So I resolved to set up the best tools and infrastructure I could, and then just myself pursue as efficiently as possible the research that I thought should be done.

In the late 1970s there began to be computer programs for large-scale Boolean minimization—the best known being Espresso .

The method can be implemented using Module[{a = Flatten[data], r, s}, {r, s} = Dimensions[data]; Partition[Do[ a 〚 i + {1, s - 1, s, s + 1} 〛 += m (a 〚 i 〛 - If[a 〚 i 〛 < 1/2, 0, 1]), {i, r s - s - 1}]; Map[If[# < 1/2, 0, 1] &, a], s]] In its original version m = {7, 3, 5, 1}/16 , as in the first row of pictures below. But even with m = {1, 0, 1, 0}/2 the method generates fairly random patterns, as in the second row below. … To give the best impression of uniform gray, one must in general minimize features detected by the human visual system.

Implementation [of network substitution rules] For many practical purposes the best representation for networks is the one given on page 1031 .

Presumably it is best to add axioms that allow the widest range of new statements to be proved.

Explanations suggested for apparent absence include: • Extraterrestrials are visiting, but we do not detect them; • Extraterrestrials have visited, but not in recorded history; • Extraterrestrials choose to exist in other dimensions; • Interstellar travel is somehow infeasible; • Colonization is somehow ecologically limited; • Physical travel is not worth it; only signals are ever sent.

After a large number of steps t , the number of distinct positions visited will be proportional to t , at least above 2 dimensions (in 2D, it is proportional to t/Log[t] and in 1D √ t ).

And in the 1830s, Charles Babbage described what he called an analytical engine, which, if built, would have been able to perform sequences of arithmetic operations under punched card control.

Almost all spin configurations with e[s] > - √ 2 (where here and below all quantities are divided by the total number of spins, so that -2 ≤ e[s] ≤ 2 and -1 ≤ m[s] ≤ + 1 ) yield m[s]  0 . … But whenever the evolution is ergodic, so that all states of a given energy are visited with equal frequency, the average behavior obtained will at least eventually correspond to the average over all states discussed above. … And from my discussion of intrinsic randomness generation it should come as no surprise that even a completely deterministic rule for the evolution of spins can make the system visit possible states in an effectively random way.

Discretizing yields lattice gauge theories with energy functions involving for example Cos[ θ i - θ j ] for color directions at adjacent sites.