Continuous informational systems

Abstract

A continuous information (c.i.) system is an informational system in which every object is described by a vector of functions of real variable. In the special case where all these functions are constant the c.i. system is equivalent to the information storage and retrieval (i.s.r.) system defined by Marek and Pawlak in [3]. The paper deals with a description language for a c.i. system. The structure of this language is analogous to the structure of the description language for i.s.r. systems introduced in [3]. Terms represent properties of objects, and formulas are statements about the system. All terms are built by means of Boolean operations from terms representing properties of the type (ajx)(I) ⊂ A, where ajx is the jth function describing the object x, I is an open interval or a point of the real line, and A is an arbitrary subset of the codomain of ajx. The axioms for the description language together with an inference rule are given. The valuation of terms and formulas is defined and the adequacy theorem is proved. Basing on a certain normal form theorem and the properties of a special system Smax, the completeness theorem is finally proved.