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Completeness and Soundness Results for χ with Intersection and Union Types

Bakel S. van

Fundamenta Informaticae

2012

Vol. 121, nr 1/4

1-41

With the eye on defining a type-based semantics, this paper defines intersection and union type assignment for the sequent calculus χ, a substitution-free language that enjoys the Curry- Howard correspondence with respect to the implicative fragment of Gentzen's sequent calculus for classical logic. We investigate the minimal requirements for such a system to be complete (i.e. closed under redex expansion), and show that the non-logical nature of both intersection and union types disturbs the soundness (i.e. closed uder reduction) properties. This implies that this notion of intersection-union 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 Call-by-Name and Call-by- Value, and prove soundness results for those.