Informational Representability: Abstract Models Versus Concrete Models

Abstract

Information logics are modal formalisms for representation of and reasoning about concepts derived from data that describe an application domain. Traditionally, concepts are determined by defining their extension or denotation and intension or connotation. The extension of a concept consists of the objects that are instances of this concept and the intension of a concept consists of the properties that are characteristic for the objects to which this concept applies. For example, to define the concept ‘organism’ we should list the earmarks of organism and the typical species of organisms [Bunge, 1967] . Let a set OB of objects be given, and suppose that properties of those objects are articulated in terms of attributes from a set AT and values of these attributes. For example, property of ‘being green’ is represented as a pair (colour, green), where ‘colour’ is an attribute, and ‘green’ is one of its values. Nondeterministic information about an object is of the form (attribute, a subset of values). For instance, if the age of a person is known approximately, say between 20 and 25, then this information is represented as a pair (age, {20,..., 25}). By an information system S we understand a pair (OB, AT) where OB is a non-empty set of objects and AT is a non-empty set of attributes. Each attribute a is a mapping a: OB → Ρ(Val _a )\{0}. For each a ∈ AT, the non-empty set Val _a is the set of values of the attribute a [Pawlak, 1983; Orlowska and Pawlak, 1984] . We write ΙS to denote the class of information systems. An information system S′ = (OB′, AT′) is said to be a subsystem of the information system S = (OB, AT) iff OB′ ⊆ OB and {a _OB′ : a ∈ AT} = AT′ where a _OB′ denotes the restriction of a to OB′.