Computing queries with higher-order logics

Abstract

In the present article, we study the expressive power of higher-order logics on finite relational structures or databases. First, we give a characterization of the expressive power of the fragments Σ j i and Π j i , for each i ⩾ 1 and each number of alternations of quantifier blocks j. Then, we get as a corollary the expressive power of HO i for each order i ⩾ 2 . From our results, as well as from the results of R. Hull and J. Su, it turns out that no higher-order logic can be complete. Even if we consider the union of higher-order logics of all natural orders, i.e., ⋃ i ⩾ 2 HO i , we still do not get a complete logic. So, we define a logic which we call variable order logic ( VO) which permits the use of untyped relation variables, i.e., variables of variable order, by allowing quantification over orders. We show that this logic is complete, though even non-recursive queries can be expressed in VO. Then we define a fragment of VO and we prove that it expresses exactly the class of r.e. queries. We finally give a characterization of the class of computable queries through a fragment of VO, which is undecidable.