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We see that the average value  of all points in the schema  (i.e., the area under the curve over the interval ½ £ x < 1 divided by ½, the length of the interval) is approximately 1.5. Similarly, for  the value is approximately 1, for  the value is approximately 2, etc. Thus instances of  will accumulate at a higher rate than those of  , and instances of  will accumulate still more rapidly. The result is an ever greater clustering of test points (instances) in intervals (schemata) of above-average value (see Figure 13 and the example of section 7.3). In this way the genetic plan locates a global optimum of f(x), exploiting false peaks (without entrapment) to rapidly increase the average value of points tested. |
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We will see that genetic plans act with a combination of simplicity and subtlety both pleasing to the eye and useful in application. They also act with robustness and efficiency, a fact that will be finally established in the next chapter. It should be emphasized that the plans (algorithms) set forth have a dual role. When the plan's parameter values (and functions) are determined from data about a particular natural process, the plan serves as an idealized model or hypothesis about that process. As such it is subject to the general observation-modification cycle applicable to physical theories in general. Because the model is already in algorithmic form, it is particularly suitable for simulations of the process. The other role occurs in relation to artificial (designed) processes. Here the plans serve as optimization procedures which can be fitted into the process to control its direction. In either role the theorems proved hereafter yield predictions which must come true if (for the natural systems) the basic model is verified or (for artificial systems) the algorithm is incorporated as a control. |
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1. Generalized Reproductive Plans |
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To embed reproductive plans in the  framework of chapter 2 we must define a class of plans (algorithms) applicable to an arbitrary set of structures  . Moreover each plan must be a mapping of the form  . It must use only the input from the environment, I(t),and the structure tried at time t,  , to determine a random variable over ,  , which is in turn sampled to determine the next trial,  . We will begin by defining a relatively narrow class of reproductive plans  . Later will be extended in ways which make some applications more natural, and we will see that the new algorithms are essentially no more powerful than those from . |
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To begin let  be the set of structures to be tested and, as in chapter 4, assume that the elements of are representations. (As long as each structure is |
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