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

Index

  1. A
  2. B
  3. C
  4. D
  5. E
  6. F
  7. G
  8. H
  9. I
  10. J
  11. K
  12. L
  13. M
  14. N
  15. O
  16. P
  17. Q
  18. R
  19. S
  20. T
  21. U
  22. V
  23. W
  24. X-Z

A

  • Abelian group, 198
  • Acknowledgements, 487
  • Action, 407, 441
  • Adjacency matrices, 181
    • and machine learning, 144
  • Adjacency tensors
    • for hypernetworks, 181
  • ADM formalism, 259
  • Age distribution
    • of edges, 181
  • Age of universe, 467
  • Algebra of operators, 449
  • Algebraic surface, 151
  • Alternating groups, 198
    • as symmetry of evolution, 361
  • Ambiguity
    • of rule updates, 86
  • Amplitudes in quantum theory, 407
    • as complex numbers, 441
  • Analyticity
    • see also Continuity
  • Ancestors
    • in multiway graphs, 262
  • Angular momentum
    • intrinsic of particles, 407
  • Apollonian packing, 156
  • Arbelos, 155
  • Archives, 487
  • Arrow of time, 413, 429
  • Associative calculi, 477
  • Assortativity, 183
  • Astrophysical processes, 470
  • Asymptotic, 93
  • Asymptotic growth, 93, 177
  • Automated theorem proving, 204, 436
  • Automorphism
    • hypergraph, 366

B

  • Ball
  • Bell numbers, 93, 95
  • Bells inequalities, 409
  • BellB, 93, 95
  • Betweenness centrality, 183
  • Binary relations
    • rules based on, 99
  • Binomial, 77
  • Binomial tree, 77
  • Black holes, 409
    • and quantum information, 461
    • as analog of quantum measurement, 446
    • galactic, 470
    • qubit as analog of, 458
  • Blockchain, 304
  • Boolean expression confluence, 205
  • Boost, 417
  • Branch pairs, 220
    • and commutators, 450
    • in hypergraphs, 313
  • Branchial distance, 451
  • Branchial graphs, 260, 370, 486
    • and entanglement, 438
    • weights in, 289
  • Branchial motion, 454
  • Branchial space, 438, 454
  • Branchlike connections, 454
  • Branchlike hypersurface, 434, 439
  • Branchlike separation, 444
  • Breadth-first scan, 255
  • Buckyball graph
    • directed, 254
  • Buckyball graphs, 174, 179, 188
  • Buckyball Sierpiński graphs, 197
  • Busy beaver, 135

