Cones as a model of intuitionistic linear logic

For overcoming the limitations of probabilistic coherence spaces which do not seem to provide natural interpretations of continuous data types such as the real line, we introduced with Pagani and Tasson a model of probabilistic higher order computation based on (positive) cones, and a class of totally monotone functions that we called "stable". Then Crubillé proved that this model is a conservative extension of the earlier probabilistic coherence space model. We continue these investigations by showing that the category of cones and linear and Scott-continuous functions is a model of intuitionistic linear logic. To define the tensor product, we use the special adjoint functor theorem, and we prove that this operation is an extension of the standard tensor product of probabilistic coherence spaces. We also show that these latter are dense in cones, thus allowing to lift the main properties of the tensor product of probabilistic coherence spaces to general cones. Finally we define in the same way an exponential of cones and extend measurability to these new operations.

Proof. We have a sequence of natural isomorphisms

This function is stable because prom is stable and ⊗ is bilinear and continuous. So we have f ∈ CLin( !(P & Q), !P ⊗ !Q) which satisfies f (x, y) ! = x ! ⊗ y ! . By Lemma 29 it follows that f is the inverse of m 2 .

Since ⊤ = 0 and 1 = R ≥0 we have g ∈ Cstab(⊤, 1) given by g(0) = 1 and hence g ∈ CLin( !⊤, 1) fully characterized by g 0 ! = 1. We define m 0 ∈ CLin(1, !⊤) by m 0 (λ) = λ0 ! . Lemma 29 shows that m 0 g = Id and g m 0 = Id is straightforward.