Then any FO formula can be expressed as a subfamily of boolean circuits in the class Size-Depth(nO(1),log n).

Our goal is to get an equivalent subfamily of boolean circuits with logarithmic depth, i.e., in the class Size-Depth([n.sup.O(1)], log n), so that we actually get an important decrease in the time needed for the evaluation of the query, when changing from sequential to parallel computation.

Translation of FO to Boolean Circuits When the atomic formula is interpreted for a given valuation, it is satisfied or no, that is, true or false.

Query Expressed as finite Subfamily of Boolean Circuits

For a r-ary query, q = [n.sup.r] -1, then we obtain a finite subfamily of boolean circuits C = {[C.sub.0],[C.sub.1],...,[C.sub.q]}, equivalent to nr boolean queries, such that for every valuation [V.sub.i]:{[x.sub.1]...,[x.sub.r]}[right arrow][D.sup.B], the boolean query [[phi].sub.i] is [phi]([x.sub.1]...,[x.sub.r])[right arrow][D.sup.B], the Boolean query [[phi].sub.i] ([x.sub.1],...[x.sub.r]) [[v.sub.i]([x.sub.1])...,[v.sub.i]([x.sub.r])],for 0[less than or equal to]I[less than or equal to]q.

In this way, we construct the finite subfamily of boolean circuits that represents [phi]([x.sub.1],...,[x.sub.r]).

It is represented in the boolean circuits formalism by a circuit Ch, for some h, equivalent to ?h.

Generation of Suitable Equivalent Formulas for Parallelism The idea is, we have a FO formula then we want to find another equivalent, such that its expression tree has special characteristics that allow to obtain a finite subfamily of boolean circuits whose width and depth are suitable for the class NC1, this is Size- Depth([n.sup.O(1)],logn).

Boolean circuits are a suitable theoretical model for the study of the computability and parallel complexity of queries to relational databases.

Given an r-ary query and a natural number n that represents the size of the domain of a given database, we showed how to build a finite subfamily of boolean circuits which preserves the property of uniformity, and which has a much better relation between size and depth, thus improving the time needed for the parallel evaluation of the query, as well as the appreciation of the parallelizability of the query.

A quite related topic, which we plan to consider, is the use of boolean circuits for the computation of queries to distributed relational databases.