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The problem with this equation is that adding it as a defining clause in the definition of the semantics of WHILE-commands will result in a definition which is not well-founded. This is because the right hand side appeals to the interpretation [while t pred t' do c], which is the same interpretation the equation is try ing to define. Schematically, the equation looks like: [W]er = · · · [W]er' · · · Denotational Semantics of WHILE 36 T hat is, the interpretation of the while-command (schematically, W) is defined in terms of itself.

To summarize: monotonic function f is continuous iff for every chain c, the value returned by f for the least upper bound of c is equal to the least upper bound of the chain obtained by applying f to each element of c. There is a common less the equation refers to the least upper bound of the chain precise but more memorable way to put this. We can refer more briefly to the least upper bound of a chain as the limit of the chain. " A positive example This example concerns the domain fined on N U { w} by f(x) = (N U { w}, s w).

2 (Omega-chains). An w-chain in a partially ordered function f from IN to X such that f(n) C f(n + l),for all n E IN. set (X, C) is a We think of w-chains as sequences of elements of X, ordered by C; that is, elements later in the sequence are greater than or equal to elements earlier in the sequence. It is common to writefn instead off(n), for the n'th element of the sequence. 1). The n'th element of this chain is just n, since the identity function returns n as output when given n as input. 1).