Analysis of Probabilistic Basic Parallel Processes

Bonnet, Rémi; Kiefer, Stefan; Lin, Anthony W.

doi:10.1007/978-3-642-54830-7_3

Cited by 4 publications

(7 citation statements)

References 19 publications

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“…For claim (2), suppose that the algorithm accepts a guess σ, τ and outputsg = v σ . Since v σ ≤ g * σ, * ≤ g * ≤ q * * ,τ , we have:…”

Section: Approximating the Value Of Bssgs And The Gfp Of Max-minppssmentioning

confidence: 99%

“…We then show that one can approximate the GFP solution g * ∈ [0, 1] n of a maxPPS (or minPPS), x = P (x), in time polynomial in both the encoding size |P | of the system of equations and in log(1/ǫ), where ǫ > 0 is the desired additive error bound of the solution. 2 (The model of computation is the standard Turing machine model.) We also show that the qualitative analysis of determining the coordinates of the GFP that are 0 and 1, can be done in P-time (and hence the same holds for the optimal reachability probabilities of BMDPs).…”

Section: Introductionmentioning

confidence: 99%

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Greatest fixed points of probabilistic min/max polynomial equations, and reachability for branching Markov decision processes

Etessami

Stewart

Yannakakis

2018

Information and Computation

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We give polynomial time algorithms for quantitative (and qualitative) reachability analysis for Branching Markov Decision Processes (BMDPs). Specifically, given a BMDP, and given an initial population, where the objective of the controller is to maximize (or minimize) the probability of eventually reaching a population that contains an object of a desired (or undesired) type, we give algorithms for approximating the supremum (infimum) reachability probability, within desired precision ǫ > 0, in time polynomial in the encoding size of the BMDP and in log(1/ǫ). We furthermore give P-time algorithms for computing ǫ-optimal strategies for both maximization and minimization of reachability probabilities. We also give P-time algorithms for all associated qualitative analysis problems, namely: deciding whether the optimal (supremum or infimum) reachability probabilities are 0 or 1. Prior to this paper, approximation of optimal reachability probabilities for BMDPs was not even known to be decidable.Our algorithms exploit the following basic fact: we show that for any BMDP, its maximum (minimum) non-reachability probabilities are given by the greatest fixed point (GFP) solution g * ∈ [0, 1] n of a corresponding monotone max (min) Probabilistic Polynomial System of equations (max/minPPS), x = P (x), which are the Bellman optimality equations for a BMDP with non-reachability objectives. We show how to compute the GFP of max/minPPSs to desired precision in P-time.We also study more general branching simple stochastic games (BSSGs) with (non-)reachability objectives. We show that: (1) the value of these games is captured by the GFP, g * , of a corresponding max-minPPS, x = P (x); (2) the quantitative problem of approximating the value is in TFNP; and (3) the qualitative problems associated with the value are all solvable in P-time.

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“…For claim (2), suppose that the algorithm accepts a guess σ, τ and outputsg = v σ . Since v σ ≤ g * σ, * ≤ g * ≤ q * * ,τ , we have:…”

Section: Approximating the Value Of Bssgs And The Gfp Of Max-minppssmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Greatest fixed points of probabilistic min/max polynomial equations, and reachability for branching Markov decision processes

Etessami

Stewart

Yannakakis

2018

Information and Computation

View full text Add to dashboard Cite

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“…Probabilistic models of concurrent processes. The design of computational models that combine concurrency and probabilities has a long tradition [54,49] and gave birth to a variety of operational approaches [50] and concrete probabilistic extensions of well-known concurrency models, such as CCS [27], CSP [40,25], Petri nets [9], Klaim [19], and name-passing process calculi [28,53,42,26]. Our language for processes can be seen as the session-based counterpart of (a synchronous version of) the simple probabilistic π-calculus [42], which features both probabilistic and non-deterministic choices.…”

Section: Related Workmentioning

confidence: 99%

Probabilistic Analysis of Binary Sessions

Inverso¹,

Melgratti²,

Padovani³

et al. 2020

Preprint

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We study a probabilistic variant of binary session types that relate to a class of Finite-State Markov Chains. The probability annotations in session types enable the reasoning on the probability that a session terminates successfully, for some user-definable notion of successful termination. We develop a type system for a simple session calculus featuring probabilistic choices and show that the success probability of well-typed processes agrees with that of the sessions they use. To this aim, the type system needs to track the propagation of probabilistic choices across different sessions. ACM Subject Classification Theory of computation → Type structuresKeywords and phrases Probabilistic choices; session types; static analysis; deadlock freedom.

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“…Moran's process [6]). The standard notion of fairness gives rise to a rather unintuitive and unrealistic strategy for the scheduler, which could delay an enabled process for as long as it desires while still being fair (see [21,Example 8] and the Herman's protocol example in Section 3). For this reason, we propose to consider Alur & Henzinger's [22] finitary fairness -a stronger notion of fairness that allows the scheduler to delaying executing an enabled process in an infinite run for at most k steps, for some unknown but fixed bound k ∈ N. Alur & Henzinger argued that this fairness notion is more realistic in practice, but it is not as restrictive as the notion of k-fairness, which fixes the bound k a priori.…”

Section: Introductionmentioning

confidence: 99%

Fair Termination for Parameterized Probabilistic Concurrent Systems

Lengál

Lin

Majumdar

et al. 2017

Tools and Algorithms for the Construction and Analysis of Systems

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Abstract. We consider the problem of automatically verifying that a parameterized family of probabilistic concurrent systems terminates with probability one for all instances against adversarial schedulers. A parameterized family defines an infinite-state system: for each number n, the family consists of an instance with n finite-state processes. In contrast to safety, the parameterized verification of liveness is currently still considered extremely challenging especially in the presence of probabilities in the model. One major challenge is to provide a sufficiently powerful symbolic framework. One well-known symbolic framework for the parameterized verification of non-probabilistic concurrent systems is regular model checking. Although the framework was recently extended to probabilistic systems, incorporating fairness in the framework -often crucial for verifying termination -has been especially difficult due to the presence of an infinite number of fairness constraints (one for each process). Our main contribution is a systematic, regularity-preserving, encoding of finitary fairness (a realistic notion of fairness proposed by Alur & Henzinger) in the framework of regular model checking for probabilistic parameterized systems. Our encoding reduces termination with finitary fairness to verifying parameterized termination without fairness over probabilistic systems in regular model checking (for which a verification framework already exists). We show that our algorithm could verify termination for many interesting examples from distributed algorithms (Herman's protocol) and evolutionary biology (Moran process, cell cycle switch), which do not hold under the standard notion of fairness. To the best of our knowledge, our algorithm is the first fully-automatic method that can prove termination for these examples.

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Analysis of Probabilistic Basic Parallel Processes

Cited by 4 publications

References 19 publications

Greatest fixed points of probabilistic min/max polynomial equations, and reachability for branching Markov decision processes

Greatest fixed points of probabilistic min/max polynomial equations, and reachability for branching Markov decision processes

Probabilistic Analysis of Binary Sessions

Fair Termination for Parameterized Probabilistic Concurrent Systems

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