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Announcement
Introduction
Materials
Q&A
Livestreams
Registry
Tools
Helping
Gallery
Archives
A Class of Models with the Potential to Represent Fundamental Physics
Introduction
Basic Form of Models
Basic Structure
First Example of a Rule
A Slightly Different Rule
Self-Loops
Multiedges
The Representation of Rules
Rules Depending on More Than One Relation
Termination
Connectedness
Typical Behaviors
The Representation of Rules
The Number of Possible Rules
Initial Conditions
Rules Depending on a Single Unary Relation
Rules Depending on a Single Binary Relation
Rules Depending on One Ternary Relation
Rules Depending on More Than One Relation: The 2
2
3
2
Case
Rules with Signature 2
2
4
2
Binary Rules with Signatures Beyond 2
2
4
2
Rules Depending on Two Ternary Relations: the 2
3
3
3
Case
Rules Involving More Ternary Relations
Rules with Mixed Arity
Multiple Transformation Rules
Rules Involving Disconnected Pieces
Termination
The Effect of Initial Conditions
Behavior without Growth
Random Rules and Overall Classification of Behavior
Limiting Behavior and Emergent Geometry
Recognizable Geometry
Hyperbolic Space
Geometry from Subdivision
Nested Patterns
The Notion of Dimension
Dimension-Related Characterizations
Curvature
Homogeneity and Local Graph Neighborhoods
Adjacency Matrices and Age Distributions
Other Graph Properties
Graph Properties Conserved by Rules
Apparent Randomness and Growth Rates
Statistical Mechanics
The Effect of Perturbations
Geodesics
Functions on Graphs
Manifolds and Model Spaces
The Updating Process for String Substitution Systems
String Substitution Systems
The Phenomenon of Causal Invariance
States Graphs
Typical Multiway Graph Structures
Testing for Causal Invariance
The Frequency of Causal Invariance
Events and Their Causal Relationships
Causal Graphs for Particular Updating Sequences
The Significance of Causal Invariance
Causal Foliations and Causal Cones
Causal Graphs for Infinite Evolutions
Typical Causal Graphs
Limits of Causal Graphs
Foliations and Coordinates on Causal Graphs
The Concept of Branchial Graphs
Typical Forms of Branchial Graphs
Foliations of the Multiway Graph and the Structure of Branchial Space
The Relationship between Graphs, and the Multiway Causal Graph
Weighted Multiway Graphs
Effective Causal Invariance
Generational Evolution
The Updating Process in Our Models
Updating Events and Causal Dependence
Multiway Systems for Our Models
Causal Invariance
Testing for Causal Invariance
Causal Graphs for Causal Invariant Rules
The Role of Causal Graphs
Typical Causal Graphs
Large-Scale Structure of Causal Graphs
Foliations of Causal Graphs
Causal Disconnection
Global Symmetries and Conservation Laws
Local Symmetries
Branchial Graphs and Multiway Causal Graphs
Equivalence and Computation in Our Models
Correspondence with Other Systems
Alternative Formulations
Computational Capabilities of Our Models
Potential Relation to Physics
Introduction
Basic Concepts
Potential Basic Translations
The Structure of Space
Time and Spacetime
Motion and Special Relativity
The Vacuum Einstein Equations
Matter, Energy and Gravitation
Elementary Particles
Reversibility and Irreversibility
Cosmology, Expansion & Singularities
Basic Concepts of Quantum Mechanics
Quantum Formalism
Quantum Measurement
Operators in Quantum Mechanics
Wave-Particle Duality, Uncertainty Relations, Etc.
Correspondence between Relativity and Quantum Mechanics
Event Horizons and Singularities in Spacetime and Quantum Mechanics
Local Gauge Invariance
Units and Scales
Specific Models of the Universe
Multiway Systems in the Space of All Possible Rules
Additional Material
Notes & Further References
Appendix: Implementation
Appendix: Graph Types
Acknowledgements
Tools, Data & Source Materials
References
Index
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technical introduction
7
Equivalence and Computation in Our Models
Correspondence with Other Systems
Alternative Formulations
Computational Capabilities of Our Models
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