ANSWERS TO THE EXERCISES 



8/8 



8/7. 1: It does not maintain all distinctions; a perfect inverter is fundamen- 

 tally impossible with it. 



R R R 

 S S S 

 (will not occur) 

 S S S 

 R R R 

 (will not occur) 

 Q Q Q 



(will not occur) 



Q Q Q 



R R 



S S 



immediate effects m -> x -> .v, with y 

 X may be of the form 



2: (h,B) 



(h,C) 

 (b,D) 

 ic,A) 

 (c,B) 

 (c,C) 

 (c,D) 

 UhA) 

 {d,B) 

 id,C) 

 id,D) 



3: It must have diagram of 

 emitting »'s values two steps later. 



R 



S 



If now V has the form 



then y will emit capital letters corresponding to «'s original values. 



If the two (x + y) are regarded as one machine with state (x,y), 

 the transformations must be 



In general, «( takes (x;,t4) to (xi,Uj), which goes, at the next step, to 

 ( — ,Ui), thus repeating the original mj. 



8/8. 1: p' = n, m' = c/d; join by putting d = n and c = p. 2: p' = n, 

 m' = \{d — c) + 2; join by putting d = n and c ^ p. 3: p\' = x. 

 Pi = y, mi = (ci + C2)l2du m2 = (c2 - Ci)/2d2; join by putting 

 di = X, d2 = y, ci = pi, C2 = P2- 4: The equation cannot be solved 

 for a and b separately, or a and b affect the equation only in the com- 

 bination a + b or their distinct effects do not appear distinctly in the 

 output, and cannot therefore be traced back — these are different ways 

 of expressing the same basic idea. Notice that the reason for the 

 impossibility lies not in the lack of suitable gadgetry but in the fact that 

 the output does not define the input — the necessary information is 



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