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Linear Cellular Automata
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Linear Cellular Automata
Contents
History
Early origins
Automata theory
The Gardner era
The Wolfram era
One dimensional automata
The vocabulary of automata theory
Probability theory
Graph theory and de Bruijn diagrams
Reversible automata
What to look for
General characteristics
Cycles
Periods
Ancestors
Subautomata
Factor automata
Product automata
Automata with memory
Idempotent rules
Membranes and macrocells
Totalistic rules
Two-cell neighborhoods
Blocking transformations
Tailor made automata
Cycles in space
Cycles in Life
Evolution for Rule 22 - one dimensional Life
Evolutionary diagrams and matrices
Evolution of a seven-cell ring
Cycles for a sixteen-member ring
Quiescent ring
Period 14
Period 12, 6 phases
Period 12, 12 phases
period 6
Period 4
Period 2
Period 1 (still life)
Cycles for Rule 22
The evolution matrix
The reduced evolution matrix
Periods in time
Characteristics of cycles
Overlapping of neighborhoods
The de Bruijn matrix
as product and sum
Periods and other properties
Superluminal configurations for Rule 22
Periods for Rule 22
The gap theorems
The Garden of Eden
The subset construction
Excluded states for Rule 22
Symbolic equations
Arden's lemma
The use of symbolic equations
Systems of symbolic equations
Ancestorless states for Rule 18
Factors
The calculus of regular expressions
Derivatives
Ideals and factors
Rule 18
Factors for Rule 18
A geometrical representation
Probabilistic de Bruijn matrix
Block probabilities
Kolmogorov conditions in matrix form
Probabilistic de Bruijn matrix
Some properties of n-block probabilities
Some simple examples
Determinant and inverse
Characteristic equation
merged product
trace
second coefficient
principal minors
determinant
Correlations
Probabilistic evolution matrix
Regularities and anomalies
Mean field theory
More refined theories
Local structure theory
Hartree-Fock approach
Kolmogorov consistency conditions
The vector subset diagram
Estimating the number of ancestors
Trivial solutions
Positive matrices
Gerschgorin's disks
Eigenvalues on the boundary
Minimax principle
Largest eigenvalue
Second largest eigenvalue
Averaging and convergence
Non-negative matrices
Zeta function
Counting loops
Traces, ,
Infinite de Bruijn matrix
Cluster expansion
Reduced evolution matrix
Cycles for Rule 22
Summary
N = 1
N = 2
N = 3
N = 4
N = 5
N = 6
N = 7
N = 8
N = 9
N = 10
N = 11
N = 12
N = 13
N = 14
N = 15
N = 16
N = 17
N = 18
N = 19
N = 20
N = 21
N = 22
N = 23
N = 24
N = 25
N = 26
N = 27
N = 28
N = 29
N = 30
N = 31
N = 32
N = 33
N = 34
A pair of isolated cells
References
About this document ...
Harold V. McIntosh
E-mail:mcintosh@servidor.unam.mx