4/7 AN INTRODUCTION TO CYBERNETICS 



under the control of whoever arranges the couphng.) Let us further 

 suppose — this is essential to the orderliness of the coupling — that the 

 two machines P and R work on a common time-scale, so that their 

 changes keep in step. 



It will now be found that the two machines form a new machine 

 of completely determined behaviour. Thus, suppose the whole is 

 started with R ni a and P at /. Because P is at /, the 7?-transforma- 

 tion will be R2 (by Z). This will turn a to b; P's i will turn to k; 

 so the states a and / have changed determinately to b and k. The 

 argument can now be repeated. With P at k, the /^-transformation 

 will again (by Z) be R2; so b will turn (under R2) to a, and k will 

 turn (under P) to /. This happens to bring the whole system back 

 to the initial state of (a,i), so the whole will evidently go on in- 

 definitely round this cycle. 



The behaviour of the whole machine becomes more obvious if 

 we use the method of S.3/5 and recognise that the state of the whole 

 machine is simply a vector with two components (x,y), where x 

 is one of a, b, c, d and y is one of /, j, k. The whole machine thus 

 has twelve states, and it was shown above that the state (a,i) under- 

 goes the transitions 



(a,i) — ^ (b,k) -^ {a,i) -^ etc. 



Ex. 1 : If Q is the transformation of the whole machine, of the twelve states 



{x,y), complete Q. 

 Ex. 2: Draw Q's kinematic graph. How many basins has it? 

 Ex. 3 : Join P and R by using the transformation Y 



y. J state of P: I / j k 

 ' \ value of « . 1 2 3 



What happens when this machine is started from {a,i) ? 

 Ex. 4: If two machines are joined to form a whole, does the behaviour of the 

 whole depend on the manner of coupling? (Hint: use the previous Ex.) 



Ex. 5 : If two machines of n\ and «2 states respectively are joined together, what 

 is the maximal length of transient that the whole can produce? 



Ex. 6: If machine M has a maximal length of transient of // states, what will be 

 the maximal length of transient if a machine is formed by joining three M's 

 together ? 



Ex. 7: Take many parts {A, B, C, . . .) each with transformation 



a 



y 



1 2 



2 



1 1 1 



2 2 2 



50 



