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Computation with classical sequents
 MATHEMATICAL STRUCTURES OF COMPUTER SCIENCE
, 2008
"... X is an untyped continuationstyle formal language with a typed subset which provides a CurryHoward isomorphism for a sequent calculus for implicative classical logic. X can also be viewed as a language for describing nets by composition of basic components connected by wires. These features make X ..."
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Cited by 16 (16 self)
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X is an untyped continuationstyle formal language with a typed subset which provides a CurryHoward isomorphism for a sequent calculus for implicative classical logic. X can also be viewed as a language for describing nets by composition of basic components connected by wires. These features make X an expressive platform on which algebraic objects and many different (applicative) programming paradigms can be mapped. In this paper we will present the syntax and reduction rules for X and in order to demonstrate the expressive power of X, we will show how elaborate calculi can be embedded, like the λcalculus, Bloo and Rose’s calculus of explicit substitutions λx, Parigot’s λµ and Curien and Herbelin’s λµ ˜µ.
Reduction in X does not agree with Intersection and Union Types
, 2008
"... This paper defines intersection and union type assignment for the calculus X, a substitution free language that enjoys the CurryHoward correspondence with respect to Gentzen’s sequent calculus for classical logic. We show that this notion is the minimal one closed for subjectexpansion, and show th ..."
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Cited by 2 (2 self)
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This paper defines intersection and union type assignment for the calculus X, a substitution free language that enjoys the CurryHoward correspondence with respect to Gentzen’s sequent calculus for classical logic. We show that this notion is the minimal one closed for subjectexpansion, and show that it needs to be restricted to satisfy subjectreduction as well, making it unsuitable to define a semantics.
Classical Cutelimination in the πcalculus
"... We study the πcalculus, enriched with pairing, and define a notion of type assignment that uses the type constructor →. We encode the terms of the calculus X into this variant of π, and show that all reduction (cutelimination) and assignable types are preserved. Since X enjoys the CurryHoward iso ..."
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We study the πcalculus, enriched with pairing, and define a notion of type assignment that uses the type constructor →. We encode the terms of the calculus X into this variant of π, and show that all reduction (cutelimination) and assignable types are preserved. Since X enjoys the CurryHoward isomorphism for Gentzen’s calculu LK, this implies that all proofs in LK have a representation in π. We then enrich the logic with the connector ¬, and show that this also can be represented in π.