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Let us define the simple inversion operator as an inversion with both the structure selected for inversion and the two points x1 and x2 determined by uniform random selection. To see the combined effect of simple inversion, simple crossover, and reproduction we need only refer to Theorem 6.2.3. The theorem guarantees that, if inversion has produced a permutation x' of x where l(x') < l(x), then the proportion of x' in  increases more rapidly than the proportion of x. For example, if Pc =1 and P(x, t), P(x', t) << 1 we can expect |
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since µx = µx'. Or, after T generations |
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As a result, any time inversion yields a shorter permutation x' of a schema x of above-average performance, that permutation will rapidly predominate. Because the rate of reproduction of a schema is dependent upon its length, there is a constant "pressure" toward tighter linkage of the defining alleles of schemata. Because only schemata exhibiting above-average performance occupy substantial fractions of , the "pressure" is only important for such schemata. Inversion, by repeatedly varying the linkage, gives this pressure a chance to act. |
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A great many schemata are affected by each inversion, but tightly linked schemata are much less likely to be affected than loosely linked ones, so that variations are primarily in the loosely linked schemata. That is, changes in linkage are concentrated in the loosely linked (long) schemata of above-average performance, where changes are desirable. More precisely, if PI is the proportion of the population undergoing inversion in a given generation, then the probability of a schema x of length l(x) being affected is |
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where the second factor comes from the fact that an inversion wholly inside a schema does not affect its length. Hence, if l(x) = b·l(x') < l/4, b > 1, for two schemata x and x', x is almost b times as likely to have its length altered. |
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One new restriction must be made upon the crossover operator when it is used in combination with inversion. Because of inversion, two l-tuples in will not always have the alleles for a given detector at the same position. Crossing-over can thus produce resultants with two (or more) alleles for a given detector, or resultants with no alleles for a given detector. For example, crossing |
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