Towards a Logical Reconstruction of Relational Database Theory

Abstract

Insofar as database theory can be said to owe a debt to logic, the currency on loan is model theoretic in the sense that a database can be viewed as a particular kind of first order interpretation, and query evaluation is a process of truth functional evaluation of first order formulae with respect, to this interpretation. It is this model theoretic paradigm which leads, for example, to many valued propositional logics for databases with null values.

In this chapter I argue that a proof theoretic view of databases is possible, and indeed much morefiruitful. Specifically, I show how relational databases can be seen as special theories of gist order logic, namely theories incorporating the fiollowing assumptions:

• 1. The domain closure assumption. The individuals occurring in the database are all and only the existing individuals.

• 2. The unique name assumption. Individuals with distinct names are distinct.

• 3. The closed world assumption. The only possible instances of a rela¬tion are those implied by the database.

It will follow that a proof theoretic paradigm for relational databases provides a correct treatment of:

• 1. Query evaluation,for databases that have incomplete information, including null values.

• 2. Integrity constraints and their enforcement.

• 3. Conceptual modelling and the extension of the relational model to incorporate more real world semantics.