José María Turull-Torres

Computing queries with higher-order logics

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.

Expressibility of Higher Order Logics

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 Σij and πij, for each order i ≥ 2 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 D. Leivant and 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., Ui≥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.

A Second-Order Logic in Which Variables Range over Relations with Complete First-Order Types

We introduce a restriction of second order logic, SO

On Fragments of Higher Order Logics that on Finite Structures Collapse to Second Order

^F

Complete problems for higher order logics

, for finite structures. In this restriction the quantifiers range over relation closed by the equivalence relation

Systematic Refinement of Abstract State Machines with Higher-Order Logic

\equiv^{FO}

Relational Complexity and Higher Order Logics

. In this equivalence relation the equivalence classes are formed by

Semantic Restrictions over Second-Order Logic

k

Capturing relational NEXPTIME with a fragment of existential third order logic

-tuples whose \textit{FO type} is the same, for some integer

Foundations of information and knowledge systems : third International Symposium, FoIKS 2004 : Wilheminenburg Castle, Austria, February 17-20, 2004 : proceedings

k\geq 1