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at x = 2 yields ((1, a1), (2, a2), (2, a'2)) as one of the resultants. The simplest way to remedy this is to permit crossing-over only between homologous representations, where two representations are defined to be homologous if the detector indices (first number of each pair in the representation) are in the same order. For example, ((1, a1), (3, a3), (2, a2)) is homologous to ((1, a'1), (3, a'3), (2, a'2)), even if aj¹ a'jfor some or all j, while ((1, a1), (2, a2), (3, a3)) is not homologous to either of the foregoing. This remedy requires that the probability of inversion PI be small so that there will exist substantial homologous subpopulations for the crossover operator to act upon. A second alternative (with a biological precedent) would be to temporarily make the second of the l-tuples chosen for crossover homologous to the first by reordering it, returning it to the population in its original order after the resultants of the crossing-over are formed. Under this alternative inversion can be unrestricted, i.e., PI can be as large as desired. |
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Summing up: Inversion, in combination with reproduction and crossover, selectively increases the linkage (decreases the length) of schemata exhibiting above-average performance, and it does this in an intrinsically parallel fashion. |
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4. Generalized Genetic OperatorsMutation |
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Though mutation is one of the most familiar of the genetic operators, its role in adaptation is frequently misinterpreted. In genetics mutation is a process wherein one allele of a gene is randomly replaced by (or modified to) another to yield a new structure. Generally there is a small probability of mutation at each gene in the structure. In the formal framework this means that, each structure A = a1a2 . . . al in the population  , is operated upon as follows: |
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1. The positions x1, x2, . . , xh to undergo mutation are determined (by a random process where each position has a small probability of undergoing mutation, independently of what happens at other positions). |
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2. A new structure  is formed where  , is drawn at random from the range Vi of the detector di corresponding to position x1, each element in Vi being an equilikely candidate;  are determined in the same way. |
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If 1 PM is the probability of mutation at each position, then the probability of h mutations in a single representation is given by the Poisson distribution with parameter 1 PM. |
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