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 There are at least three possible orderings of the subsets of a powerdomain:
X (“The other domain always contains a related element”).
Hoare or Partial Correctness or Safety:
X (“The bigger domain always contains a bigger element”).
Smyth or Total Correctness or Liveness:
X (“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.
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