In the 1950s and 1960s—quite independent of symbolic dynamics—there was a certain amount of work done in connection with ideas about self-reproduction (see page 876 ) on the question of what configurations one could arrange to produce in 1D and 2D cellular automata. And this led for example to the study of so-called Garden of Eden states that can appear only in initial conditions—as well as to some general discussion of properties such as surjectivity.

Initially several different problems were considered, but after a while the only ones to survive were those such as the RSA system discussed below based essentially on the problem of factoring integers.

At first it was thought that the visual system might be sensitive only to the overall autocorrelation of an image, given by the probability that randomly selected points have the same color.

Groups require that not only rules but also symbols come in pairs. … The simplest non-trivial commutative semigroup has rules "AB"  "BA" and "BA"  "AB" , so that strings of generators with A 's and B 's in different orders are equivalent and the Cayley graph is a 2D grid. … The icosahedral group A 5 defined by the rules x 2  y 3  (x y) 5  1 has 60 elements.

General arguments for discrete space were also sometimes made—notably by Edward Fredkin , Marvin Minsky and to some extent Richard Feynman —on the basis of analogies to computers and in particular the idea that a given region of space should contain only a finite amount of information.

But the notion of abstract functions in mathematics reached its modern form only near the end of the 1800s.

And also around this time Friedrich Hayek (following ideas of Ludwig Mises in the early 1900s) suggested—presumably influenced by work in mathematical logic—that human behavior might be fundamentally unpredictable because in effect brains can explain only systems simpler than themselves, and can thus never explain their own operation.

Note, however, that in predicate logic the expressions that appear on each side of any rule are required to be so-called well-formed formulas (WFFs) consisting of variables (such as a ) and constants (such as 0 or ∅ ) inside any number of layers of functions (such as + , · , or Δ ) inside a layer of predicates (such as  or ∈ ) inside any number of layers of logical connectives (such as ∧ or ⇒ ) or quantifiers (such as ∀ or ∃ ). … But in predicate logic rules can be applied only to whole expressions, always in effect using Replace[expr, rules] .

Artificial radio signals In current technology radio signals are essentially always based on carriers of the form Sin[ ω t] with frequencies ω /(2 π ) . … In the 1940s it also became popular to use frequency modulation (FM) Sin[(1 + s[t]) ω t] , and in the 1970s pulse code modulation (PCM) (pulse trains for IntegerDigits[s[t], 2] ). … And when CDMA methods are used there can be spreading over a significant range of frequencies—with regularities being recognizable only if one knows or can cryptanalyze LFSR sequences.

And in a case like page 518 —with spacetime always effectively having a fixed finite dimension—points that are a distance t apart tend to have common ancestors only at least t steps back.