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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| Vydáno v: | Logical methods in computer science Ročník 21, Issue 1; číslo 1 |
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| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
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Logical Methods in Computer Science Association
01.01.2025
Logical Methods in Computer Science e.V |
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| ISSN: | 1860-5974, 1860-5974 |
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| 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. |
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| AbstractList | 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. 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 functions introduced in earlier work and thesecond based on a new notion of integrable analytic function on cones. |
| Author | Ehrhard, Thomas Geoffroy, Guillaume |
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| Keywords | Linear logic Measurable spaces Integration Probabilistic coherence spaces Denotational semantics Probabilistic programming languages Positive cones |
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| Title | Integration in Cones |
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