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A logical interpretation of the λcalculus into the πcalculus, preserving spine reduction and types
, 2009
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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
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.
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.
The πcalculus as a Universal Model of Computation
, 2010
"... Using a novel approach, we define a compositional outputbased encoding of the λcalculus with explicit substitution into a variant of the πcalculus with pairing, and show that this encoding preserves full βreduction (i.e. not just lazy) as well as assignable types. Furthermore, we apply this new ..."
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Using a novel approach, we define a compositional outputbased encoding of the λcalculus with explicit substitution into a variant of the πcalculus with pairing, and show that this encoding preserves full βreduction (i.e. not just lazy) as well as assignable types. Furthermore, we apply this new approach to the context of sequent calculi (which embody both parameter and context call) by defining a new encoding of the calculus λµ ˜µ; we show that this encodings is strong, in the sense that it too preserves all λµ ˜µreduction (i.e. not just outermost reduction), and in the sense that the implicative fragment of classical logic can be embedded in to the πcalculus via the preservation of assignable contexts. These results show that the πcalculus is indeed a universal model of computation for both concurrent and sequential paradigms, and fully represents both parameter and context call, as well as functional composition.
Implicative Logic based translations of the λcalculus into the πcalculus
"... We study an outputbased translation of the λcalculus with explicit substitution into the synchronous πcalculus – enriched with pairing – that has its origin in mathematical logic, and show that this translation respects reduction. We will define the notion of (explicit) head reductionwhich encom ..."
Abstract
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We study an outputbased translation of the λcalculus with explicit substitution into the synchronous πcalculus – enriched with pairing – that has its origin in mathematical logic, and show that this translation respects reduction. We will define the notion of (explicit) head reductionwhich encompasses (explicit) lazy reduction and show that the translation fully represents this reduction in that termsubstitution as well as each single reduction step are modelled. We show that all the main properties (soundness, completeness, and adequacy) hold for these notions of reduction, as well as that termination is preserved with respect to a notion of call by need reduction for the πcalculus. 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 translation. We will also show that the πcalculus gives a semantics for the (standard) λcalculus by defining an encoding that will fully represent reduction with explicit substitution, βreduction, and equality, mapping equivalent term to equivalent processes.
Fully abstract semantics of λµ in the πcalculus
"... We study the λµcalculus, extended with explicit substitution, and study a logicbased compositional outputbased translation into a variant of the πcalculus with pairing that preserves singlestep explicit head reduction with respect to contextual equivalence. We will define two notions of equalit ..."
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We study the λµcalculus, extended with explicit substitution, and study a logicbased compositional outputbased translation into a variant of the πcalculus with pairing that preserves singlestep explicit head reduction with respect to contextual equivalence. We will define two notions of equality for λµ, modelling explicit headreduction, head reduction, and show they coincide. We define four notions of equivalence, two as extensions of the equalities, one based on reduction (considering terms without headnormal form equivalent as well), and one based on approximation, and show they all coincide. We will also define four notions of weak equivalence, where now we consider terms without weak headnormal form equivalent, that all coincide as well. We will then show full abstraction results for our translation for the weak equivalences with respect to contextually equivalence on processes.