DESIGN FOR A BRAIN 22/10 



all being understood to refer only to the region P. (The equiva- 

 lences follow readily from the properties of the equations of S. 19/9 

 and their integrals.) 



22/10. Given a state-determined system and two transitions 

 from two initial states which differ only in their values of x°, (the 

 difference being A#9), the variable x k is independent of x t if x k s 

 transition is identical in the two cases. Analytically, x k is inde- 

 pendent of Xj in the conditions given if 



F k {x° v . . . , 4 . . .; dt) = F k {x\, . . . , x] + Aaf, . . .; dt) (1) 



In other words, x k is independent of Xj if x k s behaviour is invariant 

 when the initial state is changed by Ax9. (This ' change ' by 



A#j must not be confused with the change dt.) 



This narrow definition provides the basis for further develop- 

 ment. In practical application, the identity (1) may hold over all 

 values of Ax® (within some finite range, perhaps); and may also 

 hold for all initial states of x k (within some finite range, perhaps). 

 In such cases the test whether x k is independent of Xj is whether 



faoF k (x<>; t) = 0. 



The range over which the relation or equation holds must always 

 be specified (either explicitly or by implication). 



Diagrams of effects 



22/11. The diagram of immediate effects and the canonical 

 representation have a simple relation. Starting with the prag- 

 matic and empirical point of view of S. 2/7, we assume that the 

 observer gets his basic knowledge of the system by primary 

 operations. These operations will give him the functions F t of 

 S. 19/7 and also (by S. 4/12) the diagram of immediate effects. 

 Now the test for whether to draw an arrow from x, to x k is essen- 

 tially the same as the test applied algebraically to see whether x°, 

 occurs effectively in F k , and the outcomes must correspond. But 

 (by the Corollary, S. 19/9) whether F k does or does not contain x° 

 effectively over a single step dt must correspond with whether f k 

 does or does not contain x, effectively. Thus, in the diagram of 

 immediate effects an arrow will run from Xj to x k if and only if, 



278 



