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Causal Graph Dynamics and Kan Extensions

Luidnel Maignan, Antoine Spicher

TL;DR

This paper addresses whether Causal Graph Dynamics (CGDs) can be modeled as Global Transformations (GTs) via Kan extensions. It reveals that the CGD-to-GT correspondence is partial: monotonic CGDs precisely correspond to left Kan extensions along the subgraph order, and any CGD can be simulated by a monotonic CGD through a constructive encoding $\omega$, enabling a universal monotonic representation. A categorical treatment of renaming-invariance is developed, turning graphs into a category where renamed graphs are isomorphic, and CGDs are shown to arise as Kan extensions in this setting under a unique-conjugate assumption. The results thus unify CGDs with GTs in a Kan-extension framework, provide a concrete monotonic simulation of general CGDs, and illuminate how renaming-invariance can be folded into the categorical structure to obtain a more robust, finite-disk description. These findings advance a principled understanding of dynamic graph transformations and offer a path toward a canonical, universal theory of local-to-global graph dynamics with categorical guarantees.

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.

Causal Graph Dynamics and Kan Extensions

TL;DR

This paper addresses whether Causal Graph Dynamics (CGDs) can be modeled as Global Transformations (GTs) via Kan extensions. It reveals that the CGD-to-GT correspondence is partial: monotonic CGDs precisely correspond to left Kan extensions along the subgraph order, and any CGD can be simulated by a monotonic CGD through a constructive encoding , enabling a universal monotonic representation. A categorical treatment of renaming-invariance is developed, turning graphs into a category where renamed graphs are isomorphic, and CGDs are shown to arise as Kan extensions in this setting under a unique-conjugate assumption. The results thus unify CGDs with GTs in a Kan-extension framework, provide a concrete monotonic simulation of general CGDs, and illuminate how renaming-invariance can be folded into the categorical structure to obtain a more robust, finite-disk description. These findings advance a principled understanding of dynamic graph transformations and offer a path toward a canonical, universal theory of local-to-global graph dynamics with categorical guarantees.

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.
Paper Structure (34 sections, 29 theorems, 39 equations, 4 figures)

This paper contains 34 sections, 29 theorems, 39 equations, 4 figures.

Table of Contents

  1. Initial Motivation.
  2. Initial Plan.
  3. Actual Plan.
  4. Preliminaries
  5. Notations
  6. Causal Graph Dynamics
  7. Global Transformations & Kan Extensions
  8. Unifying Causal Graph Dynamics and Kan Extensions
  9. The Underlying Partial Order
  10. Comparing Disks and Subgraphs
  11. Monotonic and General Causal Graph Dynamics
  12. Monotonic CGDs as Kan Extensions.
  13. The Non-Monotonic Case.
  14. Universality of Monotonic Causal Graph Dynamics
  15. Key Ideas of the Simulation
  16. ...and 19 more sections

Key Result

Proposition 1.7

Let $F: {\mathcal{G}}_{\Sigma,\Delta,\pi} \rightarrow {\mathcal{G}}_{\Sigma,\Delta,\pi}$ be a CGD. For any renaming $R$, $F$ admits at least a conjugate renaming $R'$, i.e., $F \circ R = R' \circ F$. Moreover, for any local rule $f$ of $F$, $\mathop{\mathrm{Conj}}\nolimits_f(R) \subseteq \mathop{\ma

Figures (4)

  • Figure 1: Moving particle CGD - non-monotonic behavior. Each row represents an example of evolution with a graph $G$ on the left and $F(G)$ on the right. Colors correspond to vertex names.
  • Figure 2: Moving particle CGD - monotonic behavior. Compared to Figure \ref{['fig:non-monotonic-particle']}, the vertices have two additional ports ($l'$ and $r'$) and unlabeled vertices are now labeled by $\star$.
  • Figure 3: Different classes of graphs: from left to right, a total graph, a coherent partial graph, an incoherent graph.
  • Figure 4: We need to determine for each missing entity attached to a vertex or edge of $f(G^r_c)$, whether all other $f(G^r_v)$ agree to consider this entity as missing. Property 2 of local rules (Definition \ref{['def:graph_local_rule']}) tells us that only disks $G^r_v$ that intersect $G^r_c$ need to be checked. The furthest such $v$ are at distance $2r+2$. From there, we need to ask for radius $r$ (so radius $r'=3r+2$ from $c$) to include the entirety of $G^r_v$ (including its border at $r+1$ from $v$, and therefore $3r+3$ at most from $c$).

Theorems & Definitions (75)

  • Definition 1.1: Labeled Graph with Ports
  • Definition 1.2: Renaming
  • Definition 1.3: Consistency, Union, Intersection
  • Definition 1.4: Disk
  • Definition 1.5: Local Rule
  • Definition 1.6: Causal Graph Dynamics (CGD)
  • Proposition 1.7: Renaming Invariance
  • proof
  • Definition 1.8: Pointwise Left Kan Extension for Posets
  • Remark 1.9
  • ...and 65 more