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If successive populations are produced by mutation alone (without reproduction), the result is a random sequence of structures drawn from  . The process is evidently enumerative (see section 1.5) since the order in which structures are generated is unaffected by the observed performances of the structures. Even a reproductive plan of type  using only the mutation operator is little more than an enumerative plan retaining the best structure encountered to each point in time. That is, if 1 PMis small enough, reproduction will assure that structures with above-average performance predominate in successive generations thus retaining the better structures generated by the mutation operator. There is actually a bit of history dependence since, with 1 PMsmall, the most likely structures resulting from mutation will differ by one or two alleles from the current ''best" structures. Thus, the sequence of tests is not entirely random, though the dependence on observations is very unsophisticated compared to that generated by crossing-over. |
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Since enumerative plans are, at best, useful in very limited situations, it would seem that mutation's primary role is not one of generating new structures for triala role very efficiently filled by crossing-over. It might be objected that crossing-over cannot generate all possible combinations of alleles unless the population  contains at least one copy of every allele. However this is not a burdensome requirement. If k is the maximum number of alleles for any detector, then as few as k strings will suffice to provide a copy of each allele. (E.g., if Vi = {0, 1}, i = 1, . . ., l, then the two l-tuples 00 . . . 0 and 11 . . . 1 suffice.) There is nevertheless a difficulty which is remedied by mutation. In a population that is small relative to , there is always the possibility that the last copy of some allele will be eliminated during the deletion phase of a plan of type . Alleles which occur in structures of below-average performance are particularly susceptible; yet at some later stage these same alleles may be required in a combination necessary for further improvement. Stated another way, the lost allele may be necessary for the adaptive plan to escape a false peak. Once an allele is lost from a population, the crossover operator has no way of reintroducing it. Here, then, is a role uniquely filled by mutation, because it assures that no allele permanently disappears from the population. |
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Mutation introduces an additional source of loss for schemata undergoing reproduction. If the probability of mutation at each position is less than or equal to 1 PM, then a schema x defined on l0(x) positions can expect to undergo one or more mutations with probability |
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