Modelling environments in call-by-value programming languages

Abstract

In categorical semantics, there have traditionally been two approaches to modelling environments, one by use of finite products in cartesian closed categories, the other by use of the base categories of indexed categories with structure. Each requires modifications in order to account for environments in call-by-value programming languages. There have been two more general definitions along both of these lines: the first generalising from cartesian to symmetric premonoidal categories, the second generalising from indexed categories with specified structure to κ-categories. In this paper, we investigate environments in call-by-value languages by analysing a fine-grain variant of Moggi's computational λcalculus, giving two equivalent sound and complete classes of models: one given by closed Freyd categories, which are based on symmetric premonoidal categories, the other given by closed κ-categories.

Canonical Models for Computational Effects

“…The following result is proved but only stated implicitly in [21]; it is stated explicitly in [8,20].…”

“…Our starting point has typically been Eugenio Moggi's computational λ-calculus or λ c -calculus, which was introduced in [10,11], with four distinct sound and complete classes of category theoretic models explained in [18], and with further abstract semantic development in [8,19,20,21]. The ideas surrounding the calculus have been applied extensively by the functional programming community, albeit typically using the computational metalanguage rather than the λ c -calculus: a recent overview appears as [1].…”

Substitution, jumps, and algebraic effects

“…We use the monad S to give a denotational semantics for our metalanguage, essentially following Moggi's pattern [30,34]. We interpret types as context-indexed-sets:…”

Generic models for computational effects

“…It allows one to model environments in call-by-value programming languages containing computational effects, notably the c -calculus [25,21,9], a variant of the call-by-value -calculus designed specifically to allow one to account for computational effects. Starting with the notion of category with finite products, one obtains the notion of a symmetric monoidal category by dropping insistence upon the existence of diagonals and projections [10]: in such situations, one usually speaks of a tensor product rather than a product, corresponding to the relaxation from cartesian logic to linear logic.…”

“…Just as one has cartesian closed categories and symmetric monoidal closed categories, one can speak of closedness for a symmetric premonoidal category too [21]. Finally, if one reinstates the assumption of finite product structure but only on a specified subcategory of a putative symmetric premonoidal category, one has the notions of Freyd-category and closed Freyd-category [9]: we recall the definitions in Section 2. In this paper, motivated by computational effects, we further develop the theory of Freyd-categories.…”