Homework 2 is due before class on Monday 21 February.

1. Complete the proof that the denotational semantics is correct by
   showing that when (s,s') in C[c] there exists a derivation
   concluding with <c,s> => s'.

2. [Clayton's question:] Suppose we defined the meaning of while loops
   as follows:

   C[while e do c] = U(F^i({}))
     where F(S) = {(s,s') | (s,0) in E[e] && s' = s}
                U {(s,s') | (s,i) in E[e] && i != 0 && exists s1.
                             (s,s1) in C[c] && (s1,s') in S}

   i.e., without the extra "S" on the right-hand side.  Show that it
   is still the case that C[while e do c] <= F(C[while e do c]).

3. Using the denotational semantics, show that the following programs
   are equivalent when run in states such that x and y are non-negative.

   s := 0;                      s := 0;
   while x do {                 while y do {
     s := s + y;                  s := s + x;
     x := x - 1;                  y := y - 1;
   }                            }

   That is, show that for all states s1 where s1(x) > 0 and s2(x) > 0,
   running the first program in s1 yields a state s2, and running
   the second command yields a state s2' such that s2(s) = s2'(s).
   
4. In the lecture notes (lecture 4) we gave 3 different potential ways
   to analyze while-loops using abstract stores.  The first loop did not
   terminate and we argued informally that the third loop always terminates,
   but is more conservative than the second loop.  Either prove that the
   second loop terminates or give a counter example.  

5. Using analysis loop (3), prove that the analysis is sound.  That is,
   show that for all commands c, stores s, and abstract stores S representing
   s, that C[c]s in C'[c]S.