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But it turns out that this is no longer true if one allows systems which themselves can access the oracle in the course of their evolution.
Multivalued logic
As noted by Jan Łukasiewicz and Emil Post in the early 1920s, it is possible to generalize ordinary logic to allow k values Range[0, 1, 1/(k - 1)] , say with 0 being False , and 1 being True .
And even with a more traditional definition, this is at least true of much of my discussion of systems based on numbers in Chapter 4 .
Over the past century or so, a variety of definitions of true randomness have been proposed.
And as the pictures on the facing page demonstrate, the same is true even in cellular automata with very simple rules.
And indeed when it comes to large-scale fluid behavior, it seems to be true that in almost all cases there is no discernible difference between what happens with different detailed initial conditions.
But the true picture is more complicated.
But despite some confusion in the past it is certainly not true that just checking equality of frequencies of blocks of colors—even arbitrarily long ones—is sufficient to ensure that no regularities at all exist.
But this seems to be considerably less true when one is dealing with descriptions in which information can be lost.
But what the results in this book establish is that in fact this is far from true.