# 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.

References in periodicals archive ?
"Judgments of moral issues are justified in terms of harm or fairness that actions would cause, while judgments of conventions are justified in terms of norms and the expectations of authority." (22) Nucci concludes that "the core of human morality is a concern for fairness and human welfare." (23) In other words, domain theory has a powerful focus: social relationships--"the very ability of people to get along with one another." (24) Said differently, where the choice continuum centers on vertical respect (or respect for the institution), domain theory centers on horizontal respect (that is, respect for one another).
A truth maintenance system (TMS) [28] may be used to maintain the network of concepts that form a domain theory. The TMS stores dependencies which, when errors are found will indicate where other potential weaknesses in the theory lie.
Well, one can quite naturally give the relational meaning a finer granularity, so that it records every step which a program makes from one memory access to the next--and this can be done without leaving domain theory. But the phenomenon taught me a more radical lesson: Once the memory is no longer at the behest of a single master, then the master-to-slave (or: function-to-value) view of the program-to-memory relationship becomes a bit of a fiction.
Domain theory of cognitive and moral development states that "cognitive structure or organization is a feature of the individual's knowledge systems rather than "the thought" of the individual" (Turiel & Davidson, p.
Unlike the AP physics work, the system uses a domain theory of kinematics equations defined using equation schemas, which specify the equation's types, quantities, and applicability conditions.
They begin with the theory and phenomenology of glasses, including the Adam-Gibbs entropic theory, then thoroughly explain two-temperature thermodynamics and solvable models for the glassy state, aging urn models, such as the backgammon model, glassiness in a directed polymer model, including cooling and heating, the potential energy landscape approach, including the thermodynamics of supercooled liquids and many-body glass models, and theories of the glassy state, including the frustration limited domain theory and the random first order transition theory.
In this respect, the domain theory served as a paradigm (ref.
The PSL ontology makes a distinction between the axioms of the ontology and the axioms of a domain theory that uses the ontology, which are characterized as syntactic classes of sentences that are satisfied by elements of the models.
The unmodified version of HAMB with the complete domain theory was also run on this set of cases, as was a version of HAMB that used no domain knowledge.
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. As indicated in figure 1, they can be categorized broadly into one of three groups: (1) learning to speed up planning, (2) learning to elicit or improve the planning domain theory, or (3) learning to improve the quality of the plans produced (where quality can have a wide range of definitions).
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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