Abstract. We propose ⊤⊤-lifting as a technique for extending operational predicates to Moggi’s monadic computation types, independent of the choice of monad. We demonstrate the method with an application to Girard-Tait reducibility, using this to prove strong normalisation for the computational metalanguage λml. The particular challenge with reducibility is to apply this semantic notion at computation types when the exact meaning of “computation ” (stateful, side-effecting, nondeterministic, etc.) is left unspecified. Our solution is to define reducibility for continuations and use that to support the jump from value types to computation types. The method appears robust: we apply it to show strong normalisation for the computational metalanguage extended with sums, and with exceptions. Based on these results, as well as previous work with local state, we suggest that this “leap-frog ” approach offers a general method for raising concepts defined at value types up to observable properties of computations. 1

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 define a differential λµ-calculus which is an extension of both Parigot’s λµ-calculus and Ehrhard-Régnier’s differential λ-calculus. We prove some basic properties of the system: reduction enjoys Church-Rosser and simply typed terms are strongly normalizing. Contents 1

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Termination for classical natural deduction is difficult in the presence of commuting/permutative conversions for disjunction. An approach based on reducibility candidates is presented that uses non-strictly positive inductive definitions. It covers second-order universal quantification and also the extension of the logic by fixed-points of non-strictly positive operators, which appears to be a new result. Finally, the relation to Parigot’s strictly-positive inductive definition of his set of reducibility candidates and to his notion of generalized reducibility candidates is explained. Key words: PACS:

Abstract. This paper revisits the results of Barendregt and Ghilezan [3] and generalizes them for classical logic. Instead of λ-calculus, we use here λµ-calculus as the basic term calculus. We consider two extensionally equivalent type assignment systems for λµ-calculus, one corresponding to classical natural deduction, and the other to classical sequent calculus. Their relations and normalisation properties are investigated. As a consequence a short proof of Cut elimination theorem is obtained.

. A new proof of strong normalization of Parigot's (second order) -calculus is given by a reduction-preserving embedding into system F (second order polymorphic -calculus). The main idea is to use the least stable supertype for any type. These non-strictly positive inductive types and their associated iteration principle are available in system F, and allow to give a translation vaguely related to CPS translations (corresponding to the Kolmogorov embedding of classical logic into intuitionistic logic). However, they simulate Parigot's -reductions whereas CPS translations hide them. As a major advantage, this embedding does not use the idea of reducing stability (:: ! ) to that for atomic formulae. Therefore, it even extends to non-interleaving positive xed-point types. As a non-trivial application, strong normalization of -calculus, extended by primitive recursion on monotone inductive types, is established. 1 Introduction -calculus [12] essentially is the extension of nat...

We define a very natural restriction of the λµ-calculus which is stable under reduction and whose type system is a restriction of the Classical Natural Deduction to intuitionistic logic. However, we show that this system is in some sense degenerated unless we provide a native disjunction. We prove that the system with native disjunction is conservative over DIS-logic and also that DIS-logic is constructive. From a computational standpoint, this restriction on λµ-terms prevents a coroutine from accessing the local environment of another coroutine.

Abstract. The intuitionistic fragment of the call-by-name version of Curien and Herbelin’s λµ˜µ-calculus is isolated and proved strongly normalising by means of an embedding into the simply-typed λ-calculus. Our embedding is a continuation-and-garbage-passing style translation, the inspiring idea coming from Ikeda and Nakazawa’s translation of Parigot’s λµ-calculus. The embedding simulates reductions while usual continuation-passing-style transformations erase permutative reduction steps. For our intuitionistic sequent calculus, we even only need “units of garbage ” to be passed. We apply the same method to other calculi, namely successive extensions of the simply-typed λ-calculus leading to our intuitionistic system, and already for the simplest extension we consider (λ-calculus with generalised application), this yields the first proof of strong normalisation through a reduction-preserving embedding. 1

Λµ-calculus is a Böhm-complete extension of Parigot's λµ-calculus closely related with delimited control in functional programming. In this paper, we investigate the meta-theory of untyped Λµ-calculus by proving con uence of the calculus and characterizing the basic observables for the Separation theorem, canonical normal forms. Then, we de ne ΛS, a new type system for Λµ-calculus which contains a special type construction for streams, and prove that strong normalization and type preservation hold. Thanks to the new typing discipline of ΛS, new computational behaviors can be observed, which were forbidden in previous type systems for λµ-calculi. Those new typed computational behaviors witness the stream interpretation of Λµ-calculus.