But that does not mean that there cannot exist higher forms of perception and analysis that succeed in recognizing at least some regularities that our existing methods do not.

Region (a) shows a block separator—corresponding to a dashed line in picture (d) on page 679 —hitting the single black element in the sequence that exists at the first step.

But despite these successes, one might still imagine that other systems could exist in nature that are based, say, on continuous mathematics, and which would allow computations more sophisticated than those in systems like cellular automata to be done.

And in fact there is a particularly abstruse result in mathematical logic that might be thought to show that systems can exist that exhibit some features of arbitrarily sophisticated computation, but which are nevertheless not universal.

For although there are exceptions—like crystals, bubbles and animal horns—the majority of objects that exist in nature have irregular and often very intricate forms that seem much more complex than typical artifacts.

Practical arguments [about exterrestrials] If extraterrestrials exist at all an obvious question—notably asked by Enrico Fermi in the 1940s—is why we have not encountered them. … 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.

And one immediate slight difference from other class 4 rules that we have discussed is that structures in rule 110 do not exist on a blank background: instead, they appear as disruptions in a regular repetitive pattern that consists of blocks of 14 cells repeating every 7 steps.

Binary decision diagrams One can specify a Boolean function of n variables by giving a finite automaton (and thus a network) in which paths exist only for those lists of values for which the function yields True .

In general the density for an arrangement of white squares with offsets v is given in s dimensions by (no simple closed formula seems to exist except for the 1 × 1 case) Product[With[{p = Prime[n]}, 1 - Length[Union[Mod[v, p]]]/p s ], {n, ∞ }] White squares correspond to lattice points that are directly visible from the origin at the top left of the picture, so that lines to them do not pass through any other integer points.

Extraterrestrial life Conditions thermally and chemically similar to those on Earth have presumably existed on other bodies in the solar system.