guldstrand Larsen
, Radu Mardare

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

12 Citations (Scopus)

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
Title of host publication	Foundations of Software Science and Computation Structures
Subtitle of host publication	18th International Conference, FOSSACS 2015, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2015, Proceedings
Editors	Andrew Pitts
Place of Publication	Berlin
Publisher	Springer
Pages	185-199
Number of pages	15
Volume	9034
ISBN (Electronic)	978-3-662-46678-0
ISBN (Print)	978-3-662-46677-3
DOIs	https://doi.org/10.1007/978-3-662-46678-0_12
Publication status	Published - 2015

Publication series

Name	Lecture Notes in Computer Science
Volume	9034
ISSN (Print)	0302-9743

Keywords

model check
measurable space
coupling structure
arbitrary precision
total variation distance

Access to Document

10.1007/978-3-662-46678-0_12

https://www.scopus.com/pages/publications/84944096394

Cite this

Bacci, G., Bacci, G., Larsen, K. G., & Mardare, R. (2015). On the total variation distance of Semi-Markov chains. In A. Pitts (Ed.), Foundations of Software Science and Computation Structures: 18th International Conference, FOSSACS 2015, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2015, Proceedings (Vol. 9034, pp. 185-199). (Lecture Notes in Computer Science; Vol. 9034). Springer. https://doi.org/10.1007/978-3-662-46678-0_12

Bacci, Giorgio ; Bacci, Giovanni ; Larsen, Kim Guldstrand et al. / On the total variation distance of Semi-Markov chains. Foundations of Software Science and Computation Structures: 18th International Conference, FOSSACS 2015, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2015, Proceedings . editor / Andrew Pitts. Vol. 9034 Berlin : Springer, 2015. pp. 185-199 (Lecture Notes in Computer Science).

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title = "On the total variation distance of Semi-Markov chains",

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.",

keywords = "model check, measurable space, coupling structure, arbitrary precision, total variation distance",

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Bacci, G, Bacci, G, Larsen, KG & Mardare, R 2015, On the total variation distance of Semi-Markov chains. in A Pitts (ed.), Foundations of Software Science and Computation Structures: 18th International Conference, FOSSACS 2015, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2015, Proceedings . vol. 9034, Lecture Notes in Computer Science, vol. 9034, Springer, Berlin, pp. 185-199. https://doi.org/10.1007/978-3-662-46678-0_12

On the total variation distance of Semi-Markov chains. / Bacci, Giorgio; Bacci, Giovanni; Larsen, Kim Guldstrand et al.
Foundations of Software Science and Computation Structures: 18th International Conference, FOSSACS 2015, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2015, Proceedings . ed. / Andrew Pitts. Vol. 9034 Berlin: Springer, 2015. p. 185-199 (Lecture Notes in Computer Science; Vol. 9034).

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

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AU - Bacci, Giovanni

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N2 - 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.

AB - 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.

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KW - measurable space

KW - coupling structure

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KW - total variation distance

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Bacci G, Bacci G, Larsen KG, Mardare R. On the total variation distance of Semi-Markov chains. In Pitts A, editor, Foundations of Software Science and Computation Structures: 18th International Conference, FOSSACS 2015, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2015, Proceedings . Vol. 9034. Berlin: Springer. 2015. p. 185-199. (Lecture Notes in Computer Science). doi: 10.1007/978-3-662-46678-0_12

On the total variation distance of Semi-Markov chains

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