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Note (b) for The Relationship of Space and Time

Network constraint systems Cases (a), (f) and (p) allow all networks that do not contain respectively cycles of length 1 (self-loops), cycles of length 3 or less, and cycles of length 5 or less.

Note (b) for Undecidability and Intractability

But if the evolution is computationally irreducible then to find its outcome will involve explicitly following each of its t steps—thereby effectively finding results for each of the 2^Log[2, t] possible arrangements of digits corresponding to numbers less than t .

Note (f) for Fundamental Issues in Biology

But in fact, as discussed in Chapter 7 , randomness and complexity tend to lead to more, rather than less, robustness in overall behavior.

Biological Pigmentation Patterns

But in cases where the stripes are less regular they typically look very much like the patterns generated in the pictures at the bottom of this page using a version of the simple mechanism described in the first caption.

Computational Irreducibility

But what traditional theoretical science in a sense implicitly relies on is that much of this effort is somehow unnecessary—and that in fact it should be possible to find the outcome of the evolution with much less effort.

Note (d) for Fundamental Issues in Biology

But in fact there are always many somewhat arbitrary details, particularly centering around exactly how to prune less fit organisms.

Note (a) for Biological Pigmentation Patterns

But in more realistic models patterns emerge from long time behavior with generic initial conditions, making boundary conditions—and effects such as changes in them associated with growth of an embryo—much less important.

Note (b) for Irreversibility and the Second Law of Thermodynamics

In a sense what I do is just to require that the operation of coarse graining correspond to a computation that is less sophisticated than the actual evolution of the system being studied.

Note (e) for The Threshold of Universality in Cellular Automata

[Cellular automaton] rule emulations The network below shows which quiescent symmetric elementary rules can emulate which with blocks of length 8 or less. … And it turns out also to be possible to determine the color of a particular cell from slices at essentially any rational angle corresponding to a propagation speed less than r .

Note (c) for Implications for Mathematics and Its Foundations

The only cases of 2 or less operators that appear with k = 2 are {{}, {10}, {12}, {1, 7}, {3, 12}, {5, 10}, {6, 9}, {10, 12}} .