# domain theory

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## domain theory

[dō′mān ‚thē·ə·rē]
(solid-state physics)
A theory of the behavior of ferromagnetic and ferroelectric crystals according to which changes in the bulk magnetization and polarization arise from changes in size and orientation of domains that are each polarized to saturation but which point in different directions.

## domain theory

(theory)
A branch of mathematics introduced by Dana Scott in 1970 as a mathematical theory of programming languages, and for nearly a quarter of a century developed almost exclusively in connection with denotational semantics in computer science.

In denotational semantics of programming languages, the meaning of a program is taken to be an element of a domain. A domain is a mathematical structure consisting of a set of values (or "points") and an ordering relation, <= on those values. Domain theory is the study of such structures.

("<=" is written in LaTeX as \subseteq)

Different domains correspond to the different types of object with which a program deals. In a language containing functions, we might have a domain X -> Y which is the set of functions from domain X to domain Y with the ordering f <= g iff for all x in X, f x <= g x. In the pure lambda-calculus all objects are functions or applications of functions to other functions. To represent the meaning of such programs, we must solve the recursive equation over domains,

D = D -> D

which states that domain D is (isomorphic to) some function space from D to itself. I.e. it is a fixed point D = F(D) for some operator F that takes a domain D to D -> D. The equivalent equation has no non-trivial solution in set theory.

There are many definitions of domains, with different properties and suitable for different purposes. One commonly used definition is that of Scott domains, often simply called domains, which are omega-algebraic, consistently complete CPOs.

There are domain-theoretic computational models in other branches of mathematics including dynamical systems, fractals, measure theory, integration theory, probability theory, and stochastic processes.

See also abstract interpretation, bottom, pointed domain.
References in periodicals archive ?
23) In other words, domain theory has a powerful focus: social relationships--"the very ability of people to get along with one another.
The third and final domain in domain theory is the conventional domain.
A truth maintenance system (TMS)  may be used to maintain the network of concepts that form a domain theory.
In the nontransfer condition, after failing to solve a target problem, the equation schemas necessary to solve that problem were added directly to the system's target domain theory.
Icarus explains a provided trace from a source problem using concepts from its source domain theory.
Gunter and Scott  is a recent exposition of Scott's domain theory.
Heuristics reducing the number of uninteresting discoveries: The domain theory can contain knowledge about attributes and values that are uninteresting or meaningless to the user.
In this respect, the domain theory served as a paradigm (ref.
There is a wide variety of targets that the learning component of a planning system might aim toward, such as learning search control rules, learning to avoid dead-end or unpromising states, or improving an incomplete domain theory.
CBR techniques have been shown to be effective for two main reasons: First, frequently, there is no complete domain theory that could be used to generate plans from scratch.
The second characteristic is that CBR permits problem solving even when the underlying domain theory is incomplete.

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