We introduce the type theory ¯ v , a call-by-value variant of Parigot's ¯-calculus, as a Curry-Howard representation theory of classical propositional proofs. The associated rewrite system is Church-Rosser and strongly normalizing, and definitional equality of the type theory is consistent, compatible with cut, congruent and decidable. The attendant call-by-value programming language ¯pcf v is obtained from ¯ v by augmenting it by basic arithmetic, conditionals and fixpoints. We study the behavioural properties of ¯pcf v and show that, though simple, it is a very general language for functional computation with control: it can express all the main control constructs such as exceptions and first-class continuations. Proof-theoretically the dual ¯ v -constructs of naming and ¯-abstraction witness the introduction and elimination rules of absurdity respectively. Computationally they give succinct expression to a kind of generic (forward) "jump" operator, which may be regarded as a unif...

This paper considers the consequences of relaxing the bracketing condition on `dialogue games', showing that this leads to a category of games which can be `factorized' into a well-bracketed substructure, and a set of classically typed morphisms. These are shown to be sound denotations for control operators, allowing the factorization to be used to extend the definability result for PCF to one for PCF with control operators at atomic types. Thus we define a fully abstract and effectively presentable model of a functional language with non-local control as part of a modular approach to modelling non-functional features using games. 1.

This thesis examines the use of denotational semantics to reason about control flow in sequential, basically functional languages. It extends recent work in game semantics, in which programs are interpreted as strategies for computation by interaction with an environment. Abramsky has suggested that an intensional hierarchy of computational features such as state, and their fully abstract models, can be captured as violations of the constraints on strategies in the basic functional model. Non-local control flow is shown to fit into this framework as the violation of strong and weak ‘bracketing ’ conditions, related to linear behaviour. The language µPCF (Parigot’s λµ with constants and recursion) is adopted as a simple basis for higher-type, sequential computation with access to the flow of control. A simple operational semantics for both call-by-name and call-by-value evaluation is described. It is shown that dropping the bracketing condition on games models of PCF yields fully abstract models of µPCF.

Connections between the sequentiality/concurrency distinction and the semantics of proofs are investigated, with particular reference to games and Linear Logic.

Abstract. We consider the relation of the dual calculus of Wadler (2003) to the λµ-calculus of Parigot (1992). We give translations from the λµ-calculus into the dual calculus and back again. The translations form an equational correspondence as defined by Sabry and Felleisen (1993). In particular, translating from λµ to dual and then ‘reloading ’ from dual back into λµ yields a term equal to the original term. Composing the translations with duality on the dual calculus yields an involutive notion of duality on the λµ-calculus. A previous notion of duality on the λµcalculus has been suggested by Selinger (2001), but it is not involutive. Note This paper uses color to clarify the relation of types and terms, and of source and target calculi. If the URL below is not in blue please download the color version from

This paper proposes and studies a typed λ-calculus for classical linear logic. I shall give an explanation of a multiple-conclusion formulation for classical logic due to Parigot and compare it to more traditional treatments by Prawitz and others. I shall use Parigot's method to devise a natural deduction formulation of classical linear logic. This formulation is compared in detail to the sequent calculus formulation. In an appendix I shall also demonstrate a somewhat hidden connexion with the paradigm of control operators for functional languages which gives a new computational interpretation of Parigot's techniques.

We derive a confluent -calculus with a catch/throw mechanism (called ct -calculus) from M. Parigot's -calculus. We also present several translations from one calculus into the other which are morphisms for the reduction. We use them to show that the ct-calculus is a retract of -calculus (these calculi are isomorphic if we consider only convertibility). As a by-product, we obtain the subject reduction property for the ct -calculus, as well as the strong normalization for ct-terms typable in the second order classical natural deduction. 1 Introduction In the last four years, several extensions of the -calculus with some catch/throw mechanism have been proposed by H. Nakano (1994a; 1994b; 1995) and by M. Sato (1997) and Y. Kameyama (1997; 1998). In these papers, the authors consider the catch/throw mechanism as "intrinsically non-deterministic" and thus investigate non-confluent calculi or confine themselves to some specific evaluation strategy. For instance in (Nakano, 1994b), the non-...

Abstract. We study the equational theory of Parigot’s second-order λµ-calculus in connection with a call-by-name continuation-passing style (CPS) translation into a fragment of the second-order λ-calculus. It is observed that the relational parametricity on the target calculus induces a natural notion of equivalence on the λµ-terms. On the other hand, the unconstrained relational parametricity on the λµ-calculus turns out to be inconsistent. Following these facts, we propose to formulate the relational parametricity on the λµ-calculus in a constrained way, which might be called “focal parametricity”. Dedicated to Prof. Gordon Plotkin on the occasion of his sixtieth birthday 1.

We introduce a family of type theories as internal languages for autonomous (or symmetric monoidal closed) and -autonomous categories, in the same sense that simply-typed -calculus (augmented by appropriate constructs for products and the terminal object) is the internal language for cartesian closed categories. The rules are presented in the style of Gentzen's Sequent Calculus. A key feature is the systematic treatment of naturality conditions by explicitly representing the categorical composition, or cut in the type theory, by explicit substitution, and the introduction of new let-constructs, one for each of the three type constructors ?;\Omega and (, and a Parigot-style ¯-abstraction to give expression to the involutive negation. The commutation congruences of these theories are precisely those imposed by the naturality conditions. In particular the type theory for -autonomous categories may be regarded as a term assignment system for the multiplicative (\Omega ; (;?;?)-fragmen...

. There have traditionally been two approaches to modelling environments, one by use of ønite products in Cartesian closed categories, the other by use of the base categories of indexed categories with structure. Recently, there have been more general deønitions along both of these lines: the ørst generalising from Cartesian to symmetric premonoidal categories, the second generalising from indexed categories with speciøed structure to -categories. The added generality is not of the purely mathematical kind; in fact it is necessary to extend semantics from the logical calculi studied in, say, Type Theory to more realistic programming language fragments. In this paper, we establish an equivalence between these two recent notions. We then use that equivalence to study semantics for continuations. We give three category theoretic semantics for modelling continuations and show the relationships between them. The ørst is given by a continuations monad. The second is based on a symmetric prem...