C

  • c
    • see Speed of light
  • Cactus-like graphs, 81, 100
  • Canonical forms
    • see also Confluence
  • Canonical systems, 477
  • Canonicalization
    • of rules, 91
  • Cantor set, 196
  • Category theory, 477
  • Causal disconnection, 347
  • Causal dynamical triangulation, 477
  • Causal graphs, 230, 237, 314, 431, 484
    • disconnection in, 457
    • foliations of, 342
    • in spacetime, 414
    • large-scale structure of, 340
    • limits of, 251
    • typical, 245, 325
  • Causal invariance, 205, 208, 307
    • adding completions for, 294
    • and commutation relations, 450
    • and Einsteins equations, 419
    • and quantum mechanics, 433
    • effective, 290
    • frequency of, 226
    • in rule space, 474
    • in trivalent graphs, 392
    • testing for, 219, 311
    • total, 207, 219
  • Causal relationships
    • in substitution systems, 228
  • Causal set theory, 477
  • Cayley graphs, 197, 400
  • Cellular automata
    • additive
    • see Additive cellular automata
    • asynchronous
    • see Sequential cellular automata
    • block
    • see Block cellular automata
    • class 3, 138
    • emulation of, 384
    • foliations in, 385
    • invertible
    • see Reversible cellular automata
    • particles in, 190, 425
    • perturbations in, 190
    • reversible
    • see Reversible cellular automata
    • rule labeling in, 92
    • second-order
    • see Reversible cellular automata
    • totalistic
    • see Totalistic cellular automata
    • see also Additive cellular automata
  • Cellular automaton
    • on graph, 185
  • Centrality, 183
  • Charge, 407
  • Circle packing, 156
  • Circles
    • area of on sphere, 174
  • Class 3 behavior, 138
    • see also Rule 30, 90, etc.
  • Class 4 behavior
    • see also Rule 110, etc.
  • Classical physics, 432
  • Classical-like states, 435
  • Classification
    • of behavior, 143
  • Closed timelike curves, 346, 410, 432
  • Clustering coefficient, 183, 184
  • CMB, 431
  • Coarse-graining, 429
    • and completions, 294
  • Coherent structures
    • see also Localized structures
  • Collinearity
    • in multiway graphs, 273
  • Colored graphs, 392
  • Combinatory logic, 399
  • Common successors, 260
  • Commutation relations, 450
    • and symmetry operations, 359
    • in spacetime, 456
  • Company, 487
  • Complete graphs, 261
  • Completions, 293
    • by observers, 436
  • Complex numbers
    • as amplitudes, 441
  • Computation
    • see also Programs
  • Computation universality
    • and rule space, 474
  • Computational exploration, 477
  • Computational irreducibility, 403
    • and quantum observers, 437
  • Cones
    • rule generating, 149
  • Confluence, 208
    • see also Canonical forms
    • see also Causal invariance
    • see also ChurchRosser property
  • Conformal field theory, 477
  • Connected components, 131
  • Connections (on fiber bundles), 463
  • Connectivity, 88
  • Consciousness
    • see also Intelligence
    • see also Thinking
  • Conservation laws, 359
    • for particles, 426
  • Conservation of energy, 408, 423
  • Continuity
    • on graphs, 195
  • Continuity equation
    • on graph, 196
  • Continuous space, 148, 411
    • geodesics in, 192
  • Continuum limits
    • of networks, 167
  • Contravariant vectors, 192
  • Coordinates
    • on ellipsoid, 193
  • Correlation function, 188
  • Cosmic censorship hypothesis, 461
  • Cosmic microwave background, 431
  • Cosmological arrow of time, 413
  • Cosmological constant, 408, 410
  • Cosmology, 430
    • see also Universe
  • Covariant derivatives, 194
  • Crinkling, 157
  • Critical pairs, 220
    • see also Branch pairs
  • Crossings
    • in graphs, 186
  • CTCs
    • see Closed timelike curves
  • Cube
    • as initial condition, 137
    • from multiway graph, 213
  • Cubic graphs, 390
  • Cubic grid, 164
  • Curvature
    • and geodesics, 192
    • in fractional-dimensional space, 199
    • in networks, 174
    • in space, 413
    • in spacetime, 418
    • vs. dimension change, 430
  • Cusps, 153
  • Cycle index polynomials, 95
  • Cycles
    • in graphs, 184
  • Cyclic behavior, 132, 140
  • Cylinders
    • rule generating, 152

D

  • d-dimensional grid, 164
  • Dangling connections, 119, 390
  • Dark energy, 430
  • Dark matter, 409, 469
  • de Sitter space, 410
  • Decoherence, 436, 456
    • and computational irreducibility, 449
  • Degree centrality, 183
  • Degree distribution, 184
  • Depth-first scan, 256
  • Description frame
    • in rule space, 475
  • Diameter
    • of graphs, 183
  • Diamonds
    • in multiway graphs, 206
  • Digital physics, 477
  • Dimensions
    • local estimates of, 180
    • of networks, 162
    • of space, 163, 411
  • Diophantine equations
    • solutions of, 425
  • Disconnected behavior, 113, 126, 134
  • Disconnected rules, 96, 130
  • Disconnection
  • Distance matrices, 271
  • Distances
    • directed, 258
    • in graph, 163
    • in hypergraph, 410
    • spatial, 152
  • Divergence
    • of geodesics, 192
    • on graph, 196
  • Dodecahedron graph, 197
  • Doublepushout graph grammars, 477
  • DPO graph grammars, 477
  • Dynamical systems theory, 189

