The shift and reset operators, proposed by Danvy and Filinski, are powerful control primitives for capturing delimited continuations. Delimited continuation is a similar concept as the standard (unlimited) continuation, but it represents part of the rest of the computation, rather than the whole rest of computation. In the literature, the semantics of shift and reset has been given by a CPS-translation only. This paper gives a direct axiomatization of calculus with shift and reset, namely, we introduce a set of equations, and prove that it is sound and complete with respect to the CPS-translation. We also introduce a calculus with control operators which is as expressive as the calculus with shift and reset, has a sound and complete axiomatization, and is conservative over Sabry and Felleisen's theory for first-class continuations.

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). 1

Abstract. We propose a semantic framework for modelling the linear usage of continuations in typed call-by-name programming languages. On the semantic side, we introduce a construction for categories of linear continuations, which gives rise to cartesian closed categories with “linear classical disjunctions ” from models of intuitionistic linear logic with sums. On the syntactic side, we give a simply typed call-by-name λµcalculus in which the use of names (continuation variables) is restricted to be linear. Its semantic interpretation into a category of linear continuations then amounts to the call-by-name continuation-passing style (CPS) transformation into a linear lambda calculus with sum types. We show that our calculus is sound for this CPS semantics, hence for models given by the categories of linear continuations.

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. 1

Abstract. We present a rewriting system for the linear lambda calculus corresponding to the {!, ⊸}-fragment of intuitionistic linear logic. This rewriting system is shown to be strongly normalizing, and Church-Rosser modulo the trivial commuting conversion. Thus it provides a simple decision method for the equational theory of the linear lambda calculus. As an application we prove the strong normalization of the simply typed computational lambda calculus by giving a reduction-preserving translation into the linear lambda calculus. 1

Continuations can be used to explain a wide variety of control behaviours, including calling/returning (procedures), raising/handling (exceptions), labelled jumping (goto statements), process switching (coroutines), and backtracking. However, continuations are often manipulated in a highly stylised way, and we show that all of these, bar backtracking, in fact use their continuations linearly ; this is formalised by taking a target language for cps transforms that has both intuitionistic and linear function types.

We show that the f!;(g-fragment of Intuitionistic Linear Logic is full in the f!; (g-fragment, both formulated as linear lambda calculi. The proof is a mild extension of our previous technique used for showing the fullness of Girard's translation from Intuitionistic Logic into Intuitionistic Linear Logic, and makes use of double-parameterized logical predicates.

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. 1.

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. 1