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system is thus faced with a specific problem of compact storage, access, and effective use of information about extremely large numbers of schemata. Chapter 6 ("Reproductive Plans and Genetic Operators") sets forth a resolution of these difficulties, but a closer look at schemata (the remainder of this chapter) and the optimal allocation of trials to sets of schemata (the next chapter) provides the proper setting. |
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Let us begin with a concrete, but fairly general, interpretation of schemata stemming from the earlier discussion of control and function optimization (section 3.5, p. 57). Consider an arbitrary bounded function f(x), 0 £ x < 1, and assume that x is specified to an accuracy of one part in a million or, equivalently, that values of x are discretely represented by 20 bits. Define  to be the set of 220 discrete values of x represented with 20 detectors  where dj(x),  , assigns to x the value of the jth bit in the binary expansion of x. The schema  then is just the right half-plane ½ £ x < 1, while the schema  is a set of four strips  ,  and the schema  is the intersection of the two previous schemata  (see Figure 10). |
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With this representation there are 320 distinct schemata since any 20-tuple over the set  defines a schema. (More technically, the schemata are simply hyperplanes, of dimension 20 or less, in the 20-dimensional space of detector-value combinations.) Note that there are many points, such as  which are instances of all three of the schemata just singled out. Note also that f has a well-defined average value fx on each schema x (for any weighting of the values f(x), as by a probability distribution). Clearly, for any x, knowledge of f(x) is relevant to estimating fx for any schema for which  . Moreover, observations |
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Fig. 10.
Some schemata for a one-dimensional function |
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