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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.

Environmental bisimulations for probabilistic higher-order languages are studied. In contrast with applicative bisimulations, environmental bisimulations are known to be more robust and do not require sophisticated techniques such as Howe's in the proofs of congruence. As representative calculi, call-by-name and call-by-value λcalculus, and a (call-by-value) λ-calculus extended with references (i.e., a store) are considered. In each case full abstraction results are derived for probabilistic environmental similarity and bisimilarity with respect to contextual preorder and contextual equivalence, respectively. Some possible enhancements of the (bi)simulations, as 'up-to techniques', are also presented. Probabilities force a number of modifications to the definition of environmental bisimulations in non-probabilistic languages. Some of these modifications are specific to probabilities, others may be seen as general refinements of environmental bisimulations, applicable also to non-probabilistic languages. Several examples are presented, to illustrate the modifications and the differences.

Abstract. We study how applicative bisimilarity behaves when instantiated on a call-by-value probabilistic λ-calculus, endowed with Plotkin's parallel disjunction operator. We prove that congruence and coincidence with the corresponding context relation hold for both bisimilarity and similarity, the latter known to be impossible in sequential languages.

Environmental bisimulations for probabilistic higher-order languages are studied. In contrast with applicative bisimulations, environmental bisimulations are known to be more robust and do not require sophisticated techniques such as Howe's in the proofs of congruence.As representative calculi, call-by-name and call-by-value λcalculus, and a (call-by-value) λ-calculus extended with references (i.e., a store) are considered. In each case full abstraction results are derived for probabilistic environmental similarity and bisimilarity with respect to contextual preorder and contextual equivalence, respectively. Some possible enhancements of the (bi)simulations, as 'up-to techniques', are also presented.Probabilities force a number of modifications to the definition of environmental bisimulations in non-probabilistic languages. Some of these modifications are specific to probabilities, others may be seen as general refinements of environmental bisimulations, applicable also to non-probabilistic languages. Several examples are presented, to illustrate the modifications and the differences.

This paper studies the discriminating power offered by higherorder concurrent languages, and contrasts this power with those offered by higher-order sequential languages (à la λ-calculus) and by first-order concurrent languages (à la CCS). The concurrent higherorder languages that we focus on are Higher-Order π-calculus (HOπ), which supports higher-order communication, and an extension of HOπ with passivation, a simple higher-order construct that allows one to obtain location-dependent process behaviours.The comparison is carried out by providing embeddings of firstorder processes into the various languages, and then examining the resulting contextual equivalences induced on such processes. As first-order processes we consider both ordinary Labeled Transition Systems (LTSs) and Reactive Probabilistic Labeled Transition Systems (RPLTSs).The hierarchy of discriminating powers so obtained for RPLTSs is finer than that for LTSs. For instance, in the LTS case, the additional discriminating power offered by passivation in concurrency is captured, in sequential languages, by the difference between the call-by-name and call-by-value evaluation strategies of an extended typed λ-calculus.

Allegories were introduced by Freyd and Scedrov; they form a fragment of Tarski's calculus of relations. We show that their equational theory is decidable by characterising it in terms of a specific class of graph homomorphisms. We actually do so for an extension of allegories which we prove to be conservative: allegories with top. This makes it possible to exploit a correspondence between terms and K 4-free graphs, for which isomorphisms were known to be finitely axiomatisable.

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