Most are quick to point out at least anecdotal cases in which features of organisms do not seem to have been shaped by natural selection.

Most clusters that can overlap will be able to do so in an infinite number of possible networks.

Ultimately what one wants to do is to find what possible types of forms for local regions are inequivalent under the application of the underlying rules.

The Anthropic Principle It is sometimes argued that the reason our universe has the characteristics it does is because otherwise an intelligence such as us could not have arisen to observe it.

And perhaps the explanation for this is in part that most of those who one can now see made contributions to the kinds of foundational issues I address were capable enough to have been successful at something—but without the whole context of this book they tended to view the types of results I discuss largely as curiosities, and so never tried to do much with them. Note that in mentioning people in connection with ideas and results, I have tried to concentrate on those who seemed to make the most essential contributions for my purposes—even when this does not entirely agree with traditions or criteria in particular academic fields.

Given two numbers x and y their product can be computed in base k by ( FromDigits does the carries) FromDigits[ListConvolve[IntegerDigits[x, k], IntegerDigits[y, k], {1, -1}, 0], k] For numbers with n digits direct evaluation of the convolution would take about n 2 steps. … Note that even though one may only want to find a single digit in m t , I know of no way to do this without essentially computing all the other digits in m t as well.

And if one always does computations using systems that have only nearest-neighbor rules then just combining 2t + 1 bits of information can take up to t steps—even if the bits are combined in a way that is not computationally irreducible. … But I strongly suspect that computational irreducibility prevents outcomes in systems like rule 30 and rule 110 from being found by computations that are in NC—implying in effect that allowing arbitrary connections does not help much in computing the evolution of such systems.

And in general what such equations do is to specify constraints that systems must satisfy. … To do this on a computer requires constructing a discrete approximation.

Feynman diagrams The pictures below show a typical set of Feynman diagrams used to do calculations in QED—in this case for so-called Compton scattering of a photon by an electron. … To work out the total probability for a process from Feynman diagrams, what one does is to find the expression corresponding to each diagram, then one adds these up, and squares the result. … Since for QED α ≃ 1/137 , one might expect that this would give quite an accurate result—and indeed experiments suggest that it does.

And how does it relate to the complexity we have seen in systems like cellular automata?