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For over and over again we have seen that simple initial conditions are quite sufficient to produce behavior of immense complexity, and that making the initial conditions more complicated typically does not lead to behavior that looks any different.
Much as in the case of universality for complete systems, however, the Principle of Computational Equivalence does not just say that a sophisticated computation will be found somewhere in a pattern produced by a system like rule 30.
But can this detailed kind of phenomenon really be used as the basis for doing fundamentally more sophisticated computations?
And if one does this then one encounters the phenomenon of undecidability that was identified in the 1930s.
The framework I develop in this book also shows that by viewing the process of doing mathematics in fundamentally computational terms it becomes possible to address important issues about the foundations even of existing mathematics.
But how complicated an axiom system does it then need?
Statements like p ¬ ¬ p do not hold in general with more than 2 truth values.
And to do so I have often ended up compressing into a page or even a paragraph the essence of what a chapter or even a book could have been written about.
Nevertheless, some programs for circuit design such as SIS do include a few heuristics.
Spell-checking systems typically find suggested corrections by doing a succession of lookups after applying transformations based on common errors.