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1. Two structures,  and  , are selected (usually at random) from the current population  . (ai and  are elements of the set of attribute values V. Hence, if A0 is the basic structure prior to representation, si(A0) = ai. Again a1a2 . . . al abbreviates (a1, a2, . . ., al), etc.) |
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2. A number x is selected from {1, 2, . . .,l - 1} (again at random). |
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3. Two new structures are formed from A and A' by exchanging the set of attributes to the right of position x, yielding  and  . |
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(To incorporate crossing-over directly into plans of type  one of the resultant structures is discarded.) |
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The quickest way to get a feeling for the role crossing-over plays in adaptation is to look at its effect upon schemata. To do this, consider as a pool of schemata (following the suggestions of chapter 4) where the number Mx(t) of instances of x in reflects x's current "usefulness." The two direct effects of crossing-over on this pool are: |
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1. Generation of new instances of schemata already in the pool. E.g., A = a1a2. . . al is an instance of the schema  and, after crossing-over with  , we have a new instance of  , namely  (assuming ai¹ a'j for some i ³ x). Each new instance of a schema x amounts to a new trial of the random variable corresponding to x. As such it increases the likelihood that the observed average performance  of the instances of x closely approximates the expectation µx of the random variable x. |
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2. Generation of new schemata (i.e. schemata having neither A nor A' as an instance). E.g., after the crossing-over of A with A' the schema  has an instance, though neither A nor A' are instances of it (if ax¹ a'x or ax + 1¹ a'x +1). Thus  will receive its first trial with the instance a1a2 . . . axa'x + 1 . . . a'l, unless the schema has previously been introduced to the pool from another source. |
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