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A formulaeastypes interpretation of subtractive logic
 Journal of Logic and Computation
, 2004
"... We present a formulaeastypes interpretation of Subtractive Logic (i.e. biintuitionistic logic). This presentation is twofold: we first define a very natural restriction of the λµcalculus which is closed under reduction and whose type system is a constructive restriction of the Classical Natural ..."
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We present a formulaeastypes interpretation of Subtractive Logic (i.e. biintuitionistic logic). This presentation is twofold: we first define a very natural restriction of the λµcalculus 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 firstclass coroutines (a restricted form of firstclass continuations). Keywords: CurryHoward isomorphism, Subtractive Logic, control operators, coroutines. 1
Arithmetical proofs of strong normalization results for symmetric λcalculi
"... symmetric λµcalculus ..."
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Reducibility and ⊤⊤lifting for computation types
 In Proc. 7th International Conference on Typed Lambda Calculi and Applications (TLCA), volume 3461 of Lecture Notes in Computer Science
, 2005
"... 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 GirardTait reducibility, using this to prove strong normalisation for the computational meta ..."
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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 GirardTait 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, sideeffecting, 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 “leapfrog ” approach offers a general method for raising concepts defined at value types up to observable properties of computations. 1
A Computational Interpretation of the λμcalculus
, 1998
"... This paper proposes a simple computational interpretation of Parigot's calculus. The calculus is an extension of the typed calculus which corresponds via the CurryHoward correspondence to classical logic. Whereas other work has given computational interpretations by translating the calculu ..."
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Cited by 10 (0 self)
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This paper proposes a simple computational interpretation of Parigot's calculus. The calculus is an extension of the typed calculus which corresponds via the CurryHoward correspondence to classical logic. Whereas other work has given computational interpretations by translating the calculus into other calculi, I wish to propose here that the 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 singlestep semantics which, in particular, leads to a relatively simple, but powerful, operational theory.
A Computational Interpretation of the λµcalculus
 PROCEEDINGS OF SYMPOSIUM ON MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE
, 1998
"... This paper proposes a simple computational interpretation of Parigot's λµcalculus. The λµcalculus is an extension of the typedcalculus which corresponds via the CurryHoward correspondence to classical logic. Whereas other work has given computational interpretations by translating the λµca ..."
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Cited by 10 (1 self)
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This paper proposes a simple computational interpretation of Parigot's λµcalculus. The λµcalculus is an extension of the typedcalculus which corresponds via the CurryHoward correspondence to classical logic. Whereas other work has given computational interpretations by translating the λµcalculus into other calculi, I wish to propose here that the λµcalculus itself has a simple computational interpretation: it is a typedcalculus which is able to save and restore the runtime environment. This interpretation is best given as a singlestep semantics which, in particular, leads to a relatively simple, but powerful, operational theory.
A Modal Calculus for Effect Handling
, 2003
"... 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 prov ..."
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Cited by 7 (2 self)
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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 extralogical constructs to support handling. In this paper we adopt...
Unchecked Exceptions can be Strictly More Powerful than Call/CC
 HigherOrder and Symbolic Computation
, 1996
"... We demonstrate that in the context of staticallytyped purelyfunctional 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 simplytyped lambda calculus with unchecked exce ..."
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We demonstrate that in the context of staticallytyped purelyfunctional 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 simplytyped lambda calculus with unchecked exceptions is strictly more powerful than all known sound extensions of Girard's Fomega (a superset of the simplytyped 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 simplytyped 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 uncheckedexceptionreturning functions. The result concerning extensions of Fomega with call/cc stems from previous work of the author and Robert Harper.
A New Formulation of the Catch/Throw Mechanism
 Second Fuji International Workshop on Functional and Logic Programming
, 1997
"... The catch/throw mechanism in Common Lisp gives a simple control structure for nonlocal exits. Nakano[7, 9] and Sato[13] proposed intuitionistic calculi with inference rules which give logical interpretations of the catch/throwconstructs. Although the calculi are theoretically wellfounded, we c ..."
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The catch/throw mechanism in Common Lisp gives a simple control structure for nonlocal exits. Nakano[7, 9] and Sato[13] proposed intuitionistic calculi with inference rules which give logical interpretations of the catch/throwconstructs. Although the calculi are theoretically wellfounded, we cannot use the catch/throw mechanism for handling runtime errors in a meaningful way, because of the sidecondition of the implicationintroduction rule (the formulation rule of the abstract). This deficiency is critical if we use higherorder functions with the catch/throw mechanism. In this paper, we propose a new formulation of catch/throw calculi, which has no sidecondition on the implicationintroduction rule. By restricting the types of thrown terms to data types (nonfunctional types) instead, we obtain a strongly normalizing calculus for the catch/throw mechanism where we can write higherorder functions which handles runtime errors. 1. Introduction Recently, control st...
Stabilization—An Alternative to DoubleNegation Translation for Classical Natural Deduction
, 2004
"... A new proof of strong normalization of Parigot’s secondorder λµcalculus is given by a reductionpreserving embedding into system F (secondorder polymorphic λcalculus). The main idea is to use the least stable supertype for any type. These nonstrictly positive inductive types and their associate ..."
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A new proof of strong normalization of Parigot’s secondorder λµcalculus is given by a reductionpreserving embedding into system F (secondorder polymorphic λcalculus). The main idea is to use the least stable supertype for any type. These nonstrictly 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 doublenegation embedding of classical logic into intuitionistic logic). However, they simulate Parigot’s µreductions 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 fixedpoint types. The article expands on “Parigot’s SecondOrder λµCalculus and Inductive Types ” (Conference Proceedings TLCA 2001, Springer LNCS 2044) by the author. 1
On the Strong Normalisation of Natural Deduction with PermutationConversions
"... . We present a modular proof of the strong normalisation of intuitionistic logic with permutationconversions. This proof is based on the notions of negative translation and CPSsimulation. 1 Introduction Natural deduction systems provide a notion of proof that is more compact (or, quoting Girard [ ..."
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. We present a modular proof of the strong normalisation of intuitionistic logic with permutationconversions. This proof is based on the notions of negative translation and CPSsimulation. 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 proofnormalisation 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 socalled permutationconversions are needed. Strong normalisation proofs for intuitionistic logic [12] are mor...