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Strong Normalisation of CutElimination in Classical Logic
, 2000
"... In this paper we present a strongly normalising cutelimination procedure for classical logic. This procedure adapts Gentzen's standard cutreductions, but is less restrictive than previous strongly normalising cutelimination procedures. In comparison, for example, with works by Dragalin and D ..."
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Cited by 36 (4 self)
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In this paper we present a strongly normalising cutelimination procedure for classical logic. This procedure adapts Gentzen's standard cutreductions, but is less restrictive than previous strongly normalising cutelimination procedures. In comparison, for example, with works by Dragalin and Danos et al., our procedure requires no special annotations on formulae and allows cutrules to pass over other cutrules. In order to adapt the notion of symmetric reducibility candidates for proving the strong normalisation property, we introduce a novel term assignment for sequent proofs of classical logic and formalise cutreductions as term rewriting rules.
Termination Checking with Types
, 1999
"... The paradigm of typebased termination is explored for functional programming with recursive data types. The article introduces , a lambdacalculus with recursion, inductive types, subtyping and bounded quanti cation. Decorated type variables representing approximations of inductive types ..."
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Cited by 28 (6 self)
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The paradigm of typebased termination is explored for functional programming with recursive data types. The article introduces , a lambdacalculus with recursion, inductive types, subtyping and bounded quanti cation. Decorated type variables representing approximations of inductive types are used to track the size of function arguments and return values. The system is shown to be type safe and strongly normalizing. The main novelty is a bidirectional type checking algorithm whose soundness is established formally.
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 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 9 (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 Proof Theoretical Account of Continuation Passing Style
 In CSL ’02: Proceedings of the 16th International Workshop and 11th Annual Conference of the EACSL on Computer Science Logic
, 2002
"... We study "classical proofs as programs" paradigm in CallBy Value (CBV) setting. Specifically, we show the CBV normalization for CND (Parigot 92) can be simulated by the cutelimination procedure for LKQ (DanosJoinetSchellinx 93), namely the qprotocol. We use proofterm assignment s ..."
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Cited by 3 (0 self)
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We study "classical proofs as programs" paradigm in CallBy Value (CBV) setting. Specifically, we show the CBV normalization for CND (Parigot 92) can be simulated by the cutelimination procedure for LKQ (DanosJoinetSchellinx 93), namely the qprotocol. We use proofterm assignment system to prove this fact. The term calculus for CND we use follows Parigot's #calculus with new CBV normalization procedure. A new term calculus for LKQ is presented as a variant of #calculus with a letconstruct. We then define a translation from CND into LKQ and prove simulation theorem. We also show the translation we use can be thought of a familiar CBV CPStranslation without translation on types.
A ThirdOrder Representation of the λμCalculus
, 2001
"... Higherorder logical frameworks provide a powerful technology to reason about object languages with binders. This will be demonstrated for the case of the calculus with two different binders which can most elegantly be represented using a thirdorder constant. Since cases of third and highero ..."
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Cited by 1 (0 self)
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Higherorder logical frameworks provide a powerful technology to reason about object languages with binders. This will be demonstrated for the case of the calculus with two different binders which can most elegantly be represented using a thirdorder constant. Since cases of third and higherorder encodings are very rare in comparison with those of second order, a secondorder representation is given as well and equivalence to the thirdorder representation is proven formally.
The λμcalculus: Function and Control
, 1998
"... Parigot's calculus is an intriguing extension of the typed calculus which corresponds via the CurryHoward correspondence to classical logic. Following the seminal work of Grin, it is known that certain control operators can be given types which, when viewed as formulae, are classical but ..."
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Parigot's calculus is an intriguing extension of the typed calculus which corresponds via the CurryHoward correspondence to classical logic. Following the seminal work of Grin, it is known that certain control operators can be given types which, when viewed as formulae, are classical but not intuitionistic tautologies. Previous computational explanations of the calculus have simply translated terms into an existing control calculus, or presented an operational semantics from which it is hard to determine what is going on. In particular the treatment of a callbyvalue strategy has appeared problematic.
Termination Checking with Types: Strong Normalization for MendlerStyle CourseOfValue Recursion
"... Gimenez' type system for structural recursion in the Calculus of Constructions is adapted to typed functional programming. As core language, the #calculus with inductive types, subtyping and bounded quantification is introduced. Decorated type variables, which represent approximations of i ..."
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Gimenez' type system for structural recursion in the Calculus of Constructions is adapted to typed functional programming. As core language, the #calculus with inductive types, subtyping and bounded quantification is introduced. Decorated type variables, which represent approximations of inductive types, enable the type system to track the size of arguments to recursive functions and the size of the result of function calls. Novel are an algorithm for type checking and proofs of type preservation and strong normalization.