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represented by a finite string of attributes,  can be made countably infinite without affecting the presentation of  . This will be discussed in chapter 8.) Each plan in is an algorithm which acts at each instant t upon a small set of structures  from (interpretable, for instance, as a population or data base). The algorithm uses a single basic cycle to modify elements of the small set, one at a time, thereby producing a sequence of new structures for trial. In general terms, the basic steps of the cycle are: |
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1. Select one structure from probabilistically, after assigning each structure a probability proportional to its observed performance. |
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2. Copy the selected structure, then apply operators to the copy to produce a new structure. |
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3. Select a second element from  at random (all elements equally likely) and replace it by the new structure produced in step 2. |
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4. Observe and record the performance of the new structure. |
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5. Return to step 1. |
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Note that the number of elements in remains constant. (From the point of view of genetics, it is convenient to look upon the size of as an upper bound on population size determined, say, by the "carrying capacity" of the environment.) The number of structures in can be varied up to the maximum number by allowing null structures or vacancies. |
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With this outline as a guide, we can now go on to the rigorous definition of the algorithms in . The following symbols and definitions will be used with the interpretations given: |
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| the set of basic structures being tested. |  | the set of all M-tuples of structures corresponding to possible compositions of  . | | the particular set of M structures {A1(t), A2(t), . ., AM(t)) available to the adaptive plan at time t. |  | the first M positive integers, used as an index set for . |  | the set of stochastic operators for modifying structures. |  | compositions of with one structure selected (for modification by an operator); i.e. (i, A1(t),. . .,  corresponds to with the ith structure, Ai(t)selected. |
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