In satisfiability what one does is to start with a collection of rows of black, white and gray squares. And then what one asks is whether any sequence of just black and Translation between an NP-complete problem about non-deterministic Turing machines and about cellular automata.

And within such statements one can refer, say, to infinite sets of integers just by a symbol like s . … But the remarkable fact that follows from Gödel's Theorem is that whatever one does there will always be cases where the approach must ultimately fail.

But if one looks at the history of science many of its greatest advances have come precisely from identifying ways in which we are not special—for this is what allows science to make ever more general statements about the universe and the things in it. … For if one thinks in computational terms the issue is essentially whether we somehow show a specially high level of computational sophistication.

But I suspect that one will often have a much better chance of capturing fundamental mechanisms for phenomena in the social sciences by using instead the new kind of science that I develop in this book based on simple programs. … There will be new questions formulated, but it will take time before it becomes clear when general theories are possible, and when one must instead inevitably rely on the details of judgement for specific cases.

Starting in the 1930s the idea of symbolic dynamics began to emerge, in which one partitions continuous values in a system into bins represented by discrete symbols, and then looks at the sequences of such symbols that can be produced by the evolution of the system. … One approach was to try to find invariants that would remain unchanged in different partitionings—and this is what led, for example, to the study of topological entropy in the 1960s. … 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.

To set up second-order logic, however, one imagines also being able to use ∀ f and ∃ f where f is a function (say the head of f[x] ). And then in third-order logic one imagines using ∀ g and ∃ g where g appears in g[f][x] . … For even to enumerate theorems in second-order logic is in general impossible for a system like a Turing machine unless one assumes that an oracle can be added.

String transformations An example of a rule that allows one to go from any string of A 's and B 's to any other is {"A"  "AA", "AA"  "A", "A"  "B", "B"  "A"} (Compare page 1038 .)

Implementation of totalistic cellular automata To handle totalistic rules that involve k colors and nearest neighbors, one can add the definition CAStep[TotalisticCARule[rule_List, 1], a_List] := rule 〚 -1 - (RotateLeft[a] + a + RotateRight[a]) 〛 to what was given on page 867 . The following definition also handles the more general case of r neighbors: CAStep[TotalisticCARule[rule_List, r_Integer], a_List] := rule 〚 -1 - Sum[RotateLeft[a, i], {i, -r, r}] 〛 One can generate the representation of totalistic rules used by these functions from code numbers using ToTotalisticCARule[num_Integer, k_Integer, r_Integer] := TotalisticCARule[IntegerDigits[num, k, 1 + (k - 1)(2r + 1)], r]

But generally my experience has been that the further one goes from those involved with specific molecular or other details of biological systems the more one encounters a fundamental conviction that natural selection must be the ultimate origin of any important feature of biological systems.

If one looks at projections of surfaces, it is common to see lines of discontinuity at which a surface goes, say, from having three sheets to one.