Core Concepts
Monotonic CGD are universal among all CGD, bridging the gap between CGD and GT frameworks.
Abstract
The article explores the relationship between Causal Graph Dynamics (CGD) and Global Transformations (GT), focusing on monotonic CGD. It delves into the formalism of both frameworks, highlighting their similarities and differences. The study aims to unify these concepts by showing that monotonic CGD can simulate any CGD, providing a deeper understanding of their interplay.
Structure:
Introduction to CGD and GT frameworks.
Initial Motivation for the study.
Comparison of technical features of CGD with CA studies.
Definitions of Labeled Graphs with Ports, Isomorphism, Consistency, Union, Intersection.
Definition of Disk in relation to locality in graphs.
Local Rule definition for describing local evolutions consistently.
Formal definitions of Causal Graph Dynamics (CGD).
Explanation of Global Transformations & Kan Extensions using category theory.
Unifying Causal Graph Dynamics and Kan Extensions through Monotonicity.
Simulation process for encoding original graphs into a larger universe for simulation purposes.
Conclusion on the universality of Monotonic CGD among all types of CGD.
Stats
"A function F : GΣ,∆,π → GΣ,∆,π is a causal graph dynamics (CGD) if there exists a radius r and a local rule f of radius r such that F(G) = { f(Gr_v) | v ∈ V(G) }."
"Given three posets A, B and C, and two monotonic functions i : A → B, f : A → C, the function Φ : B → C given by Φ(b) = sup { f(a) ∈ C | a ∈ A s.t. i(a) ⪯ b } is called the pointwise left Kan extension."
"The main result is the proof that causal graph dynamics are localisable functions."
"For any vertex v ∈ V(G), Gr_v ∈ Iv."
"Two graphs G and H are consistent precisely when they admit an upper bound in (GΣ,∆,π)."
Quotes
"This work uncovers the interesting class of Monotonic Causal Graph Dynamics."
"All non-monotonic CGDs are not left Kan extensions as we have developed so far."