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If perturbations are indeed important, it implies that a traditional fluid description is not adequate.
But when the wave theory of light finally became popular in the mid-1800s it seemed to imply that no similar principle could be true for light.
And although at some level computational irreducibility and NP completeness (see page 766 ) imply that in general only a limited amount of this computational work can be saved, there are in practice quite often important optimizations that can be made.
(For rules 90 and 30 the functions obtained after one step are respectively 2 p (1 - p) and p (2 p 2 - 5 p + 3) , both of which turn out to imply correct final densities of 1/2 ).
Grand unified models typically do this for all known gauge bosons (except gravitons) and for corresponding families of quarks and leptons—and inevitably imply the existence of various additional particles more massive than those known, but with properties that are somehow intermediate.
No particularly rigorous treatments of the system were given, and the mere existence of configurations that can act for example like logic gates was often assumed immediately to imply universality.
The last axiom is a schema (see page 1156 ) that states the principle of mathematical induction: that if a statement is valid for a = 0 , and its validity for a = b implies its validity for a = b + 1 , then it follows that the statement must be valid for all a .
Most Greek philosophers seem to have believed that their various mechanical or moral theories implied rigid determination of human actions.
But the fact that there are fairly few short such equivalences for Nand (see page 818 ) implies that there can be no axiom system for Nand with 6 or less Nand s except the ones discussed above.
All of these effects are implied by the standard mathematical formalism of quantum theory.