On the Kolmogorov Expressive Power of Boolean Query Languages

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 nite structures. This turns the above informal question into a precisely deened mathematical one. This notion gives a meaningful deenition of the expressive power of a Boolean query language in a single nite database. The notion of Kolmogorov expressive power of a Boolean query language L in a nite database A is deened by considering two values: the Kolmogorov complexity of the isomor-phism 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 rst, the more expressive is the query language. After giving the deenitions and proving that they are correct, we concentrate our eeorts on rst order logic and its powerful extensions: innationary xpoint logic and partial xpoint 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 deenability of order. We show that, except of being of interest for its own, our notion may have important diagnostic value for database query optimization. Preliminary version of this paper appeared in the form of a conference paper 22].