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.

String diagrammatic calculi have become increasingly popular in fields such as quantum theory, circuit theory, probabilistic programming, and machine learning, where they enable resource-sensitive and compositional algebraic analysis.Traditionally, the equations of diagrammatic calculi only axiomatise exact semantic equality.However, reasoning in these domains often involves approximations rather than strict equivalences.In this work, we develop a quantitative framework for diagrammatic calculi, where one may axiomatise notions of distance between string diagrams.Unlike similar approaches, such as the quantitative theories introduced by Mardare et al., this requires us to work in a monoidal rather than a cartesian setting.We define a suitable notion of monoidal theory, the syntactic category it freely generates, and its models, where the concept of distance is established via enrichment over a quantale.To illustrate the framework, we provide examples from probabilistic and linear systems analysis.

We present a generalisation of the theory of quantitative algebras of Mardare, Panangaden and Plotkin where (i) the carriers of quantitative algebras are not restricted to be metric spaces and can be arbitrary fuzzy relations or generalised metric spaces, and (ii) the interpretations of the algebraic operations are not required to be nonexpansive. Our main results include: a novel sound and complete proof system, the proof that free quantitative algebras always exist, the proof of strict monadicity of the induced Free-Forgetful adjunction, the result that all monads (on fuzzy relations) that lift finitary monads (on sets) admit a quantitative equational presentation.

The framework of quantitative equational logic has been successfully applied to reason about algebras whose carriers are metric spaces and operations are nonexpansive. We extend this framework in two orthogonal directions: algebras endowed with generalised metric space structures, and operations being nonexpansive up to a lifting. We apply our results to the algebraic axiomatisation of the Łukaszyk–Karmowski distance on probability distributions, which has recently found application in the field of representation learning on Markov processes.

We study monads resulting from the combination of nondeterministic and probabilistic behaviour with the possibility of termination, which is essential in program semantics. Our main contributions are presentation results for the monads, providing equational reasoning tools for establishing equivalences and distances of programs.