A Call-By-Push-Value FPC and its interpretation in Linear Logic

We present and study a functional calculus similar to Levy’s CallBy-Push-Value lambda-calculus, extended with fix-points and recursive types. We explain its connection with Linear Logic by presenting a denotational interpretation of the language in any model of Linear Logic equipped with a notion of embedding retraction pairs. We consider the particular case of the Scott model of Linear Logic from which we derive an intersection type system for our CBPV FPC and prove an adequacy theorem. Last, we introduce a fully polarized version of CBPV which is closer to Levy’s original calculus, turns out to be a term language for a large fragment of Laurent’s LLP and refines Parigot’s lambda-mu.