The enriched effect calculus: syntax and semantics

This paper introduces the enriched eect calculus , which extends established type theories for computational eects with primitives from linear logic. The new calculus provides a formalism for expressing linear aspects of computational eects; for example, the linear usage of imperative features such as state and/or continuations. The enriched eect calculus is implemented as an extension of a basic eect calculus without linear primitives, which is closely related to Moggi’s computational metalanguage, Filinski’s eect PCF and Levy’s call-by-push-value. We present syntactic results showing: the delity of the behaviour of the linear connectives of the enriched eect calculus; the conservativity of the enriched eect calculus over its non-linear core (the eect calculus); and the non-conservativity of intuitionistic linear logic when considered as an extension of the enriched eect calculus. The second half of the paper investigates models for the enriched eect calculus, based on enriched category theory. We give several examples of such models, relating them to models of standard eect calculi (such as those based on monads), and to models of intuitionistic linear logic. We also prove soundness and completeness.