THE DETERMINATE MACHINE 3/1 



system, started at a given "state", will thus repeatedly pass through 

 the same succession of states. 



By a state of a system is meant any well-defined condition or 

 property that can be recognised if it occurs again. Every system 

 will naturally have many possible states. 



When the beads are released, their positions (P) undergo a series 

 of changes, Pq, Pi, P2 - ', this point of view at once relates the 

 system to a transformation: 



\ 



Po Pi P2 P3 

 Pi Pi P3 P4 



Clearly, the operands of the transformation correspond to the 

 states of the system. 



The series of positions taken by the system in time clearly corres- 

 ponds to the series of elements generated by the successive powers 

 of the transformation (S.2/14). Such a sequence of states defines a 

 trajectory or line of behaviour. 



Next, the fact that a determinate machine, from one state, cannot 

 proceed to both of two different states corresponds, in the trans- 

 formation, to the restriction that each transform is single-valued. 



Let us now, merely to get started, take some further examples, 

 taking the complications as they come. 



A bacteriological culture that has just been inoculated will increase 

 in "number of organisms present" from hour to hour. If at first 

 the numbers double in each hour, the number in the culture will 

 change in the same way hour by hour as n is changed in successive 

 powers of the transformation n' = 2«. 



If the organism is somewhat capricious in its growth, the system's 

 behaviour, i.e. what state will follow a given state, becomes somewhat 

 indeterminate. So "determinateness" in the real system evidently 

 corresponds, in the transformation, to the transform of a given 

 operand being single-valued. 



Next consider a clock, in good order and wound, whose hands, 

 pointing now to a certain place on the dial, will point to some deter- 

 minate place after the lapse of a given time. The positions of its 

 hands correspond to the transformation's elements. A single 

 transformation corresponds to the progress over a unit interval of 

 time; it will obviously be of the form n' = n + k. 



In this case, the "operator" at work is essentially undefinable, 

 for it has no clear or natural bounds. It includes everything that 

 makes the clock go: the mainspring (or gravity), the stiffness of the 



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