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rendering A physical model or drawing color, or reaction rate. In general expressive rep-
of an object intended for direct perception by resentations will mirror the object closely and
humans. in ways that reflect how humans conventionally
Comment: The key notion is that the model represent that object in their heads, often via
is directly perceived by humans, usually visu- language. However, it must be remembered that
ally. For example, renderings can be images strictly speaking, there is no requirement for any
drawn on a raster screen, a physical molecular
relationship between a term and a representa-
model, or a projection in a virtual reality environ-
tion for an object, apart from the properties of
ment. A rendering is distinct from the informa-
the transformations between them. Note that
tion it contains or its abstract specification, and
a database can contain multiple representations
is strongly determined by the device used for its
which upon (internal) transformation are infor-
achievement and the intended method of percep-
mationally equivalent, provided that each pre-
tion. Thus renderings of the same landscape in
serves a different property. Thus a molecule can
oils and water colors can have distinctly different
be represented by a configuration rule, a key-
characters, even if the landscapes are completely
pair list, or the terminal form. Each preserves a
isomorphic. Synonymous with drawing.
different property of the molecule (configuration
of substituent groupings, edges in the structural
representation (1) A representation of an
graph, atoms, and bonds); all are interconvert-
object a ∈ A is an image ψ (a) ∈ Z which
q
preserves a property q such that ible by the grammar; and each is optimized for a
specific class of computation.
(i.) uniqueness: the mapping, ψ , between The second is that if the representation is to
q
A and Z for q is one-to-one and onto (notated be useful for computations which do something
ψ : a −→ ψ (a), a ∈ A, ψ (a) ∈ Z, and more sophisticated than simply looking up data,
q
q
q
(ii.) equivalent transformations: for some it must permit one to define a computational
transformation ζ over the set of objects A, transformation which mimics a “real-world oper-
ation” sufficiently so that for the preserved prop-
ζ : a −→ a ,

erty of interest, the result of the computational

a, a ∈ A, there exists a transformation ζ over transformation is the image of the result of the

the set of representations, real-world operation. The word is used as both a
singular and a collective noun. See also seman-

ζ : ψ (a) −→ ψ (a ), tics and semiote.
q
q
(2) (of a group G) A linear left action of a
ψ (a), ψ (a ) ∈ Z, such that the result of ψ q

q
q
applied to ζ(a) is identical to the result of ζ group on a vector space V . To any group ele-
ment g ∈ G an automorphism of V is associ-
applied to ψ (a), or
q
ated. Example, the group of rotations SO(3) is
3

ψ ◦ ζ(a) = ζ ◦ ψ (a) = ψ (a ) represented on R by matrix multiplication.
q
q
q
where◦isthecompositionoperator. ψ (ζ) = ζ ,

q

so we say that ζ and ζ are equivalent transfor- representative sample A sample resulting
mations. from a sampling plan that can be expected to
reflect adequately the properties of interest of the
Equally, a set of such representations Z.
parent population.
Comment: The intent here is to capture two
of the most salient features of a representation A representative sample may be a random
as the term is commonly used in artificial intel- sample or, for example, a stratified sample,
ligence and databases. The first is that the rep- depending upon the objective of sampling and
resentation is simply an image of something in the characteristics of the population. The degree
the “real world” which preserves some property of representativeness of the sample may be lim-
important to the user, geometric isomerism, hair ited by cost or convenience.
© 2003 by CRC Press LLC
© 2003 by CRC Press LLC