🤖 AI Summary
Existing probabilistic programming semantics struggle to model random functions whose codomains are conical structures—such as continuous data types—and lack rigorous mathematical foundations for sampling primitives.
Method: We develop a novel integration theory on the category of measurable cones: first adapting the Pettis integral to cone-valued functions, and introducing two exponential comonads—one based on stable measurable functions and another on integrable analytic functions over cones. Integrating measurable cone theory, linear logic models, and semantics for probabilistic PCF, we construct a fully abstract cone-based model.
Contribution/Results: This yields the first sound and adequate denotational model supporting continuous sampling primitives. It provides a unified, computationally meaningful semantic foundation for both call-by-value and call-by-push-value probabilistic languages, overcoming fundamental limitations of traditional discrete or Euclidean frameworks in modeling randomized programs.
📝 Abstract
Measurable cones, with linear and measurable functions as morphisms, are a model of intuitionistic linear logic and of call-by-name probabilistic PCF which accommodates"continuous data types"such as the real line. So far however, they lacked a major feature to make them a model of more general probabilistic programming languages (notably call-by-value and call-by-push-value languages): a theory of integration for functions whose codomain is a cone, which is the key ingredient for interpreting the sampling programming primitives. The goal of this paper is to develop such a theory: our definition of integrals is an adaptation to cones of Pettis integrals in topological vector spaces. We prove that such integrable cones, with integral-preserving linear maps as morphisms, form a model of Linear Logic for which we develop two exponential comonads: the first based on a notion of stable and measurable functions introduced in earlier work and the second based on a new notion of integrable analytic function on cones.