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String Diagrams for Graded Monoidal Theories, with an Application to Imprecise Probability

Authors Ralph Sarkis , Fabio Zanasi

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  • Theory of computation → Equational logic and rewriting
  • Mathematics of computing → Probabilistic representations
Keywords
  • string diagrams
  • graded categories
  • probability
  • nondeterminism

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Abstract

We introduce string diagrams for graded symmetric monoidal categories. Our approach includes a definition of graded monoidal theory and the corresponding freely generated syntactic category. Also, we show how an axiomatic presentation for the graded theory may be modularly obtained from one for the grading theory and one for the base category. The Para construction on monoidal actegories is a motivating example for our framework. As a case study, we show how to axiomatise a variant of the graded category ImP, recently introduced by Liell-Cock and Staton to model imprecise probability [Liell-Cock and Staton, 2025]. This culminates in a representation, as string diagrams with grading wires, of programs with primitives for nondeterministic and probabilistic choices and conditioning.

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Ralph Sarkis and Fabio Zanasi. String Diagrams for Graded Monoidal Theories, with an Application to Imprecise Probability. In 11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 342, pp. 5:1-5:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CALCO.2025.5

Author Details

Ralph Sarkis
  • Department of Computer Science, University College London, UK
Fabio Zanasi
  • Department of Computer Science, University College London, UK

Funding

This research was supported by the ARIA Safeguarded AI programme.

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