Intuitionistic linear logic regains the expressive power of intuitionistic logic through the ! (`of course') modality. Benton, Bierman, Hyland and de Paiva have given a term assignment system for ILL and an associated notion of categorical model in which the ! modality is modelled by a comonad satisfying certain extra conditions. Ordinary intuitionistic logic is then modelled in a cartesian closed category which arises as a full subcategory of the category of coalgebras for the comonad. This paper attempts to explain the connection between ILL and IL more directly and symmetrically by giving a logic, term calculus and categorical model for a system in which the linear and non-linear worlds exist on an equal footing, with operations allowing one to pass in both directions. We start from the categorical model of ILL given by Benton, Bierman, Hyland and de Paiva and show that this is equivalent to having a symmetric monoidal adjunction between a symmetric monoidal closed category and a cartesian closed category. We then derive both a sequent calculus and a natural deduction presentation of the logic corresponding to the new notion of model.

Moggi's computational lambda calculus is a metalanguage for denotational semantics which arose from the observation that many different notions of computation have the categorical structure of a strong monad on a cartesian closed category. In this paper we show that the computational lambda calculus also arises naturally as the term calculus corresponding (by the Curry-Howard correspondence) to a novel intuitionistic modal propositional logic. We give natural deduction, sequent calculus and Hilbert-style presentations of this logic and prove a strong normalisation result. 1 Introduction The computational lambda calculus was introduced by Moggi as a metalanguage for denotational semantics which more faithfully models real programming language features such as non-termination, differing evaluation strategies, non-determinism and side-effects than does the ordinary simply typed lambda calculus [17, 18]. The starting point for Moggi's work is an explicit semantic distinction between compu...

Models of intuitionistic linear logic also provide models of Moggi's computational metalanguage. We use the adjoint presentation of these models and the associated adjoint calculus to show that three translations, due mainly to Moggi, of the lambda calculus into the computational metalanguage (direct, call-by-name and call-by-value) correspond exactly to three translations, due mainly to Girard, of intuitionistic logic into intuitionistic linear logic. We also consider extending these results to languages with recursion. 1. Introduction Two of the most significant developments in semantics during the last decade are Girard's linear logic [10] and Moggi's computational metalanguage [14]. Any student of these formalisms will suspect that there are significant connections between the two, despite their apparent differences. The intuitionistic fragment of linear logic (ILL) may be modelled in a linear model -- a symmetric monoidal closed category with a comonad ! which satisfies some extr...

. We study the longstanding problem of semantics for input /output (I/O) expressed using side-effects. Our vehicle is a small higher-order imperative language, with operations for interactive character I/O and based on ML syntax. Unlike previous theories, we present both operational and denotational semantics for I/O effects. We use a novel labelled transition system that uniformly expresses both applicative and imperative computation. We make a standard definition of bisimilarity and prove it is a congruence using Howe's method. Next, we define a metalogical type theory M in which we may give a denotational semantics to O. M generalises Crole and Pitts' FIX-logic by adding in a parameterised recursive datatype, which is used to model I/O. M comes equipped both with judgements of equality of expressions, and an operational semantics; M itself is given a domain-theoretic semantics in the category CPPO of cppos (bottom-pointed posets with joins of !-chains) and Scott continuous functions...

We show that an enriched version of Freyd's principle of versality holds in the Kleisli category of a commutative strong monad with fixed-point object. This gives a general categorical setting in which it is possible to model recursive types involving the usual datatype constructors.

We study the longstanding problem of semantics for input/output (I/O) expressed using side-effects. Our vehicle is a small higher-order imperative language, with operations for interactive character I/O and based on ML syntax. Unlike previous theories, we present both operational and denotational semantics for I/O effects. We use a novel labelled transition system that uniformly expresses both applicative and imperative computation. We make a standard definition of bisimilarity. We prove bisimilarity is a congruence using Howe's method. Next, we define a metalanguage M in which we may give a denotational semantics to O. M generalises Crole and Pitts' FIX-logic by adding in a parameterised recursive datatype, which is used to model I/O. M comes equipped both with an operational semantics and a domain-theoretic semantics in the category CPPO of cppos (bottom-pointed posets with joins of !-chains) and Scott continuous functions. We use the CPPO semantics to prove that M is computationally...

This article 1 has two parts: In the first part, we present some general results about fixpoint objects. The minimal categorical structure required to model soundly the equational type theory which combines higher order recursion and computation types (introduced by [4]) is shown to be precisely a let-category possessing a fixpoint object. Functional completeness for such categories is developed. We also prove that categories with fixpoint operators do not necessarily have a fixpoint object. In the second part, we extend Freyd's gluing construction for cartesian closed categories to cartesian closed let-categories, and observe that this extension does not obviously apply to categories possessing fixpoint objects. We solve this problem by giving a new gluing construction for a limited class of categories with fixpoint objects; this is the main result of the paper. We use this category-theoretic construction to prove a type-theoretic conservative extension result. A version of this pap...

This paper presents computational adequacy results for the FIX logical system introduced by Crole and Pitts in LICS '90. More precisely, we take two simple PCF style languages (whose dynamic semantics follow a call-by-value and call-by-name regime) give translations of the languages into suitable judgements in the FIX-logic and prove that the translations are adequate for the static and dynamic semantics. This shows that the FIX-logic can be regarded as a programming metalogic which will uniformly interpret both call-by-value and call-by-name languages. The proofs of dynamic adequacy make use of a logical relations technique which is based on the methods of Plotkin and Tait. We also show that there is some choice in the translation of recursion; certain translations make use of an existence property of the FIX-logic to prove computational adequacy.

We present an inference system for translating programs in a PCF-like source language into a variant of Moggi's computational lambda calculus. This translation combines a simple strictness analysis with its associated optimising transformations into a single system. The correctness of the translation is established using a logical relation between the denotational semantics of the source and target languages. 1 Introduction 1.1 Background Strictness analysis of lazy functional programs has been studied extensively during the the last 15 years or so, usually with the justification that the results of the analysis can be used in an optimising compiler [Myc81, BHA86, Ben92]. There has, however, been surprisingly little serious work on just how the results of strictness analysis can be used as the basis for optimising transformations. This is probably because it turns out to be rather more difficult to express and justify these optimisations than one might at first imagine. Roughly speak...

This work expounds the notion that (structured) categories are syntax free presentations of type theories, and shows some of the ideas involved in deriving categorical semantics for given type theories. It is intended for someone who has some knowledge of category theory and type theory, but who does not fully understand some of the intimate connections between the two topics. We begin by showing how the concept of a category can be derived from some simple and primitive mechanisms of monadic type theory. We then show how the notion of a category with finite products can model the most fundamental syntactical constructions of (algebraic) type theory. The idea of naturality is shown to capture, in a syntax free manner, the notion of substitution, and therefore provides a syntax free coding of a multiplicity of type theoretical constructs. Using these ideas we give a direct derivation of a cartesian closed category as a very general model of simply typed λ-calculus with binary products and a unit type. This article provides a new presentation of some old ideas. It is intended to be a tutorial paper aimed at audiences interested in elementary categorical type theory. Further details can be found in [Cro93]. 1 1