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operators can actually be subsumed in the stochastic selection of offspring. See below.) The operators are computation procedures using random numbers; generally, they use at most one other member of the population, in addition to Ai(t)(t), in the determination of A'(t).(For instance, the operator may randomly select a "mate" for Ai(t)(t).)The argument of each  includes the whole population, because any structure in the population is a conceivable candidate for the second operand, even when wis essentially a binary operator. (E.g., the probable outcomes of a "mating" will depend upon the range of "mates" available.) |
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It should be noted that the state of the algorithm at the beginning of any cycle includes not only the population  , but also the retained performances µE(Ah(t)), h = 1, . . ., M, of the structures in . Thus, in the general formalism of chapter 2, |
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where [0, r]is the interval of possible payoffs (performances), i.e. [0, r] is the range of µE, |
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The new information I(t),from the environment  at each time t,is simply the payoff µE(A'(t))of the new structure A'(t).Thus any adaptive plan  has the required form |
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Informally, a reproductive plan is one under which the better an individual performs the more offspring it has. For plans a precise counterpart of this property can be established with the help of the following |
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LEMMA 6.1: If, at any time-step, p1 is the probability that a structure A produces an "offspring" during that time-step and p2 is the probability that A is deleted during that time-step, then the expected number of "offspring" of A is p1/p2. |
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