On Quantitative Algebraic Higher-Order Theories

Authors Ugo Dal Lago, Furio Honsell, Marina Lenisa, Paolo Pistone

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Author Details

Ugo Dal Lago
  • Department of Computer Science and Engineering, University of Bologna, Italy
Furio Honsell
  • Department of Mathematical Sciences, Informatics and Physics, University of Udine, Italy
Marina Lenisa
  • Department of Mathematical Sciences, Informatics and Physics, University of Udine, Italy
Paolo Pistone
  • Department of Computer Science and Engineering, University of Bologna, Italy

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Ugo Dal Lago, Furio Honsell, Marina Lenisa, and Paolo Pistone. On Quantitative Algebraic Higher-Order Theories. In 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 228, pp. 4:1-4:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


We explore the possibility of extending Mardare et al.’s quantitative algebras to the structures which naturally emerge from Combinatory Logic and the λ-calculus. First of all, we show that the framework is indeed applicable to those structures, and give soundness and completeness results. Then, we prove some negative results clearly delineating to which extent categories of metric spaces can be models of such theories. We conclude by giving several examples of non-trivial higher-order quantitative algebras.

Subject Classification

ACM Subject Classification
  • Theory of computation → Program semantics
  • Theory of computation → Type theory
  • Quantitative Algebras
  • Lambda Calculus
  • Combinatory Logic
  • Metric Spaces


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