THE MACHINE WITH INPUT 4/14 



What of v's transitions? We can rc-classify them, with x as 

 parameter, by representing, e.g. "00-^01" as "when x = 0, y goes 

 from to 1". This gives the table 



X 





 I 



2 



y 



1 



1 1 

 1 1 

 1 1 



It shows at once that y's transitions do not depend on the value 

 of X. So X has no immediate effect on )'. 

 Now classify .v's transitions similarly. We get: 



y 



What X will do (i.e. .v's transition) does depend on y's value, so y 

 has an immediate effect on .v. 



Thus, the diagram of immediate effects can be deduced from a 

 statement of the primary transitions. It is, in fact, 



y 



and y has been proved to dominate x. 



Ex. : A system has three variables— .y, y, z — each of which can take only the 

 values or 1. If the transformation is 



I 000 001 010 Oil 100 101 110 HI 

 ^ 110 111 100 101 110 Oil 100 001 

 what is the diagram of immediate effects? (Hint: First find how z's transi- 

 tions depend on the values of the others.) 



4/14. Reducihility. In S.4/1 1 we noticed that a whole system may 

 consist of two parts each of which has an immediate effect on the 

 other: 



Q 



We also saw that the action may be only one way, in which case one 

 part dominates the other: 



59 



