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

Authors Davide Barbarossa , Paolo Pistone

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Davide Barbarossa
  • Università di Bologna, Italy
Paolo Pistone
  • Università di Bologna, Italy

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Davide Barbarossa and Paolo Pistone. Tropical Mathematics and the Lambda-Calculus I: Metric and Differential Analysis of Effectful Programs. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 14:1-14:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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 semantics 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Lambda calculus
  • Theory of computation → Categorical semantics
  • Theory of computation → Linear logic
  • Relational semantics
  • Differential lambda-calculus
  • Tropical semiring
  • Program metrics
  • Lawvere quantale


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