Geometrical aspect of databases and knowledge bases.
The paper is devoted to a universal-algebraic approach to formal foundations of databases and knowledge bases. The main goal is to represent non-elementary knowledge about elementary knowledge (first-order knowledge). To this end, the authors invoke methods of algebraic logic and universal-algebraic geometry. The basic assumptions are that knowledge is represented by three components: 1) description of knowledge (syntactical part of knowledge, formulas); 2) subject of knowledge (an object in the field of application, model); 3) content of knowledge (semantics). The content of knowledge is determined by the description of knowledge (a set \(T\) of formulas) and a subject of knowledge (a triple \((G,O,f)\), where \(G\) is a multisorted algebra, \(O\) is a set of relation symbols and \(f\) the interpretation of \(O\) in \(G\)). The paper shows how to equip the content of knowledge, \(Tf\), with a structure to study algebraic and geometric aspects of this structure. Emphasized are three aspects of representation of knowledge: logical (knowledge, description), algebraic (the subject of knowledge), and geometric (the content of knowledge). Preliminary notions are surveyed in Section 1. Section 2 presents basics of logic, Section 3 is about categories of algebraic sets, Section 4 is about Galois theory and so-called Halmos algebras. Sections 5--7 contain the main result, Section 5 deals with categories of elementary knowledge, Section 6 presents a formal model of databases and knowledge bases, Section 7 presents results on equivalences of databases and knowledge bases.
scientific article; zbMATH DE number 2158737
scientific article