Linear logic in normed cones: probabilistic coherence spaces and beyond

Mathematical Structures in Computer Science. 2021. Vol. 31. No. 5. P. 495–534.

Sergey Slavnov

Ehrhard et al. (2018. Proceedings of the ACM on Programming Languages, POPL 2, Article 59.) proposed a model of probabilistic functional programming in a category of normed positive cones and stable measurable cone maps, which can be seen as a coordinate-free generalization of probabilistic coherence spaces (PCSs). However, unlike the case of PCSs, it remained unclear if the model could be refined to a model of classical linear logic. In this work, we consider a somewhat similar category which gives indeed a coordinate-free model of full propositional linear logic with nondegenerate interpretation of additives and sound interpretation of exponentials. Objects are dual pairs of normed cones satisfying certain specific completeness properties, such as existence of norm-bounded monotone weak limits, and morphisms are bounded (adjointable) positive maps. Norms allow us a distinct interpretation of dual additive connectives as product and coproduct. Exponential connectives are modeled using real analytic functions and distributions that have representations as power series with positive coefficients. Unlike the familiar case of PCSs, there is no reference or need for a preferred basis; in this sense the model is invariant. PCSs form a full subcategory, whose objects, seen as posets, are lattices. Thus, we get a model fitting in the tradition of interpreting linear logic in a linear algebraic setting, which arguably is free from the drawbacks of its predecessors.

Keywords: linear logic, probabilistic coherence spaces, линейная логика, вероятностные пространства когерентности, denotational semantics, денотационная семантика