Reflective Relational Machines Working on Homogeneous Databases

Abstract

¹² We define four different properties of relational databases which are related to the notion of homogeneity in classical Model Theory. The main question for their definition is, for any given database, which is the minimum integer k, such that whenever two k-tuples satisfy the same properties which are expressible in First Order Logic with up to k variables (FO^k), then there is an automorphism which maps each of these k-tuples onto each other? We study these four properties as a means to increase the computational power of sub-classes of Reflective Relational Machines (RRM) of bounded variable complexity. For this sake we give first a semantic characterization of the sub-classes of total RRM with variable complexity k, for every natural k, with the classes of queries which we denote as $\mathcal{Q}\mathcal{C}\mathcal{Q}^k$. We prove that these classes form a strict hierarchy in a strict sub-class of total$\left( {\mathcal{Q}\mathcal{C}} \right)$. And it follows that it is orthogonal with the usual classification of computable queries in Time and Space complexity classes. We prove that the computability power of RRM^k machines is much bigger when working with classes of databases which are homogeneous, for three of the properties which we define. As to the fourth one, we prove that the computability power of RRM with sub-linear variable complexity also increases when working on databases which satisfy that property. The strongest notion, pairwise k-homogeneity, allows RRM^k machines to achieve completeness.

The work presented here was partially done during a visit to the Department of Mathematics of the University of Helsinki.

This work has been partially supported by Grant 1011049 of the Academy of Finland and by Grants of Universidad CAECE and Universidad de Luján, Argentina.