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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.
Soundness and Principal Contexts for a Shallow Polymorphic Type System based on Classical Logic
"... In this paper we investigate how to adapt the wellknown notion of MLstyle polymorphism (shallow polymorphism) to a term calculus based on a CurryHoward correspondence with classical sequent calculus, namely, theX icalculus. We show that the intuitive approach is unsound, and pinpoint the precise ..."
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In this paper we investigate how to adapt the wellknown notion of MLstyle polymorphism (shallow polymorphism) to a term calculus based on a CurryHoward correspondence with classical sequent calculus, namely, theX icalculus. We show that the intuitive approach is unsound, and pinpoint the precise nature of the problem. We define a suitably refined type system, and prove its soundness. We then define a notion of principal contexts for the type system, and provide an algorithm to compute these, which is proved to be sound and complete with respect to the type system. In the process, we formalise and prove correctness of generic unification, which generalises Robinson’s unification to shallowpolymorphic types. Key words: CurryHoward, classical logic, generic unification, principal types, cut elimination 1.
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.
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.
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.