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Additional info for Algorithms + Data Structures = Programs (Prentice-Hall Series in Automatic Computation)
First, the 0 is true. T his is because the interpretation of the first term 0 is equal to the interpretation of the second, which is also Next, the interpretation [skip]CT [O]CT [O]CT. of the body is not -1. So we are in the third case of the above equation for while, which simplifies to: [while But [skip]CT = 0 = skip]CT 0 do CT by = [while 0 = skip]([skip]CT) 0 do the definition of the semantics for skip, so this equation in turn can be simplified to: [while 0 = 0 do skip]CT = [while 0 = 0 do skip]CT Now this might seem innocuous, since this equation is merely stating a valid fact, of the form W = W.
If f is a monotonic function from predomain (X1, Ci) to predomain (X2, �2), and if c is an w-chain in (X1, �1), then f c (the composition of functions f and c) is an w-chain in (X2, �2). 2 (Preservation of chains). o Proof First, we need to understand why it makes sense to write f o c here. Re call from basic discrete mathematics the definition of function composition: the Denotational Semantics of WHILE 44 f o c returns f(c(x)) when called with input x. In our case, since c is an w-chain in (X1, �1) by assumption, it is a function from N to X1.
1) = Let us make sure, first, that it is clear what this equation say s, and why that is intuitively plausible, even though it won't work as a defining clause. First, if the interpretation of the guard t pred t' is false, then the meaning is just er. That is, the command does not change the state in that situation, since the meaning of the command is the final state resulting from execution, and in this case, execution has not altered the state in any way. Next, the equation say s that if the interpretation of the guard is true but the interpretation of the body c is -1, meaning that the body has diverged; then in that case, the meaning of the whole while-command is -1.