E

  • E=c^2m, 422
  • Early universe, 431
    • oligons in, 469
  • Edge effects, 165
  • Effective causal invariance, 290
  • Eigenmodes
    • of graph, 185
  • Eigenvector centrality, 183
  • Einsteins equations, 418, 423
    • and units, 465
  • Electromagnetic fields
    • and gauge invariance, 463
  • Electrons
    • radius of, 468
  • Elementary energy, 464, 468
  • Elementary length, 464, 468
  • Elementary particles, 424
  • Elementary time, 464, 468
  • Ellipsoid, 193
    • as model for curvature, 199
  • Encodings
    • of models, 404
  • Energy, 407, 420
  • Energy-momentum tensor, 422
  • Ensembles
    • difficulty of using, 188
  • Entanglement, 407, 437
  • Entanglement cones, 455
  • Entanglement horizon, 460
  • Entanglement speed (ζ), 456, 469
  • Entropic arrow of time, 413
  • Entropy
    • of subgraphs, 178
    • see also Second Law of thermodynamics
  • Equational logic, 204, 400
  • Equations
    • see also Differential equations
    • see also Diffusion equation
    • see also Dirac equation
    • see also Einstein equations
    • see also KleinGordon equation
    • see also Laplace equation
    • see also Maxwells equations
    • see also NavierStokes equations
    • see also Schrödinger equation
  • Equilibrium
    • and multiway systems, 429
    • in graph evolution, 183
  • Equivalence
    • Principle of Computational
    • see Principle of Computational Equivalence
  • Equivalence to substitution systems, 377
  • ER=EPR, 461
  • Ergodicity
    • for space, 411
  • Error bars
    • for dimension estimate, 166
  • Escape velocity
    • branchial, 461
  • Euclidean distance, 191
  • Euclidean space, 148
    • as model space, 196
  • Event horizons, 431, 457
    • as analog of quantum measurement, 446
    • causal and branchial, 461
    • branchial, 457
  • Events
  • Excluded subgraphs, 186
  • Expansion of universe, 408, 430
    • and energy, 423
    • units and, 465

F

  • Factorial (n!), 256
  • Fano plane, 272
  • FeatureSpacePlot, 144
  • Feynman path integral, 441
  • Fiber bundles, 463
  • Fibonacci sequence, 131, 261, 301
  • Fibonacci tree, 201
  • Field theory
    • see also Quantum field theory
  • Finite groups, 198
  • Finite-size effects
    • on dimension, 165
  • Finite-size systems, 140
  • Flat (associative), 478
  • Fluid flow, 441
  • Folds
    • in surfaces, 153

  •  
  • Foliations
    • and branchial graphs, 275
    • and coordinates, 255
    • and fiber bundles, 463
    • and realistic observers, 415
    • causal, 237
    • in cellular automata, 385
    • of causal graphs, 342
  • Fractal pattern
    • dimension of, 167
    • rule generating, 110, 160
  • Fractals
    • see also Nesting
  • Free group, 197
  • Freezing time
    • in quantum measurement, 446
  • Frilled structure, 103
  • Functions on graphs, 195

G

  • Gases
    • see also Fluid flow
  • Gauge invariance, 408, 462
  • Gausss law
    • on graph, 196
  • Gaussian distributions
    • of weights, 290
  • General covariance, 415
  • General relativity, 408
    • coordinates in, 259
  • Generating function, 164
  • Generational evolution, 294
  • Generational multiway system, 435, 443
  • Generational states, 443
  • Generators
    • of groups, 197
  • Genus
    • for graph drawing, 186
  • Geodesic balls, 163
  • Geodesic sphere, 174
  • Geodesics, 191
    • bundle shapes of, 419
    • in multiway space, 439
    • in space, 413
  • Geometric group theory, 198
  • Geometry
    • emergent, 147
    • of branchial space, 438
    • of causal graphs, 321
  • Git, 304
  • Global clustering coefficient, 183
  • Global counter
    • for elements, 304
  • Global symmetries, 359
  • Globular behavior, 143
  • Globular structures, 115, 138
    • and causal graphs, 330
    • with unordered hypergraphs, 397
  • God
    • see also Theology
  • Golden ratio, 131
  • Gorard, Jonathan (Singapore/UK, 1997 ), 487
  • Gradient
    • on graph, 196
  • Graph diameter, 183
    • in branchial graph, 261
  • Graph grammars, 477
  • Graph radius, 183
  • Graph rewriting systems, 477
  • Graph sizes, 187
  • Graphons, 189
  • Graphs
    • branchial, 260
    • colored, 392
    • correspondence to hypergraphs, 387
    • functions on, 195
    • infinite, 189
    • non-overlapping, 392
    • ordered, 392
    • rendering of, 163
    • types of, 483
    • undirected, 392
    • vertex transitive, 196
  • Gravitational constant (G), 409, 465
  • Gravity, 420
  • Grids
    • as causal graphs, 252, 331
    • dimension of, 163
    • emergence of, 123
    • from causal graphs, 249
    • rule generating, 148
  • Groups
  • Growth
    • based on rules, 129

