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Converging from branching to linear metrics on Markov chains

Published online by Cambridge University Press:  25 July 2017

GIORGIO BACCI ,
GIOVANNI BACCI ,
KIM G. LARSEN  and
RADU MARDARE
GIORGIO BACCI
Affiliation:
Department of Computer Science, Aalborg University, Aalborg, DK Email: grbacci@cs.aau.dk, giovbacci@cs.aau.dk, kgl@cs.aau.dk, mardare@cs.aau.dk
GIOVANNI BACCI
Affiliation:
Department of Computer Science, Aalborg University, Aalborg, DK Email: grbacci@cs.aau.dk, giovbacci@cs.aau.dk, kgl@cs.aau.dk, mardare@cs.aau.dk
KIM G. LARSEN
Affiliation:
Department of Computer Science, Aalborg University, Aalborg, DK Email: grbacci@cs.aau.dk, giovbacci@cs.aau.dk, kgl@cs.aau.dk, mardare@cs.aau.dk
RADU MARDARE
Affiliation:
Department of Computer Science, Aalborg University, Aalborg, DK Email: grbacci@cs.aau.dk, giovbacci@cs.aau.dk, kgl@cs.aau.dk, mardare@cs.aau.dk
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Abstract

We study two well-known linear-time metrics on Markov chains (MCs), namely, the strong and strutter trace distances. Our interest in these metrics is motivated by their relation to the probabilistic linear temporal logic (LTL)-model checking problem: we prove that they correspond to the maximal differences in the probability of satisfying the same LTL and LTL−X (LTL without next operator) formulas, respectively.

The threshold problem for these distances (whether their value exceeds a given threshold) is NP-hard and not known to be decidable. Nevertheless, we provide an approximation schema where each lower and upper approximant is computable in polynomial time in the size of the MC.

The upper approximants are bisimilarity-like pseudometrics (hence, branching-time distances) that converge point-wise to the linear-time metrics. This convergence is interesting in itself, because it reveals a non-trivial relation between branching and linear-time metric-based semantics that does not hold in equivalence-based semantics.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 29 , Special Issue 1: Special Issue: Best Papers Presented at ICTAC 2015 , January 2019 , pp. 3 - 37
Copyright
Copyright © Cambridge University Press 2017 

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