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| Domain | Definition |
Computing | Domain theory |
Source: compiled by the editor from various references; see credits. | |
(From Wikipedia, the free Encyclopedia)
The intuitive idea underlying the theory is that the ordering of a domain represents a hierarchy of information or knowledge. The higher an element is within the order, the more specific it is and the more information it contains. Lower elements represent incomplete knowledge or intermediate results. Computation then may consist of applying monotone functions repeatedly on elements of the domain in order to refine a result. Reaching a fixed point is equivalent to finishing a calculation. Domains provide a superior setting for these ideas since (among other advantages) fixed points of monotone functions can be guaranteed to exist.
Domain theory has close relations to topology and the theory of computation, because it formalizes the ideas of approximation and convergence in a very general way. An alternative approach to semantics in the spirit of the above intuition are metric spaces, where contracting functions may be applied to model computation and approximation.
The basic mathematical structures of domain theory are directed complete partial orders or dcpos. A dcpo is a partially ordered set where every directed subset has a least upper bound (not necessarily within the subset itself). Viewing directed subsets as a generalized concept of convergence, this guarantees that all limits of such sets exist. When a dcpo has a least element, it is sometimes called a complete partial order (cpo).
The appropriate morphisms for dcpos are continuous functions. A function f between two dcpos is said to be (Scott-) continuous iff it is monotone and preserves directed suprema.
However, one usually additionally needs a concept of approximation. This is obtained by introducing the '\approximation order' <<. For a dcpo (D, ≤) it is defined by:
We now say that some subset B of D is a base for D if for every x in D, the set {y in D| y << x} ∩ B contains a directed set with supremum x.
A continuous dcpo or continuous domain is a dcpo that has a base. This notion is central to domain theory. It provides the basic requirements for modelling computation, namely that every element can be approximated by a converging directed subset from a given base. A special case is given if the base is countable. In this case we talk about ω-continuous dcpos.
An element of a dcpo is said to be compact or finite iff it approximates itself. These elements cannot be obtained as suprema of directed sets that do not already contain them. If a dcpo has a base of compact elements, it is called an algebraic dcpo or an algebraic domain. For countable bases ω-algebraicity is defined as above.
A number of other important properties have been defined for domains, giving rise to additional classes (and categories) of dcpos. Note that the term domain itself is not exact and thus is only used as an abbreviation when a formal definition has been given before.
A poset D is a dcpo iff each chain in D has a supremum. However, directed sets are strictly more powerful than chains.
A poset D with a least element is a dcpo iff every monotone function f on D has a fixed point. If f is continuous then it has even a least fixed point, given as the least upper bound of all finite iterations of f on the least element 0: Vn in N f n(0).
Of course, there are many other important results, depending on the application area where domain theory is to be applied. Please see the literature (and contribute).
Formal definitions
One says that x approximates y or that x is way below y.Important results
Literature
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Domain theory."
Crosswords: DOMAIN THEORY |
| Specialty definitions using "DOMAIN THEORY": bottom-unique, boundedly complete ♦ coalesced sum, compactness preserving ♦ denotational semantics, disjoint union ♦ lifted domain ♦ Omega-algebraic ♦ partial ordering, pointed domain. (references) |
| Domain | Title |
Books | |
Source: compiled by the editor from various references; see credits. | |
| The following statistics estimate the number of searches per day across the major English-language search engines as identified by various trade publications. Hyperlinks lead to commercial use of the expression at Amazon.com. |
| Expression | Frequency per Day |
the domain theory | 3 |
| Source: compiled by the editor from various references; see credits. | |
Scrabble® Enable2K-Verified Anagrams | |
| Words within the letters "a-d-e-h-i-m-n-o-o-r-t-y" | |
-2 letters: admonitory, moderation. | |
-3 letters: anhydrite, arytenoid, diathermy, dominator, dynamiter, dynamotor, hardiment, hydration, mediatory, modernity, monitored, rhodamine, rhodonite, rhytidome. | |
-4 letters: aeronomy, anteroom, antherid, antihero, arointed, aroynted, dairymen, demotion, dominate, dormient, dynamite, enormity, headroom, hematoid, hoorayed, hyoidean, marooned, mediator, methadon, minatory, moderato, monetary, monitory, moronity, motioned, motioner, myotonia, odometry, ordinate, radiomen, rationed, ratooned, remotion, rhodamin, tandoori. | |
| Words containing the letters "a-d-e-h-i-m-n-o-o-r-t-y" | |
+5 letters: aerothermodynamic, dihydroergotamine. | |
| Source: compiled by the editor from various references; see credits. SCRABBLE® is a registered trademark. All intellectual property rights in and to the game are owned in the U.S.A and Canada by Hasbro Inc., and throughout the rest of the world by J.W. Spear & Sons Limited of Maidenhead, Berkshire, England, a subsidiary of Mattel Inc. Mattel and Spear are not affiliated with Hasbro. | |
Hexadecimal (or equivalents, 770AD-1900s) (references)44 4F 4D 41 49 4E 54 48 45 4F 52 59 |
| Leonardo da Vinci (1452-1519; backwards) (references)
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Binary Code (1918-1938, probably earlier) (references)01000100 01001111 01001101 01000001 01001001 01001110 00100000 01010100 01001000 01000101 01001111 01010010 01011001 |
HTML Code (1990) (references)D O M A I N   T H E O R Y |
ISO 10646 (1991-1993) (references)0044 004F 004D 0041 0049 004E 0054 0048 0045 004F 0052 0059 |
Encryption (beginner's substitution cypher): (references)3849473543482544239495259 |
| 1. Crosswords 2. Usage: Commercial 3. Expressions: Internet 4. Anagrams | 5. Orthography 6. Bibliography |
Copyright © Philip M. Parker, INSEAD. Terms of Use.