H

  • Halting, 87, 135
  • Hamiltonians
  • Hardy fields, 177
  • Hash, 304
  • Hausdorff dimension, 167
  • Hawking radiation
    • in branchial space, 458
  • Heisenberg group, 198
  • Hexagonal faces
  • Hexagonal grid, 424
    • causal graph as, 249
  • Hilbert spaces, 440, 454
    • from branchial graphs, 272
  • Holographic principle, 409
  • Homogeneity
    • in dimension estimate, 166
    • of universe, 431
  • Hubble time, 467
  • Hyperbolic space, 155
    • curvature of, 175
    • from causal graphs, 248
  • Hyperbolicity
    • failure of, 346
  • Hypercubes, 271
    • from multiway graph, 213
  • Hypergeometric series, 164
  • Hypergraph isomorphism, 91, 306
  • Hypergraphs
    • automorphisms of, 366
    • emulating graphs with, 387
    • enumeration of, 95
    • events in, 303
    • representation of, 81
    • unifications of, 311
    • unordered, 396
  • Hypersphere, 165

I

  • Icosahedral group, 369
  • Ideal gas
    • of oligons, 469
  • Implementation
    • of models, 478
  • Incidence geometry, 272
  • IndexGraph, 483
  • Inequivalent rules, 92
  • Inertial frames, 409, 415
  • Infinite graphs, 189
  • Inflation
    • in cosmology, 431
  • Initial conditions, 136
    • and disconnection, 357
    • for universe, 431
    • possible, 95
  • Initial creation event, 357
  • Intelligence
    • see also Thinking
  • Interference, 437
  • Intermediate growth
    • in groups, 199
  • Intermediate scales
    • of growth, 177
  • Intermediate strings
    • in branch resolution, 223
  • Intrinsic curvature, 193
  • Invariant measure
  • Invariants
    • in graph evolution, 185
    • see also Conserved quantities
  • Involutions, 256
  • Irreducibility
    • see Computational irreducibility
  • Irreversibility, 429
  • Isometry
    • in graphs, 179
  • Isomorphism
    • hypergraph, 91
    • of hypergraphs, 306

K

  • K33 non-planar graph, 186, 424
  • Kaons, 429
  • Kepler tiling, 159
  • Kinetic energy, 422
  • Knitting
    • in behavior of rule, 123
  • KnuthBendix algorithm, 293
  • Kuratowskis theorem, 186, 424

L

  • Lagrangian density, 407, 441
  • Lambda calculus, 205, 399
  • Lambert W function, 467
  • Laplacian
  • Lapse function, 259
  • Lattice operations
    • for group approximation, 369
  • Lattice patterns, 125
  • Lattice theory, 273
  • Left connectivity, 92
  • Lemma
    • added by observers, 436
  • Length
  • Lettuce leaf, 157
  • Lie groups, 462
    • as possible limits, 369
  • Light cones, 409
    • analog of in rule space, 474
    • and perturbations, 190
    • branchial analog of, 454
    • in substitution systems, 239
  • Lightning bolt, 468
  • Limiting shape, 153
  • Limits, 406
    • in computing dimension, 167
    • see also Continuum limits
  • Line-like behavior, 143
  • Lineage
    • for elements, 304
  • Linear recurrences, 187
    • for substitution system, 202
  • Lobed surface, 151
  • Local clustering coefficient, 184, 261
  • Local conservation laws
    • see also Conservation laws
  • Local gauge invariance, 462
  • Local symmetries, 365
  • Locality
    • and causal cones, 239
  • Localized structures, 425
  • Log differences, 165
  • Loop quantum gravity, 477
  • Loops
    • in causal graphs, 345
  • Lorentz invariance, 408
  • Lorentz transformations, 417
  • Lorentzian manifold, 259
  • LQG, 477

