Tropical Mathematics and the Lambda Calculus I: Metric and Differential Analysis of Effectful Programs

We study the interpretation of the lambda-calculus in a framework based on tropical mathematics, and we show that it provides a unifying framework for two well-developed quantitative approaches to program semantics: on the one hand program metrics, based on the analysis of program sensitivity via Li...

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Bibliographic Details
Published inarXiv.org
Main Authors Barbarossa, Davide, Pistone, Paolo
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 27.11.2023
Subjects
Differential calculus
Mathematical analysis
Metric space
Semantics
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Summary:We study the interpretation of the lambda-calculus in a framework based on tropical mathematics, and we show that it provides a unifying framework for two well-developed quantitative approaches to program semantics: on the one hand program metrics, based on the analysis of program sensitivity via Lipschitz conditions, on the other hand resource analysis, based on linear logic and higher-order program differentiation. To do that we focus on the semantic arising from the relational model weighted over the tropical semiring, and we discuss its application to the study of "best case" program behavior for languages with probabilistic and non-deterministic effects. Finally, we show that a general foundation for this approach is provided by an abstract correspondence between tropical algebra and Lawvere's theory of generalized metric spaces.
ISSN:2331-8422