Mathematical analysis of documentation systems. An attempt to a theory of classification and search request formulation

As an attempt to make a general structural theory of information retrieval, a documentation system (DS) is defined as a formal system consisting of (a) a set of objects (documents); (b) a set A^{+ +} of elementary attributes (key-words), from which further attributes may be constructed: A^{+ +} generates A; (c) a set of axioms of the form X^{+ +}(x) = M (M ε{lunate} M, M a set of constants) connecting attributes with objects: from the axioms further theorems (= true statements) may be constructed. By use of the theorems, different mappings A → B(a) (B(a) set of all subsets of a) (search question → set of documents retrieved) are defined. The type of a DS depends on two basic decisions: (1) choice of the rules for the construction of attributes and theorems, e.g., logical product in coordinate indexing; links. (2) choice of M: M may consist of the two constants "applicable" and "not applicable", or some positive integers, ... ; Further practical decisions: A^{+ +} hierarchical or not; kind of mapping; introduction of roles (= further attributes). The most simple case-ordinary two-valued Coordinate Indexing-is discussed in detail: A is a free distributive (but not Boolean) lattice, the homomorphic image a ring of subsets of a ; instead of negation which is not useful, a useful retrieval operation "praeternegation" is introduced. Furthermore these are discussed: a generalized definition of superimposed coding; some functions for the distance of objects or attributes; optimization and automatic derivation of classifications. The model takes into account term-term relations and document-document relations. It may serve as a structural framework in terms of which the functional problems of retrieval theory may be expressed more clearly.