<theory> 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.