Whither the fixed point?
More than one of you has asked about the fixed point of phi and why we found it. We are not using it at this point. In fact, we generally use the form of C[[while b do c]] that we found the fixed point from.
The answer is that the equation:
C[[w]] = {(sigma,sigma’) | B[[b]]sigma=true and (sigma,sigma’) in C[[w]] . C[[c]]} U {...}
does not give us a definition of the denotation. Specifically, C[[w]] appears in its own definition. This is not a well-founded function definition because, in affect, C[[w]] contains itself. So, we needed a value for C[[w]] that does not have this property.
If you look at the definition of R on page 59 of the text, notice that phi does not appear on the right-hand side of the equation. Thus, when we find the fixed point of R to give us R-hat, the definition is well founded. Once we have the value, we can in effect file it away for further use.
The answer is that the equation:
C[[w]] = {(sigma,sigma’) | B[[b]]sigma=true and (sigma,sigma’) in C[[w]] . C[[c]]} U {...}
does not give us a definition of the denotation. Specifically, C[[w]] appears in its own definition. This is not a well-founded function definition because, in affect, C[[w]] contains itself. So, we needed a value for C[[w]] that does not have this property.
If you look at the definition of R on page 59 of the text, notice that phi does not appear on the right-hand side of the equation. Thus, when we find the fixed point of R to give us R-hat, the definition is well founded. Once we have the value, we can in effect file it away for further use.
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