Let  S1,  S2 and  S3 be sets of states and let  f1 : S1 → 2S2 and  f2 : S2 → 2S3 be functions. Then the composition f1◦ f2
of  f1 with  f2 is deﬁned in the usual way,8 i.e.  ( f1◦ f2)(s) = f2( f1(s)) for all  s ∈ S1. Note that  f1(s) is a set of states, so 
t∈ f1(s) f2(t) according  to  our  previous  deﬁnitions.  That  is,  ( f1◦ f2)(s) is  a  set  of  states  in  S3,  not  a  set  of 
f2( f1(s)) = (cid:2)
subsets of S3. This is essential, since  f1◦ f2 could otherwise never be a transformation function from S1 to S3.
The composition  f1◦ f2 is not necessarily a transformation function, even if both  f1 and  f2 are transformation functions, 
as the following example shows.
Example 56. Let  S1 = S3 = {0, 1} and  S2 = {0, 1, 2}. Deﬁne two transformation functions  f1 : S1 → 2S2 and  f2 : S2 → 2S3
such that  f1(0) = {0, 1},  f1(1) = {2},  f2(0) = {0} and  f2(1) = f2(2) = {1}. Then ( f1◦ f2)(0) = {0, 1} and ( f1◦ f2)(1) = {1}, so 
f1◦ f2 is not a transformation function.
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