Converging from branching to linear metrics on Markov chains

Giorgio Bacci, Giovanni Bacci, Kim G. Larsen, Radu Mardare

Research output: Contribution to journalSpecial issue

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.

LanguageEnglish
Pages3-37
Number of pages35
JournalMathematical Structures in Computer Science
Volume29
Issue numberSpecial Issue 1
Early online date25 Jul 2017
DOIs
Publication statusPublished - 31 Jan 2019

Fingerprint

Linear Temporal Logic
Temporal logic
Markov processes
Branching
Markov chain
Linear Time
Metric
Semantics
Pseudometric
Model checking
Model Checking
Schema
Polynomial time
Exceed
NP-complete problem
Trace
Equivalence
Polynomials
Converge
Approximation

Keywords

  • Markov chains
  • linear-time metrics
  • probabilistic linear temporal logic
  • distances
  • convergence

Cite this

Bacci, Giorgio ; Bacci, Giovanni ; Larsen, Kim G. ; Mardare, Radu. / Converging from branching to linear metrics on Markov chains. In: Mathematical Structures in Computer Science. 2019 ; Vol. 29, No. Special Issue 1. pp. 3-37.
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Converging from branching to linear metrics on Markov chains. / Bacci, Giorgio; Bacci, Giovanni; Larsen, Kim G.; Mardare, Radu.

In: Mathematical Structures in Computer Science, Vol. 29, No. Special Issue 1, 31.01.2019, p. 3-37.

Research output: Contribution to journalSpecial issue

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