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The dynamics of belief: continuously monitoring and visualising complex systems

Edwin J. Beggs, John V. Tucker

TL;DR

This paper introduces a mode-based general system model for automation in human-centered contexts and couples it with a geometric visualization of system beliefs. It formalizes 'modes' and 'mode transitions' and realises beliefs as trajectories in simplicial complexes, enabling transparent explanations via $\\Delta_{\\mathcal{C}}$ and related constructs. By integrating Dempster–Shafer belief theory and a partition-of-unity visualization, the framework supports interpretable monitoring data, transition triggers, and scalable reasoning across social scenarios such as offender monitoring, hospital triage, and judicial processes. The approach aims to advance explainability and accountability in AI by linking data, belief, and visualization within a principled, hierarchical architecture, with implications for security, privacy, and governance in automated decision-making.

Abstract

The rise of AI in human contexts places new demands on automated systems to be transparent and explainable. We examine some anthropomorphic ideas and principles relevant to such accountablity in order to develop a theoretical framework for thinking about digital systems in complex human contexts and the problem of explaining their behaviour. Structurally, systems are made of modular and hierachical components, which we abstract in a new system model using notions of modes and mode transitions. A mode is an independent component of the system with its own objectives, monitoring data, and algorithms. The behaviour of a mode, including its transitions to other modes, is determined by functions that interpret each mode's monitoring data in the light of its objectives and algorithms. We show how these belief functions can help explain system behaviour by visualising their evaluation as trajectories in higher-dimensional geometric spaces. These ideas are formalised mathematically by abstract and concrete simplicial complexes. We offer three techniques: a framework for design heuristics, a general system theory based on modes, and a geometric visualisation, and apply them in three types of human-centred systems.

The dynamics of belief: continuously monitoring and visualising complex systems

TL;DR

This paper introduces a mode-based general system model for automation in human-centered contexts and couples it with a geometric visualization of system beliefs. It formalizes 'modes' and 'mode transitions' and realises beliefs as trajectories in simplicial complexes, enabling transparent explanations via and related constructs. By integrating Dempster–Shafer belief theory and a partition-of-unity visualization, the framework supports interpretable monitoring data, transition triggers, and scalable reasoning across social scenarios such as offender monitoring, hospital triage, and judicial processes. The approach aims to advance explainability and accountability in AI by linking data, belief, and visualization within a principled, hierarchical architecture, with implications for security, privacy, and governance in automated decision-making.

Abstract

The rise of AI in human contexts places new demands on automated systems to be transparent and explainable. We examine some anthropomorphic ideas and principles relevant to such accountablity in order to develop a theoretical framework for thinking about digital systems in complex human contexts and the problem of explaining their behaviour. Structurally, systems are made of modular and hierachical components, which we abstract in a new system model using notions of modes and mode transitions. A mode is an independent component of the system with its own objectives, monitoring data, and algorithms. The behaviour of a mode, including its transitions to other modes, is determined by functions that interpret each mode's monitoring data in the light of its objectives and algorithms. We show how these belief functions can help explain system behaviour by visualising their evaluation as trajectories in higher-dimensional geometric spaces. These ideas are formalised mathematically by abstract and concrete simplicial complexes. We offer three techniques: a framework for design heuristics, a general system theory based on modes, and a geometric visualisation, and apply them in three types of human-centred systems.
Paper Structure (33 sections, 3 theorems, 21 equations, 14 figures)

This paper contains 33 sections, 3 theorems, 21 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Automation and the challenge of explainability
  3. System theory as a tool
  4. A new general system model
  5. Contributions and structure of this paper
  6. Explainability and an anthropic view of systems
  7. What is an explanatory framework?
  8. Designing scenarios
  9. Designing components for actions
  10. Modes and visualisation
  11. What is a mode?
  12. Principles for designing modes
  13. What is visualised? Some simple examples
  14. Mathematical Tools 1: Simplicial complexes
  15. Simplicial complexes
  16. ...and 18 more sections

Key Result

proposition 1

To every abstract simplicial complex (as in Definition absc) is associated its standard realisation $\Delta_\mathcal{C}$, a simplicial complex as follows. Form a vector space $\mathbb{R}^\mathcal{M}$ with basis $e_\alpha$ for $\alpha\in \mathcal{M}$. The simplex spanned by $X\in \mathcal{C}$ is The simplex $\Delta_X$ is a $(|X|-1)$-simplex where $|X|$ is the size of $X$, and if $Y\subset X$ then

Figures (14)

  • Figure 1: Visualising a mode transition for a trigger mechanism and triage for emergency services
  • Figure 2: Triangle for Example \ref{['bvcu']}
  • Figure 3: A partition of unity for a cover by four sets
  • Figure 4: Localisation allowing computation of $\phi$
  • Figure 5: Diagram describing interventions in the alcohol-tagging problem
  • ...and 9 more figures

Theorems & Definitions (18)

  • definition 1
  • definition 2
  • proposition 1
  • definition 3
  • definition 4
  • definition 5
  • definition 6
  • remark 1
  • definition 7
  • definition 8
  • ...and 8 more