THE ERROR-CONTROLLED REGULATOR 12/8 



in industrial machinery but of the commonest occurrence and highest 

 importance in biological systems. The subject is returned to in 

 S. 12/11. Meanwhile we must turn aside to see what is involved 

 in this idea of a "non-determinate" machine. 



THE MARKOVIAN MACHINE 



12/8. We are now going to consider a class of machine more general 

 than that considered in Parts I and II. (Logically, the subject 

 should have been considered earlier, but so much of those Parts 

 was concerned with the determinate machine (i.e. one whose trans- 

 formations are single-valued) that an account of a more general 

 type might have been confusing.) 



A "machine" is essentially a system whose behaviour is sufficiently 

 law-abiding or repetitive for us to be able to make some prediction 

 about what it will do (S.7/19). If a prediction can be made, the 

 prediction may be in one of a variety of forms. Of one machine we 

 may be able to predict its next state — we then say it is "determinate" 

 and is one of the machines treated in Part I. Of another machine 

 we may be unable to predict its next state, but we may be able to 

 predict that, if the conditions are repeated many times, the frequencies 

 of the various states will be found to have certain values. This 

 possible constancy in the frequencies has already been noticed in 

 S.9/2. It is the characteristic of the Markov chain. 



We can therefore consider a new class of absolute system : it is one 

 whose states change with time not by a single-valued transformation 

 but by a matrix of transition probabihties. For it to remain the 

 same absolute system the values of tlie probabilities must be un- 

 changing. 



In S.2/10 it was shown that a single-valued transformation could 

 be specified by a matrix of transitions, with O's or I's in the cells 

 (there given for simplicity as O's or -t-'s). In S.9/4 a Markov chain 

 was specified by a similar matrix containing fractions. Thus a 

 determinate absolute system is a special case of a Markovian 

 machine; it is the extreme form of a Markovian machine in which all 

 the probabilities have become either or I. (Compare S.9/3.) 



A "machine with input" was a set of absolute systems, distin- 

 guished by a parameter. A Markovian machine with input must 

 similarly be a set of Markovian machines, specified by a set of 

 matrices, with a parameter and its values to indicate which matrix 

 is to be used at any particular step. 



The idea of a Markovian machine is a natural extension of the 



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