Weighted Lumpability on Markov Chains

Abstract. In this paper we study strong and weak bisimulation equivalences for continuous-time Markov decision processes (CTMDPs) and the logical characterizations of these relations with respect to the continuous-time stochastic logic (CSL). For strong bisimulation, it is well known that it is strictly finer than the CSL equivalence. In this paper we propose strong and weak bisimulations for CTMDPs and show that for a subclass of CTMDPs, strong and weak bisimulations are both sound and complete with respect to the equivalences induced by CSL and the sub-logic of CSL without next operator respectively. We then consider a standard extension of CSL, and show that it and its sub-logic without X can be fully characterized by strong and weak bisimulations respectively over arbitrary CTMDPs.

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“…In the remainder of this section, we shall focus on the proof of ∼ = ∼ CSL * . First, we introduce the following lemma in [33]:…”

Section: Strong Bisimulationmentioning

confidence: 99%

“…Lemma 5 (Theorem 5 [33]). Given a path formula ψ of CSL * and a state s, there exists a set of cylinder sets Cyls such that Sat (ψ) = ∪ C∈Cyls C.…”

Section: Strong Bisimulationmentioning

confidence: 99%

Bisimulations and Logical Characterizations on Continuous-Time Markov Decision Processes

Song

Zhang

Godskesen

2014

Lecture Notes in Computer Science

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“…T ϕ}) denote the probability that the CTMC C satisfies the MTL formula ϕ, for a given time bound can be shown as in [30]. Below we present an algorithm to compute Pr C T,<N (ϕ).…”

Section: Mtl Specificationsmentioning

confidence: 99%

Time-Bounded Verification of CTMCs against Real-Time Specifications

Chen

Diciolla

Kwiatkowska

et al. 2011

Lecture Notes in Computer Science

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Abstract. In this paper we study time-bounded verification of a finite continuous-time Markov chain (CTMC) C against a real-time specification, provided either as a metric temporal logic (MTL) property ϕ, or as a timed automaton (TA) A. The key question is: what is the probability of the set of timed paths of C that satisfy ϕ (or are accepted by A) over a time interval of fixed, bounded length? We provide approximation algorithms to solve these problems. We first derive a bound N such that timed paths of C with at most N discrete jumps are sufficient to approximate the desired probability up to ε. Then, for each discrete path σ of length at most N , we generate timed constraints over variables determining the residence time of each state along σ, depending on the real-time specification under consideration. The probability of the set of timed paths, determined by the discrete path and the associated timed constraints, can thus be formulated as a multidimensional integral. Summing up all such probabilities yields the result. For MTL, we consider both the continuous and the pointwise semantics. The approximation algorithms differ mainly in constraints generation for the two types of specifications.

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“…T-Lumpability is defined using four processalgebraic axioms, and allows for a more aggressive state space aggregation than ordinary lumpability. In [31] a novel structural definition of weighted lumpability (WL) has been provided on continuous-time Markov chains (CTMCs) that coincides with T-Lumpability. For WL it has been proved that probability of properties specified using deterministic timed automaton and metric temporal logic are preserved under WL quotienting.…”

Section: Introductionmentioning

confidence: 99%

A Two Step Perspective for Kripke Structure Reduction

Sharma

2012

Preprint

Self Cite

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This paper presents a novel theoretical framework for the state space reduction of Kripke structures. We define two equivalence relations, Kripke minimization equivalence (KME) and weak Kripke minimization equivalence (WKME). We define the quotient system under these relations and show that these relations are strictly coarser than strong (bi)simulation and divergence-sensitive stutter (bi)simulation, respectively. We prove that the quotient system obtained under KME and WKME preserves linear-time and stutter-insensitive linear-time properties. Finally, we show that KME is compositional w.r.t. synchronous parallel composition.

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Weighted Lumpability on Markov Chains

Cited by 16 publications

References 16 publications

Bisimulations and Logical Characterizations on Continuous-Time Markov Decision Processes

Bisimulations and Logical Characterizations on Continuous-Time Markov Decision Processes

Time-Bounded Verification of CTMCs against Real-Time Specifications

A Two Step Perspective for Kripke Structure Reduction

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