Induction of Relational Algebra Expressions

In the theory of database systems [1], a database query is defined as a function that maps relational databases to relations. This definition models the situation in practice where one applies an SQL query to a database instance and receives a set of output tuples as the answer to the query on that database. The problem of relational database query induction is then naturally stated as follows: we are presented with a finite number of examples, where each example consists of a relational database and an output relation, and we are asked to come up with an expression for a query that agrees with the given examples (and that satisfies the usual requirements placed on induction tasks, most notably generalization). Of course, we should also specify here which language we are using to express database queries. Whereas SQL is the universal query language used in practice, it is also very complex. A more tractable language to work with is the relational algebra, which can be found, together with SQL, in every database textbook, as the relational algebra is the basic underlying query language used for database query processing. Another common language in which relational database queries can be expressed is Datalog (function-free Prolog). Datalog is also used in Inductive Logic Programming (ILP), and when we restrict attention to nonrecursive programs, Datalog can be translated into the relational algebra [2, 3]. As a consequence, relational database query induction could be (and has been [4, 5]) considered to be a mere special case of ILP, where the background knowledge consists only of facts (the example databases). That approach, however, does not cover relational database query induction in its entirety, because not every relational algebra query can be expressed in Datalog. Nonrecursive Datalog without negation can only express the positive fragment of the relational algebra, i.e., the fragment without the set-theoretic difference operator. Consider, for example, the following rules over a binary relation r(A,B):