Methodology in this book Much of what is presented in this book comes from systematic enumeration of all possible systems of particular types.

[History of] truth tables The method of finding results in logic by enumerating all possible combinations of truth values seems to have been rediscovered many times since antiquity.

Inevitable regularities and Ramsey theory One might have thought that there could be no meaningful type of regularity that would be present in all possible data of a given kind. … If one has a sufficiently long sequence, therefore, just knowing that a run of equally spaced identical elements exists in it does not narrow down at all what the sequence actually is, and can so cannot ultimately be considered a useful regularity.

Computation of [recursive] sequences It is straightforward to compute the various sequences given here, but to avoid a rapid increase in computer time, it is essential to store all the values of f[n] that one has already computed, rather than recomputing them every time they are needed. … The default form of evaluation for recursive functions implemented by all standard computer languages (including Mathematica) is the so-called leftmost innermost scheme, which attempts to find explicit values for each f[k] that occurs first, and will therefore never notice if f[k] in fact occurs only in the combination f[k] - f[k] .

With distance on a network defined as the length of shortest path between nodes one immediately gets all three of these properties. And even though all distances defined this way will be integers, they still make any network formally correspond in mathematical terms to a metric space (or strictly a path metric space).

The main point of this is that typical machine instructions operate in parallel on all the bits in such a variable. And thus for example the values of all cells represented by an integer variable a can be updated in parallel according to rule 30 by the single C statement a = a > > 1 ^ (a | a < < 1); This statement, however, will only update the specific block of cells encoded in a . … If many steps of evolution are required, it is sufficient just to pack all cell values at the beginning, and unpack them at the end.

The idea of a generative grammar is that all possible expressions in a particular formal language can be produced by applying in all possible ways the set of replacement rules given by the grammar. Thus, for example, the rules {"x"  "xx", "x"  "(x)", "x"  "()"} starting with "x" will generate all expressions that consist of balanced sequences of parentheses.

So-called Polish notation given second-to-last above avoids all explicit delimiters (see page 896 ).

All possible arrangements of colors are thus eventually produced.

It turns out all numbers except powers of 2 can be represented this way.