Mackey-complete spaces and power series – a topological model of differential linear logic

Abstract

In this paper, we have described a denotational model of Intuitionist Linear Logic which is also a differential category. Formulas are interpreted as Mackey-complete topological vector space and linear proofs are interpreted by bounded linear functions. So as to interpret nonlinear proofs of Linear Logic, we have used a notion of power series between Mackey-complete spaces, generalizing the notion of entire functions in C. Finally, we have obtained a quantitative model of Intuitionist Differential Linear Logic, where the syntactic differentiation correspond to the usual one and where the interpretations of proofs satisfy a Taylor expansion decomposition.

Bifibrations of Polycategories and Classical Linear Logic

“…Furthermore, extending the case study of Banach spaces to LNL-polycategories could help understand better LL and DiLL, for example by studying models coming from functional analysis such as those considered in Kerjean's work, see [50,24,48,49] .…”

Fixing Incremental Computation

“…Families of denotational models for the differential λ-calculus have been studied in depth [12,13,16,29], and the relationship between these and change actions is the subject of ongoing work.…”

Weak topologies for Linear Logic

“…The lack of completion explains the form of our interpretation of non-linear proofs, as simple sequences of monomials. Thus, even though the model interprets differential linear logic, we are far from capturing the intuition of smoothness of proofs behind the notion of differentiation (as done for example in [BET12,KT15]).…”

“…Indeed, the latter is tied to the induced (co-Kleisli) cartesian closed category of non-linear functions. To deal with these non-linear functions, completion is usually necessary [Ehr05,BET12,KT15]. The lack of completion explains the form of our interpretation of non-linear proofs, as simple sequences of monomials.…”

“…It appears that the weakest completeness condition necessary to model quantitative linear logic should be Mackey completeness[KT15].…”