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powerdomain    
The powerdomain of a {domain} D is a domain
containing some of the {subsets} of D. Due to the asymmetry
condition in the definition of a {partial order} (and
therefore of a domain) the powerdomain cannot contain all the
subsets of D. This is because there may be different sets X
and Y such that X <= Y and Y <= X which, by the asymmetry
condition would have to be considered equal.

There are at least three possible orderings of the subsets of
a powerdomain:

Egli-Milner:

X <= Y iff for all x in X, exists y in Y: x <= y
and for all y in Y, exists x in X: x <= y

("The other domain always contains a related element").

Hoare or Partial Correctness or Safety:

X <= Y iff for all x in X, exists y in Y: x <= y

("The bigger domain always contains a bigger element").

Smyth or Total Correctness or Liveness:

X <= Y iff for all y in Y, exists x in X: x <= y

("The smaller domain always contains a smaller element").

If a powerdomain represents the result of an {abstract
interpretation} in which a bigger value is a safe
approximation to a smaller value then the Hoare powerdomain is
appropriate because the safe approximation Y to the
powerdomain X contains a safe approximation to each point in
X.

("<=" is written in {LaTeX} as {\sqsubseteq}).

(1995-02-03)


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