powerdomain - определение. Что такое powerdomain
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Что (кто) такое powerdomain - определение

Powerdomain; Power domain; History of power domains

powerdomain         
<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. ("<=" is written in LaTeX as sqsubseteq). (1995-02-03)
Power domains         
In denotational semantics and domain theory, power domains are domains of nondeterministic and concurrent computations.
Hoare powerdomain      

Википедия

Power domains

In denotational semantics and domain theory, power domains are domains of nondeterministic and concurrent computations.

The idea of power domains for functions is that a nondeterministic function may be described as a deterministic set-valued function, where the set contains all values the nondeterministic function can take for a given argument. For concurrent systems, the idea is to express the set of all possible computations.

Roughly speaking, a power domain is a domain whose elements are certain subsets of a domain. Taking this approach naively, though, often gives rise to domains that don't quite have the desired properties, and so one is led to increasingly complicated notions of power domain. There are three common variants: the Plotkin, upper, and lower power domains. One way to understand these concepts is as free models of theories of nondeterminism.

For most of this article we use the terms "domain" and "continuous function" quite loosely, meaning respectively some kind of ordered structure and some kind of limit-preserving function. This flexibility is genuine; for example, in some concurrent systems it is natural to impose the condition that every message sent must eventually be delivered. However, the limit of a chain of approximations in which a message was not delivered, would be a completed computation in which the message was never delivered!

A modern reference to this subject is the chapter of Abramsky and Jung [1994]. Older references include those of Plotkin [1983, Chapter 8] and Smyth [1978].