A Diagrammatic Axiomatisation of Behavioural Distance of Nondeterministic Processes

2026-04-29

AI Summary

This work addresses the formalization and axiomatization of behavioral distances between nondeterministic processes, moving beyond the limitations of classical Boolean equivalences. Building upon Milner’s graph-based models, it introduces string diagrams as a novel syntactic framework that dispenses with variables and binding mechanisms, enabling a variable-free representation focused on bisimulation rather than language equivalence. This approach naturally supports the decomposition of recursive structures and compositional semantic reasoning. The paper presents an axiomatic system that is both sound and complete with respect to the proposed behavioral distance, thereby laying a rigorous foundation for quantitative analysis in extended models such as weighted transition systems and significantly advancing the formal reasoning and quantitative comparison of nondeterministic processes.

Abstract

Behavioural distances provide a quantitative approach to comparing the states of transition systems, moving beyond traditional Boolean notions of equivalence. In this paper, we develop a sound and complete axiomatisation of behavioural distance for nondeterministic processes using Milner's charts, a model that generalises finite-state automata by incorporating variable outputs. Charts provide a compelling setting for studying behavioural distances because they shift the focus from language equivalence to bisimilarity. Their axiomatic study lays the groundwork for quantitative analysis of more expressive models, such as weighted transition systems. To formalise this approach, we adopt string diagrams as our syntax of choice. String diagrams closely mirror the graphical structure of charts, while providing a rigorous formalism that supports inductive reasoning and compositional semantics. Unlike traditional algebraic syntaxes, which require additional mechanisms such as binders and substitution, string diagrams offer a variable-free representation where recursion naturally decomposes into simpler components. This makes them well-suited for reasoning about behavioural distances and aligns with broader efforts to axiomatise automata-theoretic equivalences through a unified diagrammatic framework.

Problem

• behavioural distance
• nondeterministic processes
• axiomatisation
• string diagrams
• bisimilarity

Innovation

• behavioural distance
• string diagrams
• nondeterministic processes
• axiomatisation
• bisimilarity