12/8 AN INTRODUCTION TO CYBERNETICS 



idea of the ordinary, determinate machine — the type considered 

 throughout Part I. If the probabihties are all or 1 then the two 

 are identical. If the probabilities are all very near to or 1 , we get a 

 machine that is almost determinate in its behaviour but that 

 occasionally does the unusual thing. As the probabilities deviate 

 further and further from and 1, so does the behaviour at each 

 step become less and less determinate, and more and more hke 

 that of one of the insects considered in S.9/4. 



It should be noticed that the definition, while allowing some 

 indeterminacy, is still absolutely strict in certain respects. If the 

 machine, when at state x, goes on 90% of occasions to y and on 10% 

 of occasions to z, then those percentages must be constant (in the 

 sense that the relative frequencies must tend to those percentages 

 as the sequence is made longer; and the limits must be unchanging 

 as sequence follows sequence). What this means in practice is that 

 the conditions that determine the percentages must remain constant. 



The exercises that follow will enable the reader to gain some 

 familiarity with the idea. 



Ex. 1 : A metronome-pendulum oscillates steadily between its two extreme 

 states, R and L, but when at the right (R) it has a 1% chance of sticking 

 there at that step. What is its matrix of transition probabihties? 



Ex. 2 : A determinate machine a has the transformation 



i 



A B C D 

 B D D D 



A Markovian machine ^ has the matrix of transition probabilities 



How do their behaviours differ? (Hint: Draw a's graph and draw /3's 

 graph after letting the probabilities go to 1 or 0.) 

 Ex. 3 : A Markovian machine with input has a parameter that can take three 

 values — p, q, r — and has two states, a and b, with matrices 



It is started at state b, and goes one step with the input at q, then one step 

 with it at r, then one step with it at p. What are the probabilities that it 

 will now be at fl or 6? 



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