19/7 DESIGN FOR A BRAIN 



finding how it behaves after being released from an initial state 

 #J, . . . , x„. It generates one line of behaviour. 



The field of a system is its phase-space filled with such lines 

 of behaviour. 



19/7. If, on repeatedly applying primary operations to a 

 system, it is found that all the lines of behaviour which follow 

 an initial state S are equal, and if a similar equality occurs after 

 every other initial state S', S", . . . then the system is regular. 

 Such a system can be represented by equations of form 



x x = F x (x\ 9 ...,«£; 01 



x n — t n \x^ . . . , x n ; t)J 



Obviously, if the initial state is at t = 0, we must have 



Fi(x° v . . . , x° n ; 0) = a£ N,l n). 



The equations are the written form of the lines of behaviour ; 

 and the forms F{ define the field. They are obtained directly 

 from the results of the primary operations. 



19/8. If, on repeatedly applying primary operations to a system, 

 it is found that all lines of behaviour which follow a state S 

 are equal, no matter how the system arrived at S, and if a 

 similar equality occurs after every other state S\ S", . . . then 

 the system is absolute. 



19/9. A system is ' state-determined ' if the occurrence of a 

 particular state is sufficient to determine the line of behaviour 

 which follows. Reference to the preceding section shows that 

 absolute systems are state-determined, and vice versa. 



The equations of an absolute system form a group 

 19/10. Theorem. That the equations 



Xi = Fi{x\ t . . . , a£ ; t) (i = l n) 



should be those of an absolute system, it is necessary that, re- 

 garded as a substitution converting #J, . . ., #° to x v . . ., x n , 



204 



