Structural complexity 1
On the complexity of queries in the logical data model
Lecture notes in computer science on ICDT '88
Descriptive characterizations of computational complexity
Journal of Computer and System Sciences
Untyped sets, invention, and computable queries
PODS '89 Proceedings of the eighth ACM SIGACT-SIGMOD-SIGART symposium on Principles of database systems
On the expressive power of database queries with intermediate types
Journal of Computer and System Sciences
Computability, complexity, and languages (2nd ed.): fundamentals of theoretical computer science
Computability, complexity, and languages (2nd ed.): fundamentals of theoretical computer science
Information and Computation - Special issue: logic and computational complexity
Foundations of Databases: The Logical Level
Foundations of Databases: The Logical Level
Arity and alternation: a proper hierarchy in higher order logics
Annals of Mathematics and Artificial Intelligence
Complete problems for higher order logics
CSL'06 Proceedings of the 20th international conference on Computer Science Logic
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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 Σji and Πji, for each i ≥ 1 and each number of alternations of quantifier blocks j. Then, we get as a corollary the expressive power of HOi 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 HOi, 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.