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19
From X to π; representing the classical sequent calculus
"... Abstract. We study the πcalculus, enriched with pairing and nonblocking input, and define a notion of type assignment that uses the type constructor →. We encode the circuits of the calculus X into this variant of π, and show that all reduction (cutelimination) and assignable types are preserved. ..."
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Abstract. We study the πcalculus, enriched with pairing and nonblocking input, and define a notion of type assignment that uses the type constructor →. We encode the circuits 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 calculus LK, this implies that all proofs in LK have a representation in π.
A logical interpretation of the λcalculus into the πcalculus, preserving spine reduction and types
, 2009
"... ..."
Completeness and Partial Soundness Results for Intersection & Union Typing for λµ ˜µ
 Annals of Pure and Applied Logic
"... This paper studies intersection and union type assignment for the calculus λµ ˜µ [17], a proofterm syntax for Gentzen’s classical sequent calculus, with the aim of defining a typebased semantics, via setting up a system that is closed under conversion. We will start by investigating what the minima ..."
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This paper studies intersection and union type assignment for the calculus λµ ˜µ [17], a proofterm syntax for Gentzen’s classical sequent calculus, with the aim of defining a typebased semantics, via setting up a system that is closed under conversion. We will start by investigating what the minimal requirements are for a system for λµ ˜µ to be closed under subject expansion; this coincides with System M ∩ ∪ , the notion defined in [19]; however, we show that this system is not closed under subject reduction, so our goal cannot be achieved. We will then show that System M ∩ ∪ is also not closed under subjectexpansion, but can recover from this by presenting System M C as an extension of M ∩ ∪ (by adding typing rules) and showing that it satisfies subject expansion; it still lacks subject reduction. We show how to restrict M ∩ ∪ so that it satisfies subjectreduction as well by limiting the applicability to type assignment rules, but only when limiting reduction to (confluent) callbyname or callbyvalue reduction M ∩ ∪ ; in restricting the system, we sacrifice subject expansion. These results combined show that a sound and complete intersection and union type assignment system cannot be defined for λµ ˜µ with respect to full reduction.
Implicative Logic based encoding of the λcalculus into the πcalculus
, 2010
"... We study an outputbased encoding of the λcalculus with explicit substitution into the synchronous πcalculus – enriched with pairing – that has its origin in mathematical logic, and show that this encoding respects reduction. We will define the notion of (explicit) spine reductionwhich encompasse ..."
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We study an outputbased encoding of the λcalculus with explicit substitution into the synchronous πcalculus – enriched with pairing – that has its origin in mathematical logic, and show that this encoding respects reduction. We will define the notion of (explicit) spine reductionwhich encompasses (explicit) lazy reduction and show that the encoding fully encodes this reduction in that termsubstitution as well as each single reduction step are modelled up to contextual similarity. We show that all the main properties (soundness, completeness, and adequacy) hold for these four notions of reduction, as well as that termination is preserved. We then define a notion of type assignment for the πcalculus that uses the type constructor→, and show that all Curry types assignable to λterms are preserved by the encoding. Key words: the λcalculus, the πcalculus, intuitionistic logic, classical logic, encoding, type assignment
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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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.
An OutputBased Semantics of Λµ with Explicit Substitution
 in the πcalculus. IFIPTCS’12, LNCS 7604
, 2012
"... We study the Λµcalculus, extended with explicit substitution, and define a compositional outputbased translation into a variant of the πcalculus with pairing. We show that this translation preserves singlestep explicit head reduction with respect to contextual equivalence. We use this result to ..."
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We study the Λµcalculus, extended with explicit substitution, and define a compositional outputbased translation into a variant of the πcalculus with pairing. We show that this translation preserves singlestep explicit head reduction with respect to contextual equivalence. We use this result to show operational soundness for head reduction, adequacy, and operational completeness. Using a notion of implicative typecontext assignment for the πcalculus, we also show that assignable types are preserved by the translation. We finish by showing that termination is preserved.
Completeness and Soundness results forX with Intersection and Union Types
"... This paper defines intersection and union type assignment for the sequent calculus X, a substitutionfree language that enjoys the CurryHoward correspondence with respect to Gentzen’s sequent calculus for classical logic. We show that this notion is complete (i.e. closed for subjectexpansion), and ..."
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This paper defines intersection and union type assignment for the sequent calculus X, a substitutionfree language that enjoys the CurryHoward correspondence with respect to Gentzen’s sequent calculus for classical logic. We show that this notion is complete (i.e. closed for subjectexpansion), and show that the nonlogical nature of both intersection and union types disturbs the soundness (i.e. closed for reduction) properties. This implies that this notion of intersectionunion type assignment needs to be restricted to satisfy soundness as well, making it unsuitable to define a semantics. We will look at two (confluent) notions of reduction, called CallbyName and CallbyValue, and prove soundness results for those.
Sound and Complete Typing for λµ
"... In this paper we define intersection and union type assignment for Parigot’s calculus λµ. We show that this notion is complete (i.e. closed under subjectexpansion), and show also that it is sound (i.e. closed under subjectreduction). This implies that this notion of intersectionunion type assignme ..."
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In this paper we define intersection and union type assignment for Parigot’s calculus λµ. We show that this notion is complete (i.e. closed under subjectexpansion), and show also that it is sound (i.e. closed under subjectreduction). This implies that this notion of intersectionunion type assignment is suitable to define a semantics.
Subject Reduction vs Intersection / Union Types in λµ ˜µ Extended abstract
"... Abstract. This paper defines intersection and union type assignment for the calculus λµ ˜µ [9], a proofterm syntax for Gentzen’s classical sequent calculus. We show that this notion is closed for subjectexpansion, and show that it needs to be restricted to satisfy subjectreduction as well, even w ..."
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Abstract. This paper defines intersection and union type assignment for the calculus λµ ˜µ [9], a proofterm syntax for Gentzen’s classical sequent calculus. We show that this notion is closed for subjectexpansion, and show that it needs to be restricted to satisfy subjectreduction as well, even when limiting reduction to (confluent) callbyname or callbyvalue reduction, making it unsuitable to define a semantics.