The Kolmogorov expressive power of Boolean query languages

Abstract

We address the question “How much of the information stored in a given database can be retrieved by all Boolean queries in a given query language?”. In order to answer it we develop a Kolmogorov complexity based measure of expressive power of Boolean query languages over finite structures. This turns the above informal question into a precisely defined mathematical one. This notion gives a meaningful definition of the expressive power of a Boolean query language in a single finite database. The notion of Kolmogorov expressive power of a Boolean query language L in a finite database A is defined by considering two values: the Kolmogorov complexity of the isomorphism type of A, equal to the length of the shortest description of this type, and the number of bits of this description that can be reconstructed from truth values of all queries from L in A. The closer is the second value to the first, the more expressive is the query language. After giving the definitions and proving that they are correct, we concentrate our efforts on first order logic and its powerful extensions: inflationary fixpoint logic and partial fixpoint logic. We explore some connections between the proposed Kolmogorov expressive power of Boolean queries in these languages and their standard expressive power, in particular with the definability of order. We show that, except of being of interest on its own, our notion may have important diagnostic value for database query optimization.