Explicit Descriptions of Power Domains
Let D be a domain. The lower power domain can be defined by
- P = {closure | Ø∈A⊆D} where
-
-
-
- closure = {d∈D | ∃X⊆D, X directed, d = X, and ∀x∈X ∃a∈A x≤a}.
-
-
In other words, P is the collection of downward-closed subsets of D that are also closed under existing least upper bounds of directed sets in D. Note that while the ordering on P is given by the subset relation, least upper bounds do not in general coincide with unions.
It is important to check which properties of domains are preserved by the power domain constructions. For example, the Hoare powerdomain of an ω-complete domain is again ω-complete.
Read more about this topic: Power Domains
Famous quotes containing the words explicit, descriptions, power and/or domains:
“Like dreaming, reading performs the prodigious task of carrying us off to other worlds. But reading is not dreaming because books, unlike dreams, are subject to our will: they envelop us in alternative realities only because we give them explicit permission to do so. Books are the dreams we would most like to have, and, like dreams, they have the power to change consciousness, turning sadness to laughter and anxious introspection to the relaxed contemplation of some other time and place.”
—Victor Null, South African educator, psychologist. Lost in a Book: The Psychology of Reading for Pleasure, introduction, Yale University Press (1988)
“Our Lamaze instructor . . . assured our class . . . that our cervix muscles would become “naturally numb” as they swelled and stretched, and deep breathing would turn the final explosions of pain into “manageable discomfort.” This descriptions turned out to be as accurate as, say a steward advising passengers aboard the Titanic to prepare for a brisk but bracing swim.”
—Mary Kay Blakely (20th century)
“Of a truth, Knowledge is power, but it is a power reined by scruple, having a conscience of what must be and what may be; whereas Ignorance is a blind giant who, let him but wax unbound, would make it a sport to seize the pillars that hold up the long- wrought fabric of human good, and turn all the places of joy as dark as a buried Babylon.”
—George Eliot [Mary Ann (or Marian)
“I shall be a benefactor if I conquer some realms from the night, if I report to the gazettes anything transpiring about us at that season worthy of their attention,—if I can show men that there is some beauty awake while they are asleep,—if I add to the domains of poetry.”
—Henry David Thoreau (1817–1862)