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In the 1980s, however, work on cryptography had led to the study of some slightly weaker definitions of randomness based on inability to do cryptanalysis or make predictions with polynomial-time computations (see page 1089 ).

But although it so happens that essentially the same mathematical notation is in practice used all around the world by speakers of every ordinary language, I do not believe that it is in any way unique or inevitable, and in fact I think it shows most of the same issues of dependence on history and context as any ordinary language.

(Even if one can do operations on all digits in parallel it still takes of order n steps in a system like a cellular automaton for the effects of different digits to mix together—though see also page 1149 .)

For rules that do not show at least one-sided additivity there can be an infinite number of configurations that repeat with a given period.

But general recursive functions can be partial functions, that do not terminate for some inputs.) … But by reducing the results modulo 2 one gets a function that does not grow—and has seemingly quite random behavior—yet is presumably again not primitive recursive.

(So for example μ [r[p[1], f]][init] returns the smallest t for which the tag system reaches state {} —and never returns if the tag system does not halt.)

For example, to find a definite volume growth rate one does still need to take some kind of limit—and one needs to avoid sampling too many or too few nodes in the network.

The number of constraints which yield solutions of specified lengths Length[s] for r = 2 and r = 3 are as follows (the boxes at the end give the number of cases with no solution):
With r = 2 , as n increases an exponentially decreasing fraction of possible constraints have solutions; with r = 3 it appears that a fraction more than 1/4 continue to do so. … The only way is to use the sequence of pairs 2, 3, 3, 2—yet doing this will just produce another BAAB further on.

If one wants to enumerate all possible Diophantine equations there are many ways to do this, assigning different weights to numbers of variables, and sizes of coefficients and of exponents. … When solutions do exist, most are fairly small.

About actual spin systems evolving in time the Ising model itself does not make any statement. … But since the evolution does not conserve m[s] the average of this after many steps can be expected to be typical of all possible states of given e[s] .