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Each criterion in effect classifies the plans in  as "good" or "bad" according to whether or not it is satisfied. The first of these criteria is commonly met in a wide range of applications, while the second proves to be relevant to questions of survival under competition. (Once again, a plan satisfying the second criterion automatically meets the first but not vice versa.) Other criteria can be based on the cumulative payoff function and indeed criteria of a quite different kind can be useful in particular situations. Nevertheless the criteria given are representative and of general use; they will play a prominent role later. |
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A problem in adaptation will be said to be well posed once , e, and  have been specified within the foregoing framework. An adaptive system is specified within this framework by the set of objects ( , W, I, t) where |
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= {(A1,A2, .. .} is the set of attainable structures, the domain of action of the adaptive plan, |
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W = {w1,w2, . . . } is the set of operators for modifying structures with  being a function  , where  is some set of probability distributions over , |
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I is the set of possible inputs to the system from the environment, and |
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 is the adaptive plan which, on the basis of the input and structure at time t, determines what operator is to be applied at time t. |
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Under the intended interpretation |
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where  is a particular distribution over ,  is determined by drawing a random sample from according to the distribution . Given the input sequence  , t completely determines the stochastic process. (Occasionally, when the adaptive system is to be deterministic with being uniquely determined once I(t) and  are given, t will be defined without the use of operators so that  ) The structure of the adaptive system at time t, , will be required to summarize whatever aspects of the input history are to be available to the plan. Hence it will often be useful to represent as  , where  is the set of structures to be directly tested and  is the set of possible memory configurations, for retaining past history not directly incorporated in the tested structures. |
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