LOGIC OF KNOWLEDGE REPRESENTATION SYSTEM: AN ALGEBRAIC VIEW

Janis Cirulis, University of Latvia

A Pawlak style information, or knowledge representation, system may be thought of as consisting of two components: its formal part, the 
``frame'' (X,V_x)_{x \in X}, comprised of the attribute set and respective value sets, and its ``content'' (O,Delta), presented by the object set and information function. The atributes in X are thought of as formally independent, none of them is complex, and the only dependencies usually taken into account in Pawlak's model are those determined by the content. We shall deal there with (non-deterministic) systems with built-in inclusion dependencies between attributes, and pay attention mainly to some consequences concerning the frame. 

The atribute set X is endowed with an inclusion relation that makes it a lattice. To every pair (x,y) of attributes with x \le y there correspods a mapping of V_y into V_x; the information function Delta has to obey these "dependencies". A pair (x,a) with x \in A and a \subset V_a is treated as a proposition "x has a value in a". The set of all such propositions, called the inner logic of the system, is organised into a lattice with involution, a unary operation of period 2. (By the way, a derived quantifier-like operation can afterwards be associated with every attribute.) Every object from O gives rise to a filter in the logic.

 This way we come to a finetely based subvariety of lattices with involution, and a knowledge representation system can be presented as an algebra from this subvariety equipped with+ a family of filters.