And almost without exception what has been assumed is that this must come through the first mechanism discussed in Chapter 7 : that there is somehow randomness present in the environment that always gets into the system one is looking at. … (A somewhat related proposal involves quantum gravity effects in which irreversibility is assumed to be generated through analogs of the black hole processes mentioned in the previous note.)

But by the 1980s it became clear—notably through the failure of attempts to automate natural language understanding and translation—that language cannot in most cases (with the possible exception of grammar-checking software) meaningfully be isolated from other aspects of human thinking.

In the mid-1980s, particularly through the work of Alvy Ray Smith , L systems became widely used for realistic renderings of plants in computer graphics.

Occasional work on neural networks had continued through the 1960s and 1970s, with a few definite results being obtained using methods from physics.

One form of solution to the vacuum Einstein equations is a gravitational wave consisting of a small perturbation propagating through flat space.

In existing fields of science their largely closed communities tend to maintain standards of quality mostly through direct institutional and personal contact.

But by the late 1940s it had become clear, particularly through the writings of John von Neumann , that it would be convenient to be able to jump around instead of always having to follow a fixed sequence.

In addition to those with whom I have had direct contact, other individuals have provided input indirectly through my assistants or others (excluding photograph sources listed in the colophon): Bill Beyer, Sheila Blair, Victor Dan, Brent Daniel, Noam Elkies, Peter Falloon, Erich Friedman, Jochen Gerber, Branko Grünbaum, Richard Guy, Michel Janssen, Martin Kraus, Temur Kutsia, Richard Langley, Bernd Löchner, Crista Malick, Brendan McKay, Thomas Scanlon, Rob Scharein, Marjorie Senechal, Marc Sher, David Singmaster, Neil Sloane, Milton Van Dyke, Bob Veroff, Curtis Wilson, Mirek Wójtowicz.

The corresponding form for rule 110 is BitXor[BitAnd[a, 2a, 4a], BitOr[2a, 4a]] The final equation is then obtained from {1 + x 4 + x 12  2 (1 + x 3 ) (x 1 + 2 x 3 ) , x 2 + x 13  2 x 1 , 1 + x 5 + x 14  2 x 1 , 2 x 3 x 5 + 2 x 1 + 2 x 3 x 6 + 2 x 1 + x 3 x 15 + x 16  x 4 , 1 + x 15 + x 17  2 x 3 , 1 + x 16 + x 18  2 x 3 , 2 1 + x 3 (1 + x 1 + 2 x 3 ) (-1 + x 2 ) - x 10 + x 11  2 x 4 , x 7  BitAnd[x 6 , 2 x 6 ] ∧ x 8  BitOr[x 6 , 2 x 6 ], x 9  BitAnd[x 6 , 2 x 7 ] ∧ x 19  BitOr[x 6 , 2 x 7 ], x 10  BitAnd[x 9 , 2 x 8 ] ∧ x 11  BitOr[x 9 , 2 x 8 ]} where x 1 through x 4 have the meanings indicated in the main text, and satisfy x i ≥ 0 .

In the late 1980s, however, it was suggested that variables could be used corresponding roughly to gravitational fluxes through loops in space.