Abstract. We define an enriched effect calculus by extending a type theory for computational effects with primitives from linear logic. The new calculus provides a formalism for expressing linear aspects of computational effects; for example, the linear usage of imperative features such as state and/or continuations. Our main syntactic result is the conservativity of the enriched effect calculus over a basic effect calculus without linear primitives (closely related to Moggi's computational metalanguage, Filinski's effect PCF and Levy's call-by-push-value). The proof of this syntactic theorem makes essential use of a category-theoretic semantics, whose study forms the second half of the paper. Our semantic results include soundness, completeness, the initiality of a syntactic model, and an embedding theorem: every model of the basic effect calculus fully embeds in a model of the enriched calculus. The latter means that our enriched effect calculus is applicable to arbitrary computational effects, answering in the positive a question of Benton and Wadler (LICS 1996).
This paper introduces the enriched effect calculus, which extends established type theories for computational effects with primitives from linear logic. The new calculus provides a formalism for expressing linear aspects of computational effects; for example, the linear usage of imperative features such as state and/or continuations.The enriched effect calculus is implemented as an extension of a basic effect calculus without linear primitives, which is closely related to Moggi's computational metalanguage, Filinski's effect PCF and Levy's call-by-push-value. We present syntactic results showing: the fidelity of the behaviour of the linear connectives of the enriched effect calculus; the conservativity of the enriched effect calculus over its non-linear core (the effect calculus); and the non-conservativity of intuitionistic linear logic when considered as an extension of the enriched effect calculus.The second half of the paper investigates models for the enriched effect calculus, based on enriched category theory. We give several examples of such models, relating them to models of standard effect calculi (such as those based on monads), and to models of intuitionistic linear logic. We also prove soundness and completeness.
Abstract. The enriched effect calculus is an extension of Moggi's computational metalanguage with a selection of primitives from linear logic. In this paper, we present an extended case study within the enriched effect calculus: the linear usage of continuations. We show that established call-by-value and call-by name linearly-used CPS translations are uniformly captured by a single generic translation of the enriched effect calculus into itself. As a main syntactic theorem, we prove that the generic translation is involutive up to isomorphism. As corollaries, we obtain full completeness results for the original call-by-value and callby-name translations. The main syntactic theorem is proved using a category-theoretic semantics for the enriched effect calculus. We show that models are closed under a natural dual model construction. The canonical linearly-used CPS translation then arises as the unique (up to isomorphism) map from the syntactic initial model to its own dual. This map is an equivalence of models. Thus the initial model is self-dual.
We discuss cyclic star-autonomous categories; that is, unbraided starautonomous categories in which the left and right duals of every object p are linked by coherent natural isomorphism. We settle coherence questions which have arisen concerning such cyclicity isomorphisms, and we show that such cyclic structures are the natural setting in which to consider enriched profunctors. Specifically, if V is a cyclic star-autonomous category, then the collection of V-enriched profunctors carries a canonical cyclic structure. In the case of braided star-autonomous categories, we discuss the correspondences between cyclic structures and balances or tortile structures. Finally, we show that every cyclic star-autonomous category is equivalent to one in which the cyclicity isomorphisms are identities.1 holds for every ω : s s. But, of course, symmetry can only occur when #s ≤ 1. (Here, ω rev denotes the reverse of ω which is more commonly denoted ω op .)More generally, 2-valued profunctors Ô Ô (where Ô = (p, ≤), some fixed but arbitrary poset) form a cyclic star-autonomous poset: the tensor product is the usual composition of profunctors, and the dualising object is the complement of the reverse ordering ( ). It is again routine to verify thatholds for every ω : Ô Ô. Again, symmetry can only occur when #Ô ≤ 1. (Observe that, in general, neither ¬ω nor ω rev is a profunctor Ô Ô, but that ¬ω rev is.)It is well-understood, at least in principle, that the term cyclic star-autonomous category should mean a star-autonomous category equipped with a coherent natural isomorphism p * −→ * p. But this raises the question: what are the right coherence axioms? This question is complicated by the fact that there is a second approach to the phenomenon of cyclicity which does not explicitly refer to dual objects. (This is not a new observation; on the contrary, the origin of the term cyclic is tied up in this approach-see again, [Yet90].)Since there are several equivalent definitions of star-autonomous category, let use make clear that we use the one advocated in [CS97]: a linearly distributive category with chosen left and right duals for every object. We generally use for tensor and for par ; the linear distributions q (s t) −→ (q s) t and (p q) s −→ p (q s) are denoted κ and κ, respectively.1.3. Remark. Let K = (K, , e, , d, ( ) * , * ( )) be a star-autonomous category and let s, t K denote the external set of arrows s −→ t; then natural isomorphisms of the form p * −→ * p are in bijective correspondence with those of the form p t, d K −→ t p, d K . We shall denote this correspondence (summarised below) by a change of case: lower case for natural isomorphisms of the first form, upper case for those of the second. t lCurry(ω) } } z z z z z z z z z z rCurry(ψ) ! !
Abstract. The enriched effect calculus (EEC) is an extension of Moggi's computational metalanguage with a selection of primitives from linear logic. This paper explores the enriched effect calculus as a target language for continuation-passing-style (CPS) translations in which the typing of the translations enforces the linear usage of continuations. We first observe that established call-by-value and call-by name linear-use CPS translations of simply-typed lambda-calculus into intuitionistic linear logic (ILL) land in the fragment of ILL given by EEC. These two translations are uniformly generalised by a single generic translation of the enriched effect calculus into itself. As our main theorem, we prove that the generic self-translation of EEC is involutive up to isomorphism. As corollaries, we obtain full completeness results, both for the generic translation, and for the original call-by-value and call-by-name translations.
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