The semantics presented in Chapter 3 describe the effect of an update on the models of a theory; the semantics give no hints whatsoever on how to translate that effect into changes in the relational theory. An algorithm for performing updates cannot proceed by generating models from the theory and updating them directly; this is because the number of standard models may be exponential in the size of the theory, and it may be very difficult to find even one model, as that is equivalent to testing the satisfiability of the theory.

In the sections of this chapter, we consider algorithmic means of accomplishing successively more complicated updates, beginning with updates under the standard semantics, without nulls and selection clauses, operating under an open-world assumption, in Section 4.1. Section 4.2 extends this approach to updates with nulls, and Section 4.3 shows how to process selection clauses correctly. Section 4.4 shows how to enforce the closed-world assumption.

Section 4.5 shows that these algorithms are correct in the sense that the alternative worlds produced under the algorithms are the same as those produced by updating each alternative world individually.

Section 4.6 discusses the computational complexity of the algorithms. For relational theories and updates without nulls, the algorithms have the same asymptotic cost as for an ordinary complete-information database update, but may increase the size of the relational theory. For updates involving nulls, the increase in size will be severe if many data atoms in the theory unify with those in the update.