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Ralph Sarkis

Publications and source records attributed to Ralph Sarkis.

7 recordsLinked to original sources
arXiv2026

Complete Diagrammatic Axiomatisations of Relative Entropy

Relative entropy is a fundamental class of distances between probability distributions, with widespread applications in probability theory, statistics, and machine learning. In this work, we study relative entropy from a categorical perspective, viewing it as a quantitative enrichment of categories of stochastic matrices. We consider two natural monoidal structures on stochastic matrices, given by the Kronecker product and the direct sum. Our main results are complete axiomatisations of Kullback-Leibler divergence and, more generally, of R\'enyi divergences of arbitrary order, for each such structure. Our axiomatic theories are formulated within the framework of quantitative monoidal algebra, using a graphical language of string diagrams enriched with quantitative equations.

cs.LO↗
arXiv2025

Graded String Diagrams for Imprecise Probability and Causal Intervention

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. This culminates in a representation, as string diagrams with grading wires, of programs with primitives for nondeterministic and probabilistic choices and conditioning.

math.CT↗
arXiv2024

Quantitative Monoidal Algebra: Axiomatising Distance with String Diagrams

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.

cs.LO↗
arXiv2023

Universal Quantitative Algebra for Fuzzy Relations and Generalised Metric Spaces

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.

cs.LO↗
arXiv2022

Beyond Nonexpansive Operations in Quantitative Algebraic Reasoning

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.

cs.LO↗
arXiv2021

Semialgebras and Weak Distributive Laws

Motivated by recent work on weak distributive laws and their applications to coalgebraic semantics, we investigate the algebraic nature of semialgebras for a monad. These are algebras for the underlying functor of the monad subject to the associativity axiom alone-the unit axiom from the definition of an Eilenberg-Moore algebras is dropped. We prove that if the underlying category has coproducts, then semialgebras for a monad M are in fact the Eilenberg-Moore algebras for a suitable monad structure on the functor id + M , which we call the semifree monad M^s. We also provide concrete algebraic presentations for semialgebras for the maybe monad, the semigroup monad and the finite distribution monad. A second contribution is characterizing the weak distributive laws of the form M T => T M as strong distributive laws M^s T => T M^s subject to an additional condition.

cs.LO↗