# Descriptive Complexity of Deterministic Polylogarithmic Time and Space

## TL;DR

This paper introduces a logical framework that characterizes problems solvable in deterministic polylogarithmic time and space, providing new insights into the descriptive complexity of these classes.

## Contribution

It presents a novel two-sorted logic capturing PolylogTime and PolylogSpace, along with a variant of random-access Turing machines for finite ordered structures.

## Key findings

- Logic captures PolylogTime and PolylogSpace classes.
- Introduces a random-access Turing machine model.
- Highlights open problems in order-invariant queries.

## Abstract

We propose logical characterizations of problems solvable in deterministic polylogarithmic time (PolylogTime) and polylogarithmic space (PolylogSpace). We introduce a novel two-sorted logic that separates the elements of the input domain from the bit positions needed to address these elements. We prove that the inflationary and partial fixed point vartiants of this logic capture PolylogTime and