Modal Logics in the Theory of Relational Databases

Logical formalisms for representing and reasoning about relational databases have been very popular. However, fragments of the standard rst-order logic used in most of these formalisms are quite inadequate for reasoning about properties represented by a set of tuples from a relation. In this short note, we propose a formalism for reasoning about these properties by means of two operators with respect to a set of attributes. The rst operator, when applied on a given set of tuples, extracts those tuples that deenitely satisfy a property represented by the given set, and the second operator extracts those tuples that possibly satisfy this property. We associate these kinds of property with the language of some well-known systems of modal logics so that reasoning about these properties can be carried out in the domain of modal logics. 1 Motivation It is sometimes desirable to derive properties of a given set of tuples from a relation. As an example, consider a scenario within a company where a set of employees have been found to achieve the performance target set by their supervisors. There is a set of tuples which corresponds to these employees in the employee relation of the company's personnel database. The management of the company may like to know what other properties can be derived about these employees; for example, whether they possess special skills or whether they are from a particular site of the company. In another situation, consider those items in a supermarket which have exceeded their sale targets. There is a set of tuples in the item relation of the supermarket database corresponding to these items. Examples of further new properties one may like to derive are whether the items come from a speciic zone in the store or whether their prices were reduced. In this document, we propose a formalism for reasoning about these properties of a set of tuples from a relation. A property is typically associated with a combination of attributes of the relation. A set of tuples satisses a property if every tuple has the same value on each attribute representing the property. From a given set of tuples, we rst characterize how many of the set deenitely satisfy the property represented by the set and how many of the set possibly satisfy the property. This is achieved by two operators proposed in the following section. An alternative characterization of these two operators in …