THE DETERMINATE MACHINE 3/4 



3/4. Machine and transformation. The parallelism between 

 machine and transformation is shown most obviously when we 

 compare the machine's behaviour, as state succeeds state, with the 

 kinematic graph (S.2/17), as the arrows lead from element to element. 

 If a particular machine and a particular graph show full corres- 

 pondence it will be found that: 



(1) Each possible state of the machine corresponds uniquely to a 

 particular element in the graph, and vice versa. The correspondence 

 is one-one. 



(2) Each succession of states that the machine passes through 

 because of its inner dynamic nature corresponds to an unbroken 

 chain of arrows through the corresponding elements. 



(3) If the machine goes to a state and remains there (a state of 

 equiUbrium, S.5/3) the element that corresponds to the state will 

 have no arrow leaving it (or a re-entrant one, S.2/17). 



(4) If the machine passes into a regularly recurring cycle of states, 

 the graph will show a circuit of arrows passing through the corres- 

 ponding elements. 



(5) The stopping of a machine by the experimenter, and its re- 

 starting from some new, arbitrarily selected, state corresponds, in 

 the graph, to a movement of the representative point from one 

 element to another when the movement is due to the arbitrary 

 action of the mathematician and not to an arrow. 



When a real machine and a transformation are so related, the 

 transformation is the canonical representation of the machine, and 

 the machine is said to embody the transformation. 



Ex. 1 : A culture medium is inoculated with a thousand bacteria; their number 

 doubles in each half-hour. Write down the corresponding transformation. 



Ex. 2: (Continued.) Find n after the 1st, 2nd, 3rd, . . ., 6th steps. 



Ex. 3 : (Continued.) (i) Draw the ordinary graph, with two axes, showing the 

 culture's changes in number with time, (ii) Draw the kinematic graph of 

 the system's changes of state. 



Ex. 4: A culture medium contains 10^ bacteria and a disinfectant that, in each 

 minute, kills 20 per cent of the survivors. Express the change in the number 

 of survivors as a transformation. 



Ex. 5 : (Continued.) (i) Find the numbers of survivors after 1 , 2, 3, 4, 5 minutes. 



(ii) To what limit does the number tend as time goes on indefinitely? 

 Ex. 6: Draw the kinematic graph of the transformation in which n' is, in a table 



of four-figure logarithms, the rounded-off right-hand digit of logio(/J + 70). 



What would be the behaviour of a corresponding machine ? 



Ex. 7 : (Continued, but with 70 changed to 90.) 



Ex. 8: (Continued, but with 70 changed to 10.) How many basins has this 

 graph ? 



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