A general account of coinduction up-to

Bonchi, Filippo; Petrişan, Daniela; Pous, Damien; Rot, Jurriaan

doi:10.1007/s00236-016-0271-4

Cited by 33 publications

(47 citation statements)

References 50 publications

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“…Theorem 13 ( [30], [58]). The following properties hold for any coalgebra c : X→F M X and its determinisation [38] proof technique.…”

Section: Generalised Determinisationmentioning

confidence: 99%

“…for all a ∈ A. Coinduction tells us (see e.g. [58]) that for all x, y ∈ X, x ≡ i y iff there exists a bisimulation R such that x R y.…”

Section: Appendix a Coinduction Up-tomentioning

confidence: 99%

“…By virtue of the general theory in [58], one has that Ctx is a sound up-to technique, that is x ≡ i y iff there exists a bisimulation up-to context R such that x R y. Actually, the theory in [58] guarantees a stronger property known as compatibility [38], [62], [63]. Intuitively, this mean that the technique is sound and it can be safely combined with other compatible up-to techniques.…”

Section: Appendix a Coinduction Up-tomentioning

confidence: 99%

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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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“…Theorem 13 ( [30], [58]). The following properties hold for any coalgebra c : X→F M X and its determinisation [38] proof technique.…”

Section: Generalised Determinisationmentioning

confidence: 99%

“…for all a ∈ A. Coinduction tells us (see e.g. [58]) that for all x, y ∈ X, x ≡ i y iff there exists a bisimulation R such that x R y.…”

Section: Appendix a Coinduction Up-tomentioning

confidence: 99%

Section: Appendix a Coinduction Up-tomentioning

confidence: 99%

See 1 more Smart Citation

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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“…Compatibility with b of up-to context and up-to bisimilarity hold immediately by the results in [23]. For up-to substitutions, we will next prove compatibility (Theorem 6.2).…”

Section: Bisimulation Up-to Substitutionsmentioning

confidence: 70%

“…This fact also entails that up-to substitutions is not compatible. Indeed, following the general theory in [23], if it would be compatible, then open bisimilarity would be closed under substitution.…”

Section: Appendix D Non-compatibility For Non-monadic Specificationsmentioning

confidence: 99%

Bisimilarity of open terms in stream GSOS

Bonchi

Bussel

Lee

et al. 2019

Science of Computer Programming

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Stream GSOS is a specification format for operations and calculi on infinite sequences. The notion of bisimilarity provides a canonical proof technique for equivalence of closed terms in such specifications. In this paper, we focus on open terms, which may contain variables, and which are equivalent whenever they denote the same stream for every possible instantiation of the variables. Our main contribution is to capture equivalence of open terms as bisimilarity on certain Mealy machines, providing a concrete proof technique. Moreover, we introduce an enhancement of this technique, called bisimulation up-to substitutions, and show how to combine it with other up-to techniques to obtain a powerful method for proving equivalence of open terms.

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(In)finite Trace Equivalence of Probabilistic Transition Systems

Goy

Rot

2018

Coalgebraic Methods in Computer Science

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We show how finite and infinite trace semantics of generative probabilistic transition systems arises through a determinisation construction. This enables the use of bisimulations (up-to) to prove equivalence. In particular, it follows that trace equivalence for finite probabilistic transition systems is decidable. Further, the determinisation construction applies to both discrete and continuous probabilistic systems.The research leading to these results has received funding from the European Research Council (FP7/2007(FP7/ -2013, and the Erasmus+ program. It was carried out during the first author's master internship at Radboud University, reported in [9].

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A general account of coinduction up-to

Cited by 33 publications

References 50 publications

The Theory of Traces for Systems with Nondeterminism and Probability

The Theory of Traces for Systems with Nondeterminism and Probability

Bisimilarity of open terms in stream GSOS

(In)finite Trace Equivalence of Probabilistic Transition Systems

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