4/13 



AN INTRODUCTION TO CYBERNETICS 



by the addition of another arrow wherever tiiere are two joined 

 head to tail, turning 



u 



u 



/ \ 



to 



/ 



y 



y 



and continuing this process until no further additions are possible, 

 gives the diagram of ultimate effects. 



If a variable or part has no ultimate effect on another, then the 

 second is said to be independent of the first. 



Both the diagrams, as later examples will show, have features 

 corresponding to important and well-known features of the system 

 they represent. 



Ex. 1 : Draw the diagrams of immediate effects of the following absolute systems ; 

 and notice the peculiarity of each: 



(i) x' = xy, y' = 2y. 



(ii) X' = y, y' = z + 3,z' = x^. 



(iii) // = 2 + iix, v' = V — y, x' = u + x, y' = ;- + v2. 

 (iv) //' = 4u — I, x' = iix, y' = xy -\- I, z' — yz. 



(v) ii' = » + y, x' = I — y, y' = log y, z' = z + yz. 

 (vi) ;/' = sin 2», x' = x^, y' — y + 1, z' = xy + u. 



Ex. 2: If y' = luy — z, under what conditions does // have no immediate 

 effect on y ? 



Ex. 3 : Find examples of real machines whose parts are related as in the diagrams 

 of immediate effects of Ex. 1. 



Ex. 4: (Continued.) Similarly find examples in social and economic systems. 



Ex. 5: Draw up a table to show all possible ways in which the kinematic graph 

 and the diagram of immediate effects are different. 



4/13. In the discussion of the previous section, the system was 

 given by algebraic representation; when described in this form, the 

 deduction of the diagram of immediate effects is easy. It should 

 be noticed, however, that the diagram can also be deduced directly 

 from the transformation, even when this is given simply as a set of 

 transitions. 



Suppose, for instance that a system has two variables, x and y, 

 each of which can take the values 0, 1 or 2, and that its (.Y,;^)-states 

 behave as follows (parentheses being omitted for brevity): 



I 



00 01 02 10 11 12 20 21 22 



01 00 11 11 00 21 



58 



20 



