Probabilistic Bisimulation: Naturally on Distributions

Hermanns, Holger; Krčál, Jan; Křetínský, Jan

doi:10.1007/978-3-662-44584-6_18

Cited by 28 publications

(37 citation statements)

References 55 publications

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“…Systems exhibiting both nondeterministic and probabilistic behaviour are abundantly used in verification [1], [2], [3], [4], [5], [6], [7], AI [8], [9], [10], and studied from semantics perspective [11], [12], [13]. Probability is needed to quantitatively model uncertainty and belief, whereas nondeterminism enables modelling of incomplete information, unknown environment, implementation freedom, or concurrency.…”

Section: Introductionmentioning

confidence: 99%

The Theory of Traces for Systems with Nondeterminism and Probability

Bonchi

Sokolova

Vignudelli

2019

2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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This paper studies trace-based equivalences for systems combining nondeterministic and probabilistic choices. We show how trace semantics for such processes can be recovered by instantiating a coalgebraic construction known as the generalised powerset construction. We characterise and compare the resulting semantics to known definitions of trace equivalences appearing in the literature. Most of our results are based on the exciting interplay between monads and their presentations via algebraic theories. Monads and Algebraic TheoriesIn this paper, on the algebraic side, we deal with Eilenberg-Moore algebras of a monad on the category Sets of sets and functions, for which we also give presentations in terms of operations and equations, i.e., algebraic theories. MonadsA monad on Sets is a functor M : Sets → Sets together with two natural transformations: a unit η : Id ⇒ M and multiplication µ :We next introduce several monads on Sets, relevant to this paper. Each monad can be seen as giving side-effects.Nondeterminism. The finite powerset monad P maps a set X to its finite powerset PX = {U | U ⊆ X, U is finite} and a function f :The unit η of P is given by singleton, i.e., η(x) = {x} and the multiplication µ is given by union, i.e., µ(S) = U∈S U for S ∈ PPX. Of particular interest to us in this paper is the submonad P ne of non-empty finite subsets, that acts on functions just like the (finite) powerset monad, and has the same unit and multiplication. We rarely mention the unrestricted (not necessarily finite) powerset monad, which we denote by P u . We sometimes write f for P u f in this paper.Probability. The finitely supported probability distribution monad D is defined, for a set X and a function f : X → Y , asThe support set of a distribution ϕ ∈ DX is supp(ϕ) = {x ∈ X | ϕ(x) = 0}. The unit of D is given by a Dirac distribution η(x) = δ x = (x → 1) for x ∈ X and the multiplication by µ(Φ)(x) = ϕ∈supp(Φ) Φ(ϕ) · ϕ(x) for Φ ∈ DDX. We sometimes write i∈I p i x i for a distribution ϕ with supp(ϕ) = {x i | i ∈ I} and ϕ(x i ) = p i . 1. Personal communication with Gordon Plotkin.

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Section: Introductionmentioning

confidence: 99%

The Theory of Traces for Systems with Nondeterminism and Probability

Bonchi

Sokolova

Vignudelli

2019

2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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“…However, we enrich the states with an output weight that is used in the definition of the language semantics. Our language semantics is coarser than probabilistic (convex) bisimilarity [26] and bisimilarity on distributions [16]. In fact, in contrast to the hardness and undecidability results we proved for probabilistic language equivalence, bisimilarity on distributions can be shown to be decidable [16] with the help of convexity.…”

Section: Discussionmentioning

confidence: 89%

Convex Language Semantics for Nondeterministic Probabilistic Automata

Heerdt

Hsu

Ouaknine

et al. 2018

Theoretical Aspects of Computing – ICTAC 2018

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We explore language semantics for automata combining probabilistic and nondeterministic behavior. We first show that there are precisely two natural semantics for probabilistic automata with nondeterminism. For both choices, we show that these automata are strictly more expressive than deterministic probabilistic automata, and we prove that the problem of checking language equivalence is undecidable by reduction from the threshold problem. However, we provide a discounted metric that can be computed to arbitrarily high precision.

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“…The algorithm in [24] is exponential in the worst case. We will work out whether or not more efficient algorithms exist.…”

Section: Discussionmentioning

confidence: 99%

“…By doing so, all weak transitions can be handled in the same way as non-deterministic strong transitions in [24]. Not surprisingly, this will cause an exponential blow-up.…”

Section: Properties Of Late Distribution Bisimilaritymentioning

confidence: 96%

Probabilistic Bisimulation for Realistic Schedulers

Eisentraut

Godskesen

Hermanns

et al. 2015

FM 2015: Formal Methods

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Abstract. Weak distribution bisimilarity is an equivalence notion on probabilistic automata, originally proposed for Markov automata. It has gained some popularity as the coarsest behavioral equivalence enjoying valuable properties like preservation of trace distribution equivalence and compositionality. This holds in the classical context of arbitrary schedulers, but it has been argued that this class of schedulers is unrealistically powerful. This paper studies a strictly coarser notion of bisimilarity, which still enjoys these properties in the context of realistic subclasses of schedulers: Trace distribution equivalence is implied for partial information schedulers, and compositionality is preserved by distributed schedulers. The intersection of the two scheduler classes thus spans a coarser and still reasonable compositional theory of behavioral semantics.

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Probabilistic Bisimulation: Naturally on Distributions

Cited by 28 publications

References 55 publications

The Theory of Traces for Systems with Nondeterminism and Probability

The Theory of Traces for Systems with Nondeterminism and Probability

Convex Language Semantics for Nondeterministic Probabilistic Automata

Probabilistic Bisimulation for Realistic Schedulers

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