And with the traditional view that biological evolution is somehow a process of infinite power this seems to leave one little choice but to conclude that there must be fundamental limitations on possible methods of perception that can be useful.

Exponential distributions (as seen, for example, in learning curves) and power-law distributions (as in Zipf's law below) are both, for example, very easy to obtain.

Note that above degree 4, algebraic numbers cannot in general be expressed in radicals involving only Plus , Times and Power (see page 945 ).

And the success of Mathematica provides considerable evidence for the power of this kind of approach.

For at the time there seemed to be no limit to the power of mathematics, and no end to the theorems that could be proved.

This can be achieved by taking e n to be Nest[inc, zero, n] where zero = s[k] inc = s[s[k[s]][k]] With this setup one then finds plus = s[k[s]][s[k[s[k[s]]]][s[k[k]]]] times = s[k[s]][k] power = s[k[s[s[k][k]]]][k] (Note that power[x][y]//.crules is y[x] , and that by analogy x[x[y]] corresponds to y x 2 , x[y[x]] to x x y , x[y][x] to x y x , and so on.) … To go the other way, one uses the result that for all Church numerals x and y , Nest[s, k, n][x][y] is also a Church numeral—as can be seen recursively by noting its equality to Nest[s, k, n - 1][y][x[y]] , where as above x[y] is power[y][x] . And from this it follows that Nest[s, k, n] can be converted to the Church numeral for n by applying s[s[s[s[s[k][k]][k[s[s[k[s]][k]][s[k][k]]]]][ k[s[s[k[s]][k]][s[s[k[s]][k]][s[k][k]]]]]][s[s[k[s]][ s[s[k[s]][s[k[s[s[s[s[s[s[s[k][k]][k[s]]][k[k]]][k[s[s[ k[s]][k]][s[k][k]]]]][k[s[s[k[s]][k]][s[s[k[s]][k]][s[k][ k]]]]]][k[s[s[s[s[k][k]][k[s[s[k[s]][s[k[s[s[k][k]]]][s[ k[k]][s[k[s[s[k[s]][k]]]][s[s[k][k]][k[k]]]]]]][s[k[k]][s[ s[k][k]][k[k]]]]]]][k[s[s[s[k][k]][k[s[k]]]][k[s[k]]]]]][ k[s[k]]]]]]]][s[k[k]][s[s[s[k][k]][k[s[s[k[s]][k]][s[k][ k]]]]][k[s[s[k[s]][k]][s[s[k[s]][k]][s[k][k]]]]]]]]][ k[s[k[k]][s[s[k[s]][k]]]]]]][k[s[k][k]]]]][k[s[k]]] Using this one can find from the corresponding results for Church numerals combinator expressions for plus , times and power —with sizes 377, 378 and 382 respectively.

Power cellular automata Multiplication by m in base k corresponds to a local cellular automaton operation on digit sequences when every prime that divides m also divides k .

With sufficiently simple behavior, a cellular automaton repetition period can readily be determined in some power of Log[n] steps.

And indeed the question of whether the halting times for a system grow only like a power of input size is in general undecidable.

For in addition to 1/f noise effects, solitons and other collective lattice effects presumably lead to power-law decay of correlations.