An immediate generalization is to allow rules to have a form like {1  {1, 1}, 1  {1, 0}, 0  {0, 0}} in which several blocks are in effect allowed to serve as possible encodings for a single cell value. … And so, for example, with the setup shown in the main text, rule 54 can emulate rule 0 only with blocks of length b = 6 . But if either multiple blocks or δ = 1 are allowed, b can be reduced to 4, with rules being {1  {1, 1, 1, 1}, 0  {0, 0, 0, 0}, 0  {0, 1, 1, 1}} and {0  {0, 1, 0, 0}, 1  {0, 0, 1, 0}} in the two cases.

It is sometimes thought significant that living systems perpetuate the use of only one chirality of molecules, but actually this can quite easily be achieved by various forms of non-chemical input without life.

History [of extraterrestrial life] Although Greek philosophers such as Democritus believed that there must be an infinite number of worlds all with inhabitants like us, the prevailing view in antiquity—later supported by theological arguments—was that the Earth is special, and the only abode of life.

In the past, it has usually been assumed that if we could only build up in our minds an adequate model of the world, then we would immediately know whatever we want about the world.

With k = 2 , elements 6 out of the total of 8 possible semigroups appear; with k = 3 , 63 out of 113, and with k = 4 , 1140 out of 3492—all as shown on page 805 .) … Taking ∘ to be each of these operators, one can work out a representation for any given string like "ABAA" by for example constructing the expression ((a ∘ b) ∘ a) ∘ a and finding its value for each of the k 2 possible pairs of values of a and b . … But as the pictures above suggest, only some of the distinctions between sets of strings are ever captured when any specific form for the operator is used.

But when one proves a lemma one is in effect on a separate branch, which only merges with the main proof when one uses the lemma.

. • (a) At step t , the only new string produced is the one containing t black elements. • (b) All strings of length n containing exactly one black cell are produced—after at most 2n - 1 steps. • (c) All strings containing even-length runs of white cells are produced. • (d) The set of strings produced is complicated. … Those of length n appear after at most 3n - 3 steps. • (l) The same strings as in (k) are produced, taking now at most 2n + 1 steps. • (m) All strings beginning with a black element are produced, after at most 3n + 1 steps. • (n) The set of strings produced is complicated, and seems to include many but not all that do not end with . • (o) All strings that do not end in are produced. • (p) All strings are produced, except ones in which every element after the first is white. takes 14 steps. • (q) All strings are produced, with a string of length n with m white elements taking n + 2m steps. • (r) All strings are ultimately produced—which is inevitable after the lemmas  and  appear at steps 12 and 13.

[Structures in] the Game of Life The 2D cellular automaton described on page 949 supports a whole range of persistent structures, many of which have been given quaint names by its enthusiasts. … The so-called "puffer train" below which starts from 23 black cells becomes repetitive with period 140 only after more than 1100 steps.

Around the end of the 1500s Galileo began to develop more explicit connections between concepts in mathematics and in physics, and concluded that the universe could be understood only in the "language of mathematics", whose "characters are triangles, circles and other geometric figures".

• Is there any x < a such that Mod[x 2 , b]  c ? … (In cases where numbers are involved, it is usually crucial that these be represented by base 2 digit sequences, and not, say, in unary.) … (The latter is like a network with only 2 connections at each node.) • Solving quadratic Diophantine equations a x 2 + b y  c is NP-complete, but solving linear ones a x + b y  c is in P. • Finding a minimum energy configuration for a 2D Ising spin glass in a magnetic field is NP-complete, but is in P if there is no magnetic field. • Finding the permanent of a matrix is NP-complete, but finding its determinant is in P.