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another instance of x the result will also be an instance of x; otherwise the resultant may not be an instance of x. Since the probability of x crossing with x is P(x, t) no more than a proportion (1 - P(x, t)) Pcl(x)/(l - 1) of the modified offspring of x can be expected to be instances of schemata other than x; the remainder [1 - (1 - P(x, t)) Pcl(x)/(l -1)] will be instances of x. |
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(It should be noted that crossing-over applied to precursors which are not instances of x may yield a resultant which is an instance of x. Thus Mx(t +1) may be enlarged, by a small amount usually, from sources outside Bx(t);this of course only strengthens the above bound.) Q.E.D. |
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From this result we see that the proportion of (instances of) a schema x will increase as long as |
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or, using the fact that 1/(1 - c) ³ 1 + c for c £ 1, |
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Since the worst case occurs when Pc= 1 (every individual in  subjected to crossing-over) and P(x, t)is small, we see that x will always increase its representation if |
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Since 1/l £ l(x)/(l - 1) £ 1, short schemata need perform only slightly above average to increase, while the longest schemata (if they occur in small proportion) may have to exhibit a performance twice the population average to increase. |
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Theorem 6.2.3 provides the first evidence of the intrinsic parallelism of genetic plans. Each schema represented in the population increases or decreases according to the above formulation independently of what is happening to other schemata in the population. The proportion of each schema is essentially determined by its average performance in relation to the population average. Thus we see the evolution of a ranking of schemata based on observed performance, as suggested at the end of chapter 4 and amplified in section 5.4. Crossing-over serves |
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