Types of Conceptual Scales

Overview

A concept scale is a lattice-theoretic interpretation of ontological information.It is a controlled vocabulary in the intentional mode,which defines a new conceptual data type called a facet.At the base level,conceptual scales semantically partition the (single-valued) attributes of Conceptual Knowledge Processing. In conceptual knowledge processing there are three types of conceptual scales,which have correspondences in CKML listed in the table on the right. In CKP, abstract conceptual scales are represented by attribute names in a lattice frame, concrete conceptual scales are represented by queries bound to attribute names, and realized conceptual scales are represented by object-attribute incidence constructed by attribute query evaluation.

Abstract conceptual scales

Abstract conceptual scales introduce terms (attribute names), and abstractly specify attribute definitions via term-to-term relationships, as structured by the conceptualization of a concept lattice. This lattice was previously specified by a set of term (propositional) clauses.In CKML Version 0.2, abstract conceptual scales are represented as theories; terms or attributes are called types, and the clauses are now called sequents.

Concrete conceptual scales

Concrete conceptual scales attach meaning to the term definitions in abstract scales by attaching a (single-variabled) query to each term. These queries are required to respect the abstract term relationships (implications). In CKML Version 0.2, concrete conceptual scales are represented as theory interpretations; queries are logical queries, and represented by (single-variable) lambda expressions.

Realized conceptual scales

When applied to the collection of ontology objects, concrete conceptual scales produce facets, which are known in CKP as realized conceptual scales. Facets are the components which underly a faceted conceptual space. In CKML Version 0.2, realized conceptual scales are modeled as classification infomorphisms. Each facet is specified by a theory interpretation (FCA concrete conceptual scale).

Mathematical Development

Splitting Classification

Recall that an infomorphism f : A ! B between two classifications A = htok(A),typ(A),|=_Ai and B = htok(B),typ(B),|=_Bi satisfies the fundamental property

f (a) |=_A a iff a |=_B f (a).

The splitting classification of the infomorphism f is the classification Spl(f) = htok(A),typ(B),|=_fi whose types are the types of A, whose tokens are the tokens of B, and whose classification relation |=_f is defined in terms of the fundamental property: a |=_f a  when  f (a) |=_A a iff a |=_B f(a). Clearly, the infomorphism splits into a type-identical infomorphism f ' : A ! Spl(f) and a token-identical infomorphism f '' : Spl(f) ! B where f  =  f ' ^. f ''.

Powerset

Recall that for any set A the powerset classification

PA = hA,PA,2i

associated with A is the following classification: the types are subsets of A, the tokens are elements of A, and the classification relation is the membership relation. The powerset classification is extensional and separated.

When the theory functor

Th : Classification ! Regular

is applied to the powerset classification, the result is the powerset theory Th(PA) = hPA,`i over A, where G ` D entailment is expressed by \G v [D.

State Space

A state space S = htok(S),typ(S),s_Si consists of

1. a set, tok(S), of objects called the tokens of S,
2. a set, typ(S), of states of S,
3. a total function, s_S : tok(S) ! typ(S), called the state description of S.

The event classification Evt(S) associated with a state space S has the following components:

1. tokens are the tokens of S,
2. types (events) are subsets of types of S,
3. classification relation is defined by a |=_Evt(S) a iff s_S(a) 2 a.

There is a type-identity infomorphism called the state description infomorphism

s_S : P(typ(S)) ! Evt(S)

which is the state description on tokens, and whose fundamental property is expressed by the definition of the classification relation |=_Evt(S).

Conceptual Scale

Abstract Conceptual Scale

An abstract conceptual scale is (the same as) a theory T = hS,`i. Constraints of the theory are used to relate the types in the scale to have the desired lattice structure.

Concrete Conceptual Scale

A concrete conceptual scale is an interpretation of an abstract conceptual scale into a set (domain). More precisely, given (regular) theory T and set A, a concrete conceptual scale from T to A is a theory morphism

f : T ! Th(PA)

• f maps types a 2 S to subsets f(a) 2 PA, where subset f(a) is the interpretation of type a
• the image of any constraint G ` D satisfies \ f(G) v [ f(D).

Cla a Th

By the adjunction between the classification functor and the theory functor, there is a unique infomorphism equivalent to the theory interpretation (we ambiguously use the same symbol)

f : Cla(T) ! PA.

The splitting classification

Spl(f) = hA,S,|=_fi

for this infomorphism has classification relation defined by a |=_f a  when  a 2 f(a) for each domain value (token) a and each theory type a. So a concrete conceptual scale is specified as a theory interpretation from an abstract scale (as theory) to the powerset of a set of values; but this is equivalent to an infomorphism from the ideal classification of the theory (abstract scale) to the powerset classification over the set, and also equivalent to its splitting classification.

Realized Conceptual Scale

A realized conceptual scale is a concrete conceptual scale over a state space. More precisely, given a state space S, a realized conceptual scale is a concrete conceptual scale

f : T ! Th(P(typ(S))

that interprets an abstract conceptual scale T into the states of S. By the adjunction between the classification functor and the theory functor, there is a unique infomorphism equivalent to the theory interpretation (we ambiguously use the same symbol)

f : Cla(T) ! P(typ(S)).

Composing with the state description infomorphism s_S : P(typ(S)) ! Evt(S) gives us an infomorphism

g : Cla(T) ! Evt(S),

which is the same as f on types g(a) = f(a) but the composition g(a) = f(s_S(a)) on tokens. Its splitting classification

Spl(g) = htok(S),S,|=_gi

has classification relation defined by a |=_g a  when  s_S(a) 2 f(a) for each state space token a and each theory type a. So a realized conceptual scale "is" an infomorphism from the classification of a theory to the event classification of a state space, or "is" its splitting classification.

Please send questions, comments and suggestions about this page to: Robert E. Kent rekent@ontologos.org Last modification date: September 1998