On Safety of Unary and Nonunary IFP Operators

Resumo

The article investigates the safety of unary inflationary fixed point operators (IFP-operators), i.e., their computability in finitely many steps. Such operators correspond exactly to recursive SQL queries. Therefore, the problem under investigation is directly related to database theory. The problem arises from the fact that if recursion and universe relations, e.g., addition, are simultaneously used in a SQL query, then a query evaluation can fall into an infinite loop. Moreover, such a combination makes it possible to model a universal computing device, e.g., a Turing machine. Hence, the problem of finite computability for such SQL queries is undecidable. In our previous works, we established some properties of a universe those guarantee the finite computability of any IFP-operator in the universe. In this article, we investigate a connection between an arity of IFP-operators and their safety. The main result of this article is that some results for general IFP-operators do not hold for unary ones. An instance of a universe is constructed in which all unnested unary IFP-operators are safe. However, there are unsafe binary IFP-operators in this universe. Therefore, IFP-operators can become unsafe if the arity changes. In addition, there are unsafe nested unary operators. This contrasts with the general case in which this is impossible. There are also elementary equivalent universes where the same unary IFP-operators are unsafe. This behavior is also different from the behavior of general IFP-operators.