Descriptive Complexity of Deterministic Polylogarithmic Time

Abstract

We propose a logical characterization of problems solvable in deterministic polylogarithmic time (PolylogTime). We introduce a novel two-sorted logic that separates the elements of the input domain from the bit positions needed to address these elements. In the course of proving that our logic indeed captures PolylogTime on finite ordered structures, we introduce a variant of random-access Turing machines that can access the relations and functions of the structure directly. We investigate whether an explicit predicate for the ordering of the domain is needed in our logic. Finally, we present the open problem of finding an exact characterization of order-invariant queries in PolylogTime.