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In what follows it will often be convenient to treat the adaptive plan t as a stochastic process; instead of determining a unique structure  from I(t) and  , t assigns probabilities to a range of structures and then selects accordingly. That is, given I(t), may be transformed into any one of several structures A'1, A'2, . . ., A'j, . . ., the structure A'j being selected with probability P'j. More formally: Let  be a set of admissible probability distributions over  . Then |
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will be interpreted as assigning to each pair (I(t), ) a particular distribution over ,  . The structure to be tried at time t + 1 will then be determined by drawing a random sample from according to the probability distribution (t + 1) = t(I(t), ). For those cases where the plan t is to determine the next structure uniquely, the distribution (t + 1) simply becomes a degenerate, one-point distribution where a single structure in is assigned probability 1. Hence the form |
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In practice the transformation of to is usually accomplished by the application of an operator from some specified set of operators W. Thus the detailed operation of the adaptive plan t is given by a function |
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where the stochastic aspect is now embodied in the operators. If |
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designates the particular operator selected by t' at time t, then |
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gives the resulting distribution over . Hence t' determines t once the functions in W are specified: |
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