(An example would be a test for class 1 based on checking that no initial pattern of any size can survive.

But it turns out that given the basic axioms for commutative group theory any non-trivial set of additional axioms can always be reduced to a single axiom of the form a n  1 (where exponentiation is repeated application of ∘ ). Yet even given a particular number of elements k , there can be several distinct groups satisfying a n  1 for a given exponent n .

Complex powers [of numbers] The pictures below show successive powers of complex numbers z with digits extracted according to (2 d[Re[#], w] + d[Im[#], w]) & [z t ] d[x_, w_] := If[x < 0, 1 - d[-x, w], IntegerDigits[x, 2, w]] Non-trivial cases of complex number multiplication never correspond to local cellular automaton operations.

The functions And , Xor and Not are equivalent to Times , Plus and 1 - # & for variables modulo 2, and in this case algebraic functions like PolynomialReduce can be used for minimization.

[Construction of] universal objects A more direct way to create a universal object is to set up, say, a 4D array in which two of the dimensions range respectively over possible 1D cellular automaton rules and over possible initial conditions, while the other two dimensions correspond to space and time in the evolution of each cellular automaton from each initial condition.

Lyapunov exponents If one thinks of cells to the right of a point in a 1D cellular automaton as being like digits in a real number, then linear growth in the region of differences associated with a change further to the right is analogous to the exponentially sensitive dependence on initial conditions shown on page 155 .

It is known that to achieve this exactly, m must be at the least the number of either positive or negative eigenvalues of the distance matrix for the network, and can need to be as much as n - 1 , where n is the total number of nodes.

By inserting k = 6 Ceiling[Length[subs]/6] in the definition of TS1ToCT from page 1113 one can construct a cyclic tag system of this kind to emulate any one-element-dependence tag system.

. • 1986-1991: intensive Mathematica development • 1991-2001: writing this book (Wolfram Research, Inc. was founded in 1987; Mathematica 1.0 was released June 23, 1988; the company and successive versions of Mathematica continue to be major parts of my life.)

Implementation [of 2D Turing machines] With rules represented as a list of elements of the form {s, a}  {sp, ap, {dx, dy}} ( s is the state of the head and a the color of the cell under the head) each step in the evolution of a 2D Turing machine is given by TM2DStep[rule_, {s_, tape_, r : {x_, y_}}] := Apply[{#1, ReplacePart[tape, #2, {r}], r + #3} &, {s, tape 〚 x, y 〛 } /. rule]