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The reduction to steady state does not, however, proceed uniformly with respect to all schemata because the crossover operator induces a linkage phenomenon. Simply stated, linkage arises because a schema is less likely to be affected by crossover if its defining positions are close together. In more detail, let x's defining positions (those not having a " ") be i1 < i2 < . . .< ihand let the length of be defined as l(x) = (ih - i1). Then the probability of the crossover falling somewhere in x, once an instance of x has been selected for crossing-over, is just l(x)/(l - 1). E.g., if A = a1a2a3a4a5 . . . alis selected for crossing-over, the probability of the crossover point x falling within is 3/(l - 1). Clearly the smaller the length of a schema, the less likely it is to be affected by crossing-over. Thus, the smaller the length of x, the more slowly will a departure from l(x) be reduced. |
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Alleles defining a schema x of small length l(x) which exhibits above-average performance will be tried ever more frequently as a unit under an adaptive plan of type  . I.e., the alleles will be associated and tried accordingly. More modifications and tests of such schemata will be tried, and many of these trials will be of a variety of combinations with other similarly favored schemata defined at other positions. In effect such schemata serve as provisional structural elements or primitives. This observation is made precise by the following simple but important |
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THEOREM 6.2.3: Consider a reproductive plan of type using only the simple crossover operatordefined as a crossover operator with both precursors, and the single crossover point, determined by uniform random selection. Then the expected proportion of each schema represented in  changes in one generation from P(x, t) to |
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where Pc is the proportion of individuals undergoing crossover during a generation and  is the observed average performance of . (The unit of time herea generationis the expected time for an individual to produce its offspring.) |
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Proof: During one generation each individual  can be expected to produce  offspring under a reproductive plan of type . The total expected offspring of the set of instances  of x in is thus given by |
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If Pc is the proportion of selected to undergo crossover and l(x) is the length of x, then a proportion Pcl(x)/(l - 1) of the M'x(t) offspring will have a crossover falling within the defining positions of x. When an instance of x is crossed with |
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