M

  • Machine learning, 144
  • Mandelbrot set
    • rule looking like, 81, 100
  • Manhattan distance, 191
  • Manifolds, 192, 196
    • from emulating Turing machine, 382
  • Many-worlds models, 433
  • Mass, 422
  • Massenergy equivalence, 422
  • Mathematica
    • patterns
    • see Patterns (in Wolfram Language)
  • Matrices
    • adjacency, 181
  • Matter, 420
  • Maximum scan, 203
  • Mean clustering coefficient, 183
  • Measurement
    • in quantum mechanics, 442
  • Mesh
    • rule generating, 148
  • Metric tensors, 192, 419
  • Minkowski space, 259
  • Minors
  • Mixed arity, 127
  • Mobile automata
    • as analogy, 333
  • Model spaces, 196
  • Models
    • see also Simulations
    • see also Ultimate theory of physics
  • Module, 478
  • Molecular dynamics, 441
    • as analogy, 411
  • Momentum, 407, 420
  • Monoids, 400
  • Motion, 415
    • in branchial space, 454
  • Multicausal continuum, 456
  • Multiedges, 80
  • Multigraphs
    • multiway graphs as, 287
  • Multiple relations, 84, 110
  • Multiple transformation rules, 129
  • Multiway causal graphs, 233, 279, 370, 453, 486
    • growth of, 285
  • Multiway graphs, 485
    • generational, 295
    • geometry of, 216
    • growth in, 217
    • typical structures of, 212
    • weighted, 287
  • Multiway space, 438
  • Multiway systems, 305
    • and Cayley graphs, 401
    • and NDTMs, 382
    • and quantum mechanics, 432
    • and reversibility, 429
    • for substitution system, 202
    • history of, 477
    • rule-space, 471

N

  • Names
    • of elements, 305
  • Natural language
    • picturable nouns in, 144
  • NDTMs, 382, 404
  • Nearby rules, 145
  • Negative energy density, 430
  • Negatively curved space, 157, 175
  • Neighborhoods
    • in graph, 177
  • Nested pattern
    • dimension of, 167
    • rule generating, 110, 160
  • Nesting
    • see also Fractals
  • Nilpotent groups
    • Cayley graphs of, 198
  • Nodes
    • names for, 305
  • Non-Abelian groups
    • see also Groups
  • Non-commutative geometry, 477
  • Non-convex hull, 150
  • Non-deterministic Turing machines, 382, 404
  • Non-overlapping graphs, 392
  • Non-overlapping strings, 219, 226
  • Nonlocality
    • in rules, 88
  • Normal forms
    • see also Canonical forms
  • NP completeness
    • and multiway systems, 383
  • NP problems, 382
  • Numerological relations, 469

O

  • Objective reality
    • in quantum mechanics, 433
  • Observers
    • in quantum mechanics, 434, 444
    • in spacetime, 413
  • Obstructions
    • in planarity, 186
  • Octagon-square tiling, 159
  • Oligons, 469
  • Operator representation, 399
  • Operator-state correspondence, 408
  • Operators
    • in quantum mechanics, 449
  • Orbifolds, 153
  • Ordered graphs, 392
  • Orderless, 478
  • OrderlessPatternSequence, 481
  • Orthogonal components, 192
  • Orthogonal directions, 194
  • Overlaps

