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Proof: Using Corollary 6.4.1 in combination with the expression for the probability of a schema being affected by inversion (from section 6.3) we have, for any t0, |
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Or, by iteration of the relation, |
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But the expected number of instances of x at time t = t0 + h is just |
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Lemma 7.2, though simple, makes one very important point. Even though plans of type  try structures from  one at a time, it is really schemata which are being tested and ranked. There are somewhere between 2l and M2l schemata with instances in  . Each one changes its proportion in at a rate largely determined by its observed performance,  , and is largely uninfluenced by what is happening to other schemata. This is the foundation of the intrinsic parallelism of plans of type  . |
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While Lemma 7.2 is sharp enough as it stands to enable us to establish the efficiency of plans of type , some of the properties implied by the sharper inequality of the first line of its proof are also worth noting. As P(x, t) ® 1 the operator losses from crossing-over approach 0 and the first factor in brackets approaches 1. That is, as P(x, t) ® 1, the rate of change is very nearly |
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Moreover, if Mx(t) is at all large, will closely approximate the expected payoff µx(t) of x under the distribution (over at time t) corresponding to , because of the central limit theorem. Now recall that two schemata defined on different positions, but having identical sets of functional attribute pairs, designate the same subset of . Thus all permutations of x induced by inversion exhibit the same expected payoff µx(t) at any given time. If we treat these permutations as versions of the same schema, then inversion does not in fact result in instances of x being lost during the operator phase. This leaves mutation as the only important source |
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