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to generate its output, and in general with size n input it may be able to take more than 2 2 n steps.
So what this implies is that to answer questions about the t -step behavior of a multiway system can take any ordinary Turing machine a number of steps that increases faster than any power of t .
Yet to take full advantage of universality one must consider not only translations between objects but also translations between complete proofs.
In general one can think of technology as trying to take systems that exist in nature or elsewhere and harness them to achieve human purposes.
But quite often it takes a very large number of steps before one can tell for sure what is going to happen.
But this process may still take a very long time.
And in such cases one can set up an analog of logic in which one no longer takes ¬ ¬ a a .
As my methodology and intuition improved, the pace of my discoveries increased still more, and within just a couple of years I had managed to take my explorations of the world of simple programs to the point where the sheer volume of factual information I had accumulated would be the envy of many long-established fields of science.
If all blocks occur with an equal probability of 2 -b , then h takes on its maximum possible value of b .
{Take[#, k], Drop[ #, k](-1)^(Range[0, k - 1]/k)}] &, Partition[##]]] &, BitReverseOrder[data], 2^Range[Log[2, n]]]/ √ n ]
(See also page 1080 .)