P

  • Pairing functions, 92
  • Parallel transport, 194
  • Partial differential equations (PDEs)
    • on graph, 185
  • Partial ordering
    • of events, 255
  • Particles, 424
    • mass of, 469
    • perturbations and, 190
    • virtual, 428
    • see also Defects
    • see also Elementary particles
    • see also Localized structures
  • Pascals triangle, 289
  • Path independence, 206
  • Path integrals, 409, 441
  • Path weights
    • in multiway graphs, 287
  • Pattern matching, 478
  • Pattern rules, 399
  • Pattern variables
    • rules in terms of, 386
  • PDEs
    • see Partial differential equations
  • Pentagons
  • Permutations
    • as symmetry operation, 359
    • limits of, 369
    • of rule elements, 91
  • Perturbations
    • and particles, 428
    • in evolution, 189
  • Physical constants, 464
  • Pi (π)
    • randomness of digits in, 411
  • Picturable nouns, 144
  • Piskunov, Max (Russia/USA, 1992 ), 487
  • Planarity, 185, 424
  • Planck energy, 468
  • Planck length, 465
  • Planck time, 465
  • Planck units, 465
  • Plancks constant, 409, 451, 465
  • Plane
    • hyperbolic, 157
  • Pointers, 304
  • Polygons
    • created from rule, 79
    • rule generating, 147
  • Polynomial growth
  • Posets, 255
  • Post canonical system, 477
  • Power series
    • for metric, 192
  • Powers of rules, 367
  • Pregeometry, 477
  • Principle of Computational Equivalence, 143, 402
    • and rulial causal invariance, 474
  • Product log function, 467
  • Production systems, 477
  • Products of rules, 367
  • Programs
    • see also Cellular automata, etc.
    • see also Computation
  • Projective spaces, 272
  • Proofs
    • of causal invariance, 223

Q

  • QOF
    • see Quantum observation frame
  • Quantified variables
    • rules in terms of, 386
  • Quantum computers, 410
  • Quantum field theory
    • vacuum in, 428
  • Quantum frame, 434, 444
  • Quantum holography, 477
  • Quantum information paradox, 461
  • Quantum measurement, 442
  • Quantum mechanics, 407, 432
  • Quantum observation frame, 434, 444
  • Quantum relativity, 477
  • Quantum Zeno effect, 455
  • Quasiconformal analog, 197
  • Qubits, 447, 458

R

  • Radial behavior, 143
  • Radius
    • of graphs, 183
  • Ramsey theory, 189
  • Random rules, 142
  • Random sampling
    • of rules, 126
  • Random triangulation, 164
  • Random updating order, 112
  • Randomness
    • generation of, 186
    • generation of in evolution, 411
  • Rank-p tensor, 196
  • Reconstruction
    • of surface, 150
  • Reconvergence, 206
  • Recursion
    • see also Fractals
    • see also Nesting
  • Recursive construction
    • of patterns, 161
  • Reference frames, 415
  • References, 488
  • Regge calculus, 477
  • Registry of Notable Universes, 92, 487
  • Relations
    • in groups, 197
  • Relativistic invariance, 415
  • Relativity, 417
    • in rule space, 475
  • Renormalization group, 188
  • Repetitive behavior, 140
    • in causal graphs, 337
    • structures with
    • see Localized structures
  • Replacements
    • order of, 203
  • Resolution
    • of branch pairs, 220
  • Rest mass, 422
  • Reversibility, 408, 429
    • of cellular automata
    • see Reversible cellular automata
    • see also Irreversibility
  • Rewrite systems
    • see also Multiway systems
  • Ricci scalar curvature, 174, 192, 193, 413
    • in branchial space, 456
  • Ricci tensor, 192, 194
    • spacetime, 418
  • Riemann tensor, 194, 413, 419
    • in branchial space, 456
  • Riemannian manifold, 192
  • Rotation group, 369
  • Rotational invariance, 408
  • Rule 30, 411
    • emulation of, 384
  • Rule 60
    • see also Additive cellular automata
    • see also Rule 90
    • see also Sierpiński pattern
  • Rule 90
    • see also Additive cellular automata
    • see also Rule 60
    • see also Sierpiński pattern
  • Rule 110, 402
    • particles in, 425
  • Rule space, 474
    • see also Rulial space
  • Rule-space multiway system, 472
  • Rules
    • all possible, 471
    • basic form of, 73
    • canonical form of, 91
    • disconnected, 130
    • multiple transformation, 129
    • nearby, 145
    • number of, 92, 93
    • products of, 367
    • random, 142
    • random sampling of, 126
    • representation of, 81
    • signatures of, 91
  • Rulial space, 474

