Exponentials with infinite multiplicities

Authors:
Alberto Carraro;Thomas Ehrhard;Antonino Salibra
Affiliations:
Department of Computer Science, Ca' Foscari University, Venice and Laboratory Preuves, Programmes & Systèmes, UMR, University Paris Diderot, Paris 7 and CNRS;Laboratory Preuves, Programmes & Systèmes, UMR, University Paris Diderot, Paris 7 and CNRS;Department of Computer Science, Ca' Foscari University, Venice
Venue:
CSL'10/EACSL'10 Proceedings of the 24th international conference/19th annual conference on Computer science logic
Year:
2010

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Abstract

Given a semi-ring with unit which satisfies some algebraic conditions, we define an exponential functor on the category of sets and relations which allows to define a denotational model of differential linear logic and of the lambda-calculus with resources. We show that, when the semi-ring has an element which is infinite in the sense that it is equal to its successor, this model does not validate the Taylor formula and that it is possible to build, in the associated Kleisli cartesian closed category, a model of the pure lambda-calculus which is not sensible. This is a quantitative analogue of the standard graph model construction in the category of Scott domains. We also provide examples of such semi-rings.