The pictures at the bottom of the previous page show what happens when one applies run-length encoding using representation (e) from page 560 to various sequences of data. … Successive rows in each original image are placed end to end so as to give a one-dimensional sequence, then run-length encoded, and then chopped into rows again.

An example is x + y  z , which cannot reasonably be considered either true or false unless one knows what x , y and z are. … The same basic issue arises in the intuitionistic approach to mathematics, in which one assumes that any object one handles must be found by a finite construction. And in such cases one can set up an analog of logic in which one no longer takes ¬ ¬ a  a .

(Other appropriate primitives may conceivably be related to the solubility of Hilbert's Thirteenth Problem and the fact that any continuous function with any number of arguments can be written as a one-argument function of a sum of a handful of fixed one-argument functions applied to the arguments of the original function.) … If one has a table of choices, one can imagine generalizing this to a function of a real number. But to specify this function one normally has no choice but to use some type of finite formula.

Pattern-avoiding sequences As another form of constraint one can require, say, that no pair of identical blocks ever appear together in a sequence, so that the sequence does not match {___, x__, x__, ___} . … But with k = 3 possible elements, there are infinite nested sequences that can, such as the one produced by the substitution system {0  {0, 1, 2}, 1  {0, 2}, 2  {1}} , starting with {0} . … But it also known that among the infinite sequences which do this, there are always nested ones (sometimes one has to iterate one substitution rule, then at the end apply once a different substitution rule).

Two-operator logic [axioms] If one allows two operators then one can get standard logic if one of these operators is forced to be Not and the other is forced to be And , Or or Implies —or in fact any of operators 1, 2, 4, 7, 8, 11, 13, 14 from page 806 . … Given the first two axioms (commutativity and associativity) it turns out that no shorter third axiom will work in this case (though ones such as f[g[f[a, g[f[a, b]]]], g[g[b]]]  b of the same size do work).

The sequence expands by at least one digit every two steps; more rapid expansion is typically correlated with increased randomness. … If one works directly with a digit sequence of fixed length, dropping any carries on the left, then a repetitive pattern is typically obtained fairly quickly. If one always includes one new digit on the left at every step, even when it is 0, then a rather random pattern is produced.

An alternative kind of model, somewhat analogous to the ones based on constraints on page 483 , is to take the pattern of evolution of a multiway system to define directly a complete spacetime network. Instead of looking separately at strings produced at each step, one instead maintains just a single copy of each distinct string ever produced, and makes that correspond to a node in the network. Each node is then connected to the nodes associated with the strings reached by one application of the multiway rule, as on page 209 .

If a system is additive it means that one can work out how the system will behave from any initial condition just by combining the patterns ("Green's functions") obtained from certain basic initial conditions—say ones containing a single black cell. … But if one allows σ to be discontinuous then there can be some other exotic possibilities. … But one can also imagine setting up systems whose states are continuous functions of position. ϕ then defines a mapping from one such function to another.

Nearby powers [and integer equations] One can potentially find integer equations with large solutions but small coefficients by looking say for pairs of integer powers close in value. The pictures below show what happens if one computes x m and y n for many x and y , sorts these values, then plots successive differences. … Often they are large, but surprisingly small ones can sometimes occur (despite various suggestions from the so-called ABC conjecture).

If one looks at digit sequences, it is rather clear why this happens. For as the picture illustrates, the so-called shift map used in case (d) simply serves to shift all digits one position to the left at each step. … Instead, it is just that randomness that was inserted in the digit sequence of the original number shows up in the results one gets.