powerdomain
Y> 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) Yle="border-width:thin; border-color:#333333; border-stYle:dashed; padding:5px;" align="left">In addition suitable contents:
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