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over of A2 with one of the replicates of A3 at intersection  serves both to generate another (different) instance of  , and to generate a first instance of  . Clearly such a crossover becomes increasingly likely as instances of  and  proliferate. (Points from these schemata are likely to exhibit above-average values and hence will have more offspring on the average.) Similar effects will be happening to all other schemata instanced in  ,  , etc. Figure 13 displays a more elaborate example of these effects. |
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3. Generalized Genetic OperatorsInversion |
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Crossover, by inducing a linkage between alleles, offers the possibility of an adaptable net of associations between alleles. By changing the length of a schema we modify the probability of its being affected by crossover; instances of a shorter schema are less likely to have the defining alleles separated by crossover. In consequence, under a plan of type  , instances of the shorter schema proliferate more rapidly. The long-term effect is a selective increase in the linkage of various schemata exhibiting above-average performance. The corresponding alleles (attributes) are more frequently found in association (on the same string) in successive generations. Since schemata are defined for any string-representable domain  , such an adaptable network of associations can be induced for any such domain by introducing an appropriate operator for changing linkage. |
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The linkage between the alleles defining a schema can be altered only by changing the length of the schema. That is, the positions of the alleles defining the schema (particularly the end-points) must be modifiable. However, up to this point, the functional meaning of an allele has been determined by its position. The allele aiat the ith position of the representation of the structure A is the value di(A)of the ith detector when A is its argument. Thus, if linkage is to be changed, an allele must have the same functional interpretation in any position (as is the case generally in genetics). This in turn requires a change in the method of representation. |
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The simplest way to change the method of representation formally is to assign each allele an index indicating the detector with which it is associated. That is, each allele is now taken to be a pair (i, a)indicating that a = di(A). It follows that a structure A can be represented by any permutation of |
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