As illustrated in the main text, when m = 2 j the right-hand base 2 digits in numbers produced by linear congruential generators repeat with short periods; a digit k positions from the right will typically repeat with period no more than 2 k . … The pictures below show the evolution obtained for n = 30 with NestList[Nest[LFSRStep, #, n]&, Append[Table[0, {n - 1}], 1], t] Like additive cellular automata as discussed on page 951 , states in a linear feedback shift register can be represented by a polynomial FromDigits[list, x] . … In fact, the first known generator for digital computers was John von Neumann 's "middle square method" n  FromDigits[Take[IntegerDigits[n 2 , 10, 20], {5, 15}], 10] In practice this generator has too short a repetition period to be useful.

measurements, so that the amount of information needed to pick out a single arrangement is essentially the length in digits of one such number. … The entropy as a function of time for systems of the type shown in case (b) from page 447 . … Note that the plots above would be exactly symmetrical if they were continued to the left: the entropy would increase in the same way going both forwards and backwards from the simple initial conditions used.

Various representations of numbers from 1 to 30. … (c) is like (b), except that it has a specification of the number of digits at the front. … (e) uses a non-integer base derived from the Fibonacci sequence, with the property that a pair of black cells can appear only at the end of each number.

Chaos Theory and Randomness from Initial Conditions…But if we could watch the solar system for a few million years, then there should be significant randomness that could be attributed to sensitive dependence on the digit sequences of initial conditions—and whose presence in the past may explain some observed present-day features of our solar system. … The pictures are made assuming the system to be in uniform motion from left to right.

One cannot for example sort n objects in less than about n steps since one must at least look at each object, and one cannot multiply two n -digit numbers in less than about n steps since one must at least look at each digit. … And if the output from a computation can be of size 2 n then this will normally take at least 2 n steps to generate. … But this seems exponentially large if s is specified by its digit sequence in the original input regular expression.

(They thus differ from the Turing machines which Marvin Minsky and Daniel Bobrow studied in 1961 in the s = 2 , k = 2 case and concluded all had simple behavior.) … The number of steps before a machine with given rule halts can be computed from (see page 888 ) Module[{s = 1, a, i = 1, d}, a[_] = 0; MapIndexed[a[#2 〚 1 〛 ] = #1 &, Reverse[IntegerDigits[x, 2]]]; Do[{s, a[i], d} = {s, a[i]} /. rule; i -= d; If[i  0, Return[t]], {t, tmax}]] Of the 4096 Turing machines with s = 2 , k = 2 , 748 never halt, 3348 sometimes halt and 1683 always halt. … Most machines compute functions that involve digit manipulations without traditional interpretations as mathematical functions.

The pictures were made in about 1964 by Berni Alder and Frederick Reif from oscilloscope output from the LARC computer at what was then Lawrence Radiation Laboratory. … From the point of view of this book the randomization seen in these pictures is in large part just a reflection of the fact that a random sequence of digits were used in the initial conditions.

Implementation [of conserved quantity test] Whether a k -color cellular automaton with range r conserves total cell value can be determined from Catch[Do[ (If[Apply[Plus, CAStep[rule, #] - #] ≠ 0, Throw[False]] &)[ IntegerDigits[i, k, m]], {m, w}, {i, 0, k m - 1}]; True] where w can be taken to be k 2r , and perhaps smaller.

Chaos Theory and Randomness from Initial Conditions…And indeed to assume that it does is effectively just to ignore the fundamental question of where randomness in nature comes from. … And often such effects will tend to introduce new randomness from the environment. … And just as in the kneading process, there is very sensitive dependence on the details of the initial conditions, and the behavior that is seen reflects the digit sequence of these initial conditions.

[Patterns from] bitwise functions Bitwise functions typically yield nested patterns. … Note that 2n has the same digits as n , but shifted one position to the left.