THE DETERMINATE MACHINE 3/6 



Ex. 4: (Continued.) Express the transformation as a kinematic graph. 

 Ex. 5: The first operand, .v, is the vector (0,1,1); the operator Fis defined thus: 

 (i) the left-hand number of the transform is the same as the middle number 



of the operand ; 

 (ii) the middle number of the transform is the same as the right-hand 



number of the operand; 

 (iii) the right-hand number of the transform is the sum of the operand's 



middle and right-hand numbers. 

 Thus, F{x) is ( 1 , 1 ,2), and F\x) is (1 ,2,3). Find F\x), F\x), F\x). (Hint : 

 compare Ex. 2/14/9.) 



3/6. Notation. The last exercise will have shown the clumsiness 

 of trying to persist in verbal descriptions. The transformation F 

 is in fact made up of three sub-transformations that are applied 

 simultaneously, i.e. always in step. Thus one sub-transformation 

 acts on the left-hand number, changing it successively through 

 0^1^1-^2-^3-^5, etc. If we call the three components 

 a, b, and c, then F, operating on the vector {a, b, c), is equivalent 

 to the simultaneous action of the three sub-transformations, each 

 acting on one component only: 



fa' = ^ 

 F:^b' = c 

 Ic' =b + c 



Thus, a' = b says that the new value of a, the left-hand number in 

 the transform, is the same as the middle number in the operand; 

 and so on. Let us try some illustrations of this new method; no 

 new idea is involved, only a new manipulation of symbols. (The 

 reader is advised to work through all the exercises, since many 

 important features appear, and they are not referred to elsewhere.) 



Ex. 1 : If the operands are of the form (a,b), and one of them is (i,2), find the 

 vectors produced by repeated application to it of the transformation T: 



a' = b 

 b' = - a 



(Hint: find r(i2),r2(i,2), etc.) 

 Ex. 2: If the operands are vectors of the form {v,w,x,y,z) and U is 



f v' = w 



\w' = v 



uUx' = X 



y =z 



find Via), where a = (2,1,0,2,2). 



Ex. 3 : (Continued.) Draw the kinematic graph of U if its only operands are 

 a, Uia), U\a\ etc. 



3 33 



