Integration in Cones

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 g...

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Bibliographic Details
Published in:Logical methods in computer science Vol. 21, Issue 1; no. 1
Main Authors: Ehrhard, Thomas, Geoffroy, Guillaume
Format: Journal Article
Language:English
Published: Logical Methods in Computer Science Association 01.01.2025
Logical Methods in Computer Science e.V
Subjects:
Computer Science
computer science - logic in computer science
Logic in Computer Science
Linear logic
Measurable spaces
Integration
Probabilistic coherence spaces
Denotational semantics
Probabilistic programming languages
Positive cones
ISSN:1860-5974, 1860-5974
Online Access:Get full text
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Summary: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.
ISSN:1860-5974
1860-5974
DOI:10.46298/lmcs-21(1:1)2025