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That is, the range of t'can be changed from W to  with t' being redefined so that |
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With this extension t'and t become identical; for this reason one symbol ''t" will be used to designate both functions, the range being specified whenever the distinction is important. |
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The general objective of this formalism is comparison of adaptive plans, either as hypotheses about natural phenomena or as algorithms for artificial systems. The comparison naturally centers on the efficiency of different plans in locating high performance structures under a variety of environmental conditions. For a comparison to be made there must be a set of plans, given either explicitly or implicitly, which are candidates for comparison. This set will be formally designated  . Often will be the set of all possible plans employing the operators in W, but in some cases there will be constraints restricting , while in others will be enlarged to include all possible plans over  (i.e., all possible functions of the form  )., however defined, represents the set of technical or feasible options for the adaptive system under consideration. |
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As indicated in the survey, a nontrivial problem of adaptation exists only when the adaptive plan is faced with an initial uncertainty about its environment. This uncertainty is formalized by designating the set e of alternatives corresponding to characteristics of the environment unknown to the adaptive plan. The dependence of the plan's action upon the environment finds its formal counterpart in the dependence of the input I(t)upon which environment actually confronts the plan. One case of particular importance is that in which the adaptive plan receives a direct indication of the performance of each structure it tries. That is, a part of the input I(t)will be the payoff mE( )determined by the function |
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which measures the performance of each structure in the given environment. |
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Sometimes, when the performance of a structure in the environment E depends upon random factors, it is useful to treat the utility function as assigning a random variable from some predetermined set  to each structure in . Thus |
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and the payoff assigned to is determined by a trial of the random variable  . This extension does not add any generality to the framework (and hence is unnecessary at the abstract level) because any randomness involved |
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