A Class of Models with the Potential to Represent Fundamental Physics
  1. Introduction
  2. Basic Form of Models
  3. Typical Behaviors
  4. Limiting Behavior and Emergent Geometry
  5. The Updating Process for String Substitution Systems
  6. The Updating Process in Our Models
  7. Equivalence and Computation in Our Models
  8. Potential Relation to Physics
  9. Additional Material
  10. References
  11. Index

2.1 Basic Structure

At the lowest level, the structures on which our models operate consist of collections of relations between identical (but labeled) discrete elements. One convenient way to represent such structures is as graphs (or, in general, hypergraphs). The elements are the nodes of the graph or hypergraph. The relations are the (directed) edges or hyperedges that connect these elements.

For example, the graph

ResourceFunction["WolframModelPlot"][{{1, 2}, {1, 3}, {2, 3}, {4, 1}}, VertexLabels -> Automatic]

corresponds to the collection of relations

{{1, 2}, {1, 3}, {2, 3}, {4, 1}}

The order in which these relations are stated is irrelevant, but the order in which elements appear within each relation is considered significant (and is reflected by the directions of the edges in the graph). The specific labels used for the elements (here 1, 2, 3, 4) are arbitrary; all that matters is that a particular label always refer to the same element.