S

  • S-matrix, 440
  • Samosa, 159
  • Scalar curvature
    • see also Ricci scalar curvature
  • Scale invariance, 188
  • Scales (mathematical)
    • see also Limits
  • Schläfli symbol, 157
  • Schrödinger equation, 441
  • Schwarzschild radius, 460
  • Second Law of thermodynamics, 429
    • and quantum observers, 437, 449
  • Sectional curvature, 413
  • Self-loops, 79, 95
    • as fertile initial conditions, 138
  • Self-similarity, 160
    • see also Fractals
    • see also Nesting
  • Semi-Thue systems, 477
  • Semigroups, 400
  • Sensitive dependence
    • see also Chaos theory
  • Sequential substitution systems, 203
  • Sequential updating, 232
  • Shape
    • generated by rule, 153
  • Shift vector, 259
  • Short codes, 92
  • Shortest paths
    • see also Geodesics
  • Sierpiński graphs, 197, 315, 389
  • Sierpiński pattern
    • dimension of, 167
    • local subgraphs in, 179
    • rule generating, 110, 160, 161
    • see also Rule 60
    • see also Rule 90
  • Signatures
    • of rules, 91
    • of substitution systems, 212
  • Simplification order, 205
  • Simulations
    • see also Models
  • Simultaneity, 415
  • Singlepushout graph grammars, 477
  • Singularities
    • in branchial space, 461
    • in shape generated, 153
    • quantum coordinate, 446
  • Slow growth, 122
  • Smooth surface, 150
  • Snake states, 298
  • SO(3), 369
  • Sorting rule, 205, 236, 237
    • and foliations, 255
    • causal graph of, 249
    • in spacetime, 416
    • multiway causal graph for, 283
  • Space
    • structure of, 410
    • see also Geometry
  • Spacelike connections, 454
  • Spacelike hypersurfaces, 410, 416
  • Spacetime
    • see also Gravity
    • see also Space
  • Spacetime causal graph, 484
  • Spatial hypergraph, 414, 483
  • Special relativity, 408, 417
  • Speed of light (c), 418, 456, 465
  • Sphere graphs, 174, 179, 188
    • directed, 254
  • Spheres
    • approximation to, 151
    • as model for curvature, 199
    • volumes of, 165
    • volume of in curved space, 174, 193
  • Spherical coordinates, 151
  • Spin, 407
  • Spin networks, 477
  • SPO graph grammars, 477
  • Square grid, 163
  • Standard updating order, 87, 132, 304
    • not done by SubsetReplace, 483
  • State transition diagrams, 140
  • States graphs, 208, 485
  • Statistical averages, 411
  • Statistical mechanics, 188
  • Stopping, 87, 135
  • Strange attractors
    • see also Chaos theory
    • see also Class 3 behavior
  • String substitution systems, 201
    • emulation of, 377
    • in quantum mechanics, 435
  • String theory, 410, 477
  • StringReplace, 203, 248
  • Strong hyperbolicity
    • failure of, 346
  • Subdivision
    • geometry from, 158
  • SubsetReplace, 482
  • Substitution systems, 201
    • emulation of, 377
    • number of, 212
    • see also Multiway systems
  • Successors
  • Summer School, 487
  • Superpositions, 444
  • Superrelativity, 477
  • Surface
  • Surface reconstruction, 150
  • Symbolic dynamics, 369
    • on graphs, 178
  • Symbolic expressions, 478
    • to represent rules, 399
  • Symbolic names
    • for elements, 305
  • Symmetric groups, 198, 401
    • as symmetry of evolution, 360, 365
  • Symmetries, 359
  • Symmetry
    • local gauge, 462
    • of initial conditions, 138
  • Synchronization
    • in cellular automata, 384

