On the structure of knowledge base systems.

Presented is a framework for mathematical representation of knowledge, using set theoretical and algebraic structures. The basic concepts are generalized relations with valuated elements, decomposed into input-output parts, defining binary "knowledge modules", the output parts depending on the input parts, the dependences given either explicity or constructively as a result of "rule" applications. For given hierarchically structured "queries" or "premises" in the form of families of valuated objects together with logic functions to be applied to the valuations, a knowledge module deduces answers" or "conclusions" by matching the input patterns with corresponding output patterns and then performing algebraic computations with the selected objects according to the algebraic structure of the domains of the valuations. Introducing in addition uniform topologies on the domains of the valuations, the "perfect' matching can be generalized to an approximate matching. Further, the knowledge modules as input-output systems can be composed to complex systems.