When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.FSTTCS.2009.2335
URN: urn:nbn:de:0030-drops-23357
URL: http://drops.dagstuhl.de/opus/volltexte/2009/2335/
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### Recurrence and Transience for Probabilistic Automata

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### Abstract

In a context of $\omega$-regular specifications for infinite execution sequences, the classical B\"uchi condition, or repeated liveness condition, asks that an accepting state is visited infinitely often. In this paper, we show that in a probabilistic context it is relevant to strengthen this infinitely often condition. An execution path is now accepting if the \emph{proportion} of time spent on an accepting state does not go to zero as the length of the path goes to infinity. We introduce associated notions of recurrence and transience for non-homogeneous finite Markov chains and study the computational complexity of the associated problems. As Probabilistic B\"uchi Automata (PBA) have been an attempt to generalize B\"uchi automata to a probabilistic context, we define a class of Constrained Probabilistic Automata with our new accepting condition on runs. The accepted language is defined by the requirement that the measure of the set of accepting runs is positive (probable semantics) or equals 1 (almost-sure semantics). In contrast to the PBA case, we prove that the emptiness problem for the language of a constrained probabilistic B\"uchi automaton with the probable semantics is decidable.

### BibTeX - Entry

@InProceedings{tracol_et_al:LIPIcs:2009:2335,
author =	{Mathieu Tracol and Christel Baier and Marcus Gr{\"o}sser},
title =	{{Recurrence and Transience for Probabilistic Automata}},
booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science},
pages =	{395--406},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-939897-13-2},
ISSN =	{1868-8969},
year =	{2009},
volume =	{4},
editor =	{Ravi Kannan and K. Narayan Kumar},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},