Differential Relational Calculus for Integrity Maintenance

A differential calculus for first-order logic is developed to enforce database integrity. Formal differentiation of first-order sentences is useful in maintaining database integrity, since once a database constraint is expressed as a first-order sentence, its derivative with respect to a transaction provides the necessary and sufficient condition for maintaining integrity. The derivative is often much simpler to test than the original constraint since it maintains integrity differentially by assuming integrity before the transaction, and testing only for new violations. The formal differentiation requires no resolution search, but only substitution. It is more efficient than resolution-based approaches; and it provides a considerably more general solution than previous substitution-based methods since it is valid for all first-order sentences and with all transactions involving arbitrary collections of atomic changes to the database. It also produces a large number of sufficient conditions that are often less strict than those of the previous approaches; and it can be extended to accommodate many dynamic constraints.