Metrics for labelled Markov processes

Desharnais, Jules; Gupta, Vineet; Jagadeesan, Radha; Panangaden, Prakash

doi:10.1016/j.tcs.2003.09.013

Cited by 257 publications

(161 citation statements)

References 29 publications

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“…There have been several attempts to give a mathematical semantics to reactive processes which is based on quantitative metrics rather than boolean preorders [18,6]. In particular for probabilistic processes, it is natural to generalize bisimulation relations to bisimulation metrics [9,19], and similar generalizations can be pursued if quantities enter not through probabilities but through discounting [7] or continuous variables [2] (this work uses the Skorohod metric on continuous behaviors to measure the distance between hybrid systems). We consider distances between purely discrete (nonprobabilistic, untimed) systems, and our distances are directed rather than symmetric (based on simulation rather than bisimulation).…”

Section: Introductionmentioning

confidence: 99%

Simulation Distances

Černý

Henzinger

Radhakrishna

2010

CONCUR 2010 - Concurrency Theory

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Abstract. Boolean notions of correctness are formalized by preorders on systems. Quantitative measures of correctness can be formalized by realvalued distance functions between systems, where the distance between implementation and specification provides a measure of "fit" or "desirability." We extend the simulation preorder to the quantitative setting, by making each player of a simulation game pay a certain price for her choices. We use the resulting games with quantitative objectives to define three different simulation distances. The correctness distance measures how much the specification must be changed in order to be satisfied by the implementation. The coverage distance measures how much the implementation restricts the degrees of freedom offered by the specification. The robustness distance measures how much a system can deviate from the implementation description without violating the specification. We consider these distances for safety as well as liveness specifications. The distances can be computed in polynomial time for safety specifications, and for liveness specifications given by weak fairness constraints. We show that the distance functions satisfy the triangle inequality, that the distance between two systems does not increase under parallel composition with a third system, and that the distance between two systems can be bounded from above and below by distances between abstractions of the two systems. These properties suggest that our simulation distances provide an appropriate basis for a quantitative theory of discrete systems. We also demonstrate how the robustness distance can be used to measure how many transmission errors are tolerated by error correcting codes.

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Section: Introductionmentioning

confidence: 99%

Simulation Distances

Černý

Henzinger

Radhakrishna

2010

CONCUR 2010 - Concurrency Theory

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“…The following definition is taken from [14,15,19] (these contributions mostly deal with analytic spaces that represent a generalisation of the Borel measurable space we focus on).…”

Section: Labelled Markov Processes: Model and Semanticsmentioning

confidence: 99%

“…One of the earliest contributions is the notion of exact probabilistic bisimulation in [14], obtained as a generalisation of its discrete-state counterpart [34] and used to characterise the LMP model semantics. Exact bisimulation is in general considered a very conservative requirement, and approximate notions have been consequently developed [15,19], which are based on relaxing the notion of process equivalence or on distance (pseudo-)metrics. These metrics encode exact probabilistic bisimulation, in that the distance between a pair of states is zero if and only if the states are bisimilar.…”

Section: Introductionmentioning

confidence: 99%

“…Approximation techniques for LMPs can be based on metrics [15,19,42] and coalgebras [43,44]. Approximate notions of probabilistic bisimilarity are formally characterised and computed for finite-state labelled Markov processes in [17].…”

Section: Introductionmentioning

confidence: 99%

“…It is then shown that these approximations are related to the original, uncountable-state models by notions of approximate probabilistic bisimulations [41]. Regarding algorithms for computing abstractions, [9] employs Monte-Carlo techniques for the (approximate) computation of the concepts in [15,19] which relates to the randomised techniques in [4].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Probabilistic Model Checking of Labelled Markov Processes via Finite Approximate Bisimulations

Abate

Kwiatkowska

Norman

et al. 2014

Lecture Notes in Computer Science

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Abstract. This paper concerns labelled Markov processes (LMPs), probabilistic models over uncountable state spaces originally introduced by Prakash Panangaden and colleagues. Motivated by the practical application of the LMP framework, we study its formal semantics and the relationship to similar models formulated in control theory. We consider notions of (exact and approximate) probabilistic bisimulation over LMPs and, drawing on methods from both formal verification and control theory, propose a simple technique to compute an approximate probabilistic bisimulation of a given LMP, where the resulting abstraction is characterised as a finite-state labelled Markov chain (LMC). This construction enables the application of automated quantitative verification and policy synthesis techniques over the obtained abstract model, which can be used to perform approximate analysis of the concrete LMP. We illustrate this process through a case study of a multi-room heating system that employs the probabilistic model checker PRISM.

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