12/9 



TEMPORARY INDEPENDENCE 



cases of providing a clear and precise foundation. By its use 

 the independencies within a system can be proved by primary 

 operations only. 



12/9. The definition makes 'independence ' depend on how the 

 system behaves over a single unit of time (over a single step if 

 changing in steps, or over an infinitesimal time if changing con- 

 tinuously). The dependencies so defined between all pairs of 

 variables give, as defined in S. 4/12, the diagram of immediate 

 effects. 



In general, this diagram is not restricted: all geometrically 

 drawable forms may occur in a wide enough variety of machines. 

 This freedom, however, is not always possible if we consider the 

 relation between two variables over an extended period of time. 

 Thus, suppose Z is dependent on F, and Y dependent on X, so 

 that the diagram of immediate effects contains arrows: 



X may have no immediate effect on Z, but over two steps the 

 relation is not free; for two different initial values of X will lead, 

 one step later, to two different values of Y; and these two different 

 values of Y will lead (as Z is dependent on Y) to two different 

 values of Z. Thus after two steps, whether X has an immediate 

 effect on Z or not, changes at X will give changes at Z; and thus 

 X does have an effect on Z, though delayed. 



Another sort of independence is thus possible : whether changes 

 at X are followed at any time by changes at Z. These relations 

 can be represented by a diagram of ultimate effects. It must be 

 carefully distinguished from the diagram of immediate effects. It 

 is related to the latter in that it can be formed by taking the 

 diagram of immediate effects and adding further arrows by the 

 rule that if any two arrows are joined head to tail, 



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