Constructing differential categories and deconstructing categories of games

Abstract

Differential categories were introduced by Blute, Cockett and Seely to axiomatize categorically Ehrhard and Regnier's syntactic differential operator. We present an abstract construction that takes a symmetric monoidal category and yields a differential category, and show how this construction may be applied to categories of games. In one instance, we recover the category previously used to give a fully abstract model of a nondeterministic imperative language. The construction exposes the differential structure already present in this model, and shows how the differential combinator may be encoded in the imperative language. The second instance corresponds to a new Cartesian differential category of games. We give a model of a simply-typed resource calculus, Resource PCF, in this category and show that it possesses the finite definability property. Comparison with a semantics based on Bucciarelli, Ehrhard and Manzonetto's relational model reveals that the latter also possesses this property and is fully abstract.

Weighted Relational Models of Typed Lambda-Calculi

“…is isomorphic to the category MRel, known as the relational semantics. Therefore, this first result is not very surprising as MRel has been proved to characterize mayconvergence for a resource sensitive extension of PCF or [18].…”

“…Though our focus in this paper is on weighted models generalising relations, we believe that the key step -replacing matrices over Booleans with matrices over arbitrary R -is more widely applicable. Indeed, we discovered these models while considering a quantitative version of the constructions described in [18], which allow us to build not only relational models but also games models. We believe that, for instance, Danos and Harmer's probabilistic games model [19] can be recovered by an analogous construction.…”

Applying quantitative semantics to higher-order quantum computing

“…), mapping The category CPM is "almost" capable of handling this case, but not quite, because it cannot express infinite tuples of matrices. The model we propose in this paper is essentially an extension of CPM to infinite biproducts, using methods developed in [5,15,11,12].…”

“…is the result of a categorical construction applied to CPM s which is known to give, under certain circumstances, a dcpo-enriched Lafont category and to preserve the compact closed structure of CPM s . This construction was sketched in [5] and detailed in [15,11,12]. It consists in moving: (i) from CPM s to a category CPMs with symmetric tensors, which is actually a full sub-category of the Karoubi envelope of CPM s ; (ii) to a dcpo-enriched category CPMs using the D-completion defined in [23,7]; and, finally, (iii) constructing the free biproduct completion CPMs ⊕ of CPMs and applying Equation (4).…”

Constructing Differential Categories and Deconstructing Categories of Games

“…Related Works. This article is an extended version of [8]. We build and analyze differential categories as a semantic framework for Resource PCF, a programming language based on the resource calculus.…”