9/9 AN INTRODUCTION TO CYBERNETICS 



If, in another system, the transition probabiUties depend on 

 values occurring n steps back, then the new states must be defined 

 as vectors over n consecutive states. 



The method of re-defining may seem artificial and pointless. 

 Actually it is of fundamental importance, for it moves our attention 

 from a system that is not state-determined to one that is. The 

 new system is better predictable, for its "state" takes account of the 

 original system's past history. Thus, with the original form, to know 

 that the system was at state C did not allow one to say more than 

 that it might go to either C or D. With the second form, to know 

 that it was at the state (D,C) enabled one to predict its behaviour 

 with certainty, just as with the original form one could predict with 

 certainty when one knew what had happened earlier. What is 

 important is that the method shows that the two methods of 

 "knowing" a system — by its present state or by its past history — 

 have an exact relation. The theory of the system that is not com- 

 pletely observable (S.6/21) made use of this fact in essentially the 

 same way. We are thus led again to the conclusion that the 

 existence of "memory" in a real system is not an intrinsic property of 

 the system — we hypothesise its existence when our powers of observa- 

 tion are limited. Thus, to say "that system seems to me to have 

 memory" is equivalent to saying "my powers of observation do not 

 permit me to make a valid prediction on the basis of one observation, 

 but I can make a valid prediction after a sequence of observations". 



9/9. Sequence as vector. In the earlier chapters we have often 

 used vectors, and so far they have always had a finite and definite 

 number of components. It is possible, however, for a vector to 

 have an infinite, or indefinitely large number of components. Pro- 

 vided one is cautious, the complication need cause little danger. 



Thus a sequence can be regarded as a vector whose first component 

 is the first value in the sequence, and so on to the n-th component, 

 which is the 77-th value. Thus if I spin a coin five times, the result, 

 taken as a whole, might be the vector with five components (H, T, 

 T, H, T). Such vectors are common in the theory of probabihty, 

 where they may be generated by repeated sampling. 



If such a vector is formed by sampling with replacement, it has 

 only the slight peculiarity that each value comes from the same 

 component set, whereas a more general type, that of S.3/5 for 

 instance, can have a different set for each component. 

 9/10. Constraint in a set of sequences. A set of such sequences 

 can show constraint, just as a set of vectors can (S.7/11), by not 

 having the full range that the range of components, if they were 



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