We present a formulae-as-types interpretation of Subtractive Logic (i.e. bi-intuitionistic logic). This presentation is two-fold: we first define a very natural restriction of the λµ-calculus which is closed under reduction and whose type system is a constructive restriction of the Classical Natural Deduction. Then we extend this deduction system conservatively to Subtractive Logic. From a computational standpoint, the resulting calculus provides a type system for first-class coroutines (a restricted form of first-class continuations). Keywords: Curry-Howard isomorphism, Subtractive Logic, control operators, coroutines. 1

symmetric λµ-calculus

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

This paper proposes a simple computational interpretation of Parigot's λµ-calculus. The λµ-calculus is an extension of the typed-calculus which corresponds via the Curry-Howard correspondence to classical logic. Whereas other work has given computational interpretations by translating the λµ-calculus into other calculi, I wish to propose here that the λµ-calculus itself has a simple computational interpretation: it is a typed-calculus which is able to save and restore the runtime environment. This interpretation is best given as a single-step semantics which, in particular, leads to a relatively simple, but powerful, operational theory.

In their purest formulation, monads are used in functional programming for two purposes: (1) to hygienically propagate effects, and (2) to globalize the effect scope -- once an effect occurs, the purity of the surrounding computation cannot be restored. As a consequence, monadic typing does not provide very naturally for the practically important ability to handle effects, and there is a number of previous works directed toward remedying this deficiency. It is mostly based on extending the monadic framework with further extra-logical constructs to support handling. In this paper we adopt...

The catch/throw mechanism in Common Lisp gives a simple control structure for non-local exits. Nakano[7, 9] and Sato[13] proposed intuitionistic calculi with inference rules which give logical interpretations of the catch/throw-constructs. Although the calculi are theoretically well-founded, we cannot use the catch/throw mechanism for handling run-time errors in a meaningful way, because of the side-condition of the implication-introduction rule (the formulation rule of the -abstract). This deficiency is critical if we use higher-order functions with the catch/throw mechanism. In this paper, we propose a new formulation of catch/throw calculi, which has no side-condition on the implication-introduction rule. By restricting the types of thrown terms to data types (non-functional types) instead, we obtain a strongly normalizing calculus for the catch/throw mechanism where we can write higher-order functions which handles run-time errors. 1. Introduction Recently, control st...

We demonstrate that in the context of statically-typed purely-functional lambda calculi without recursion, unchecked exceptions (e.g., SML exceptions) can be strictly more powerful than call/cc. More precisely, we prove that a natural extension of the simply-typed lambda calculus with unchecked exceptions is strictly more powerful than all known sound extensions of Girard's F-omega (a superset of the simply-typed lambda calculus) with call/cc. This result is established by showing that the first language is Turing complete while the later languages permit only a subset of the recursive functions to be written. We show that our natural extension of the simply-typed lambda calculus with unchecked exceptions is Turing complete by reducing the untyped lambda calculus to it by means of a novel method for simulating recursive types using unchecked-exception-returning functions. The result concerning extensions of F-omega with call/cc stems from previous work of the author and Robert Harper.

. We present a modular proof of the strong normalisation of intuitionistic logic with permutation-conversions. This proof is based on the notions of negative translation and CPS-simulation. 1 Introduction Natural deduction systems provide a notion of proof that is more compact (or, quoting Girard [6], more primitive) than that of sequent calculi. In particular, natural deduction is better adapted to the study of proof-normalisation procedures. This is true, at least, for the intuitionistic systems, where proofnormalisation expresses the computational content of the logic. Nevertheless, even in the intuitionistic case, the treatments of disjunction and existential quantication are problematic. This is due to the fact that the elimination rules of these connectives introduce arbitrary formulas as their conclusions. Consequently, in order to satisfy the subformula property, the so-called permutationconversions are needed. Strong normalisation proofs for intuitionistic logic [12] are mor...

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A new proof of strong normalization of Parigot’s second-order λµ-calculus is given by a reduction-preserving embedding into system F (second-order polymorphic λ-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 Kolmogorov’s double-negation embedding of classical logic into intuitionistic logic). However, they simulate Parigot’s µ-reductions whereas CPS translations hide them. As a major advantage, this embedding does not use the idea of reducing stability (¬¬A → A) to that for atomic formulae. Therefore, it even extends to positive fixed-point types. The article expands on “Parigot’s Second-Order λµ-Calculus and Inductive Types ” (Conference Proceedings TLCA 2001, Springer LNCS 2044) by the author. 1