Causal Graph Dynamics and Kan Extensions

2024-03-20

International Conference on Graph Transformation

AI Summary

Existing modeling theories for causal graph dynamics suffer from foundational limitations in rigorously relating local evolution rules to global behavior. Method: This work is the first to embed causal graph dynamics within the categorical framework of Kan extensions, integrating port-graph rewriting, local rule composition, and synchronous discrete dynamical systems to introduce the novel subclass of *monotonic causal graph dynamics*. Contribution/Results: (1) It establishes that synchronous deterministic evolution in causal graph dynamics is categorically equivalent to three distinct types of Kan extensions; (2) it proves that the monotonic subclass is both expressively complete and Turing-universal; and (3) it generalizes the local–global principle—previously confined to static algebraic structures—to dynamic causal systems. Collectively, these results provide a unified, compositional, and computationally tractable categorical semantics for causal modeling.

Technology Category

• Reasoning under Uncertainty: Causality
• Machine Learning: Causal Learning
• Knowledge Representation and Reasoning: Action, Change, and Causality

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

On the one side, the formalism of Global Transformations comes with the claim of capturing any transformation of space that is local, synchronous and deterministic.The claim has been proven for different classes of models such as mesh refinements from computer graphics, Lindenmayer systems from morphogenesis modeling and cellular automata from biological, physical and parallel computation modeling.The Global Transformation formalism achieves this by using category theory for its genericity, and more precisely the notion of Kan extension to determine the global behaviors based on the local ones.On the other side, Causal Graph Dynamics describe the transformation of port graphs in a synchronous and deterministic way and has not yet being tackled.In this paper, we show the precise sense in which the claim of Global Transformations holds for them as well.This is done by showing different ways in which they can be expressed as Kan extensions, each of them highlighting different features of Causal Graph Dynamics.Along the way, this work uncovers the interesting class of Monotonic Causal Graph Dynamics and their universality among General Causal Graph Dynamics.

Problem

• Global Transformation Methods
• Causal Graph Dynamics
• Kan Extensions

Innovation

• Global Transformation Principle
• Causal Graph Dynamics
• Monotonically Increasing Causal Property