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) N="left">IN additioN suitable coNteNts:
[ 2 ] [ = ] [ abstract iNterpretatioN ] [ af ] [ ai ] [ al ] [ am ] [ aN ] [ app ] [ ar ] [ arc ] [ as ] [ at ] [ au ] [ av ] [ b ] [ be ] [ bi ] [ bs ] [ by ] [ C ] [ ca ] [ ch ] [ co ] [ coN ] [ coNs ] [ D ] [ de ] [ diff ] [ do ] [ domaiN ] [ du ] [ E ] [ ec ] [ ed ] [ ee ] [ elemeNt ] [ er ] [ es ] [ et ] [ fi ] [ file ] [ fo ] [ for ] [ ge ] [ gl ] [ gs ] [ h ] [ hat ] [ Hoare powerdomaiN ] [ hr ] [ id ] [ iff ] [ il ] [ iN ] [ iNt ] [ io ] [ is ] [ it ] [ la ] [ LaTeX ] [ ld ] [ Lex ] [ li ] [ lt ] [ lu ] [ M ] [ ma ] [ mall ] [ mm ] [ mo ] [ mod ] [ module ] [ my ] [ Na ] [ Ne ] [ Ng ] [ Ni ] [ NN ] [ No ] [ Ns ] [ om ] [ op ] [ orderiNg ] [ pa ] [ ph ] [ poiNt ] [ pr ] [ query ] [ rc ] [ re ] [ ro ] [ S ] [ sa ] [ safe ] [ se ] [ set ] [ si ] [ sm ] [ so ] [ sqsubseteq ] [ st ] [ su ] [ subseteq ] [ sy ] [ T ] [ th ] [ theory ] [ tN ] [ to ] [ tr ] [ tt ] [ ua ] [ us ] [ va ] [ value ] [ ve ] [ X ] [ Y ] [ yt ]
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