Orlowska RELATIONAL LOGICS FOR FORMALIZATION OF DATABASE

Dependencies between information items play an important role in knowledge representation. In this paper we present relational logical systems for representation of and reasoning about dependencies of attributes in information systems and we discuss relationships between these formalisms. Each logical system L discussed in this paper consists of a set FORL of well formed formulas of a formalised language and of at least one of the relations: `L, |=L⊆ P (FORL)×FORL. Derivability relation `L is determined by a deduction system for L and the concept of proof in this system. Deduction systems for our logics are of two kinds: Hilbert-style systems or relational proof systems. Accordingly, proofs are either sequences of formulas satisfying the well known conditions, or trees whose nodes consist of sequences of relational formulas, obtained in a precisely described way in a given system. Semantic consequence relation |=L is determined by semantics of the language of formulas of L. Semantics is defined in terms of a class MODL of models for L and the concept of truth of a formula in a model. For a subset X of formulas and for a formula A, X|=LA iff for every model M∈MODL if all the formulas from X are true in M, then A is true in M. Usually, models consist of either a relational system or an algebraic system and a meaning function that provides interpretation of formulas in this system. If both `L and |=L are defined for L, then L is sound whenever for all X⊆FORL and for all A∈FORL it holds X`LA implies X|=LA, and L is complete whenever X|=LA implies X`LA. Clearly, for a given set of formulas we can define several derivability relations and/or semantic consequence relations. Given logical systems L1 and L2, we say that L1 is |=-interpretable in L2 whenever |=L1 and |=L2 are defined and there is a translation mapping t: FORL1 → FORL2 such that for every set X⊆FORL1 and for every formula