T

  • Tails
    • of branch resolution, 224
  • Tangent bundle, 413
  • Tangent space, 195
  • Taylor series, 177
    • for metric, 192
  • Tensor fields, 413
  • Tensor function
    • on graph, 196
  • Tensor networks, 477
  • Term-rewriting systems, 208, 477
  • Termination, 87, 119, 130, 135
    • in substitution systems, 206
  • Ternary edges, 81
  • Ternary operators, 399
  • Ternary relations
    • rules based on, 107
  • Ternary tree, 104
  • Tetrahedron graph, 160
  • Theology
    • see also God
  • Theorem proving, 204
    • completions in, 294
  • Theorem-proving systems, 436
  • Thermal equilibrium
    • see also Equilibrium
  • Thermalization
    • and computational irreducibility, 449
  • ThueMorse sequence, 201
  • Tilings
    • hyperbolic, 157, 248
    • octagon-square, 159
    • see also Penrose tilings
  • Time
  • Time dilation, 408, 418
    • quantum analog of, 455
  • Time reversal violation, 429
  • Time travel, 432
  • Topochronology, 477
  • Topological structure
    • generated by rule, 102
  • Topology
    • generated by rule, 154
  • Topos theory, 477
  • Torus graph, 165
    • as causal graph, 252
  • Total causal invariance, 207, 219, 311
  • Transcendental equation, 467
  • Transformation rules
    • symbolic, 478
  • Transients, 141
  • Transition amplitude, 440
  • Transition matrix, 202
  • Translation cone
    • in rule space, 475
  • Tree-like behavior, 143
  • Trees
    • dimension of, 168
    • from causal graphs, 246
    • from substitution system, 201
    • generated from rule, 77
    • in hyperbolic space, 156
    • in multiway graphs, 214
    • minimal rules generating, 97
    • reductions on, 399
    • rule generating, 147
    • ternary, 104
  • Triangular tiling, 157
  • Triangulation, 164
  • Trivalent graphs, 390
    • nesting in, 160
  • Tubes
    • as causal graphs, 339
    • rule generating, 152
    • volume of, 193
  • Tupling function, 92
  • Turing machines
    • as analogy, 333
    • emulation of, 379
    • non-deterministic, 382, 404
    • rule labeling in, 92
  • Twistor space, 456
  • Twistor theory, 477
  • Two-point function, 188
  • Types
    • for tagging rules, 399

U

  • U(1), 369
  • Unary edges, 83
  • Unary relations, 134
    • rules based on, 96
  • Uncertainty principle, 408, 452
  • Undecidability, 403
    • in multiway graphs, 204
    • of causal invariance, 224
    • of classification, 143
    • of completions, 294
    • of disconnection, 133
    • of infinite branching, 352
  • Undirected graphs, 392
  • Unforgeable word constant, 226
  • Unifications, 224
    • in hypergraphs, 311
  • Units, 464, 468
  • Universal algebra, 400
  • Universal computation, 402
  • Universal Turing machine, 379, 402
  • Universe
    • see also Cosmology
  • Unordered hypergraphs, 396
  • Updating events, 303
    • in substitution systems, 228
  • Updating order, 87, 203
  • UUID
    • for elements, 304

V

  • Vacuum, 410
    • particles in, 428
  • Vector function
    • on graph, 195
  • Velocity
    • foliation representing, 416
  • Vertex counts, 183, 187
  • Vertex degrees
    • in branchial graphs, 262
  • Vertex transitive graphs, 196
  • Virtual particles, 409, 428
  • Visual inspection, 144
  • Visualization
  • Volumes
    • of geodesic ball, 165

W

  • Wave-particle duality, 408, 451
  • Waves
    • dimension, 430
  • Weighted branchial graphs, 439
  • Weighted multiway graphs, 287, 439
  • Weyl tensor, 419
  • Wolfram Function Repository, 487
  • Wolfram Language, 478, 487
    • patterns
    • see Patterns (in Wolfram Language)
    • updating order in, 203
  • Wolfram Research, 487
  • Wolfram Summer School, 487
    • see also Summer School
  • wolframphysics.org, 487
  • World lines, 422

X-Z

  • Young tableaux, 256
  • Zeno effect, 455
  • Zeta
    • maximum entanglement speed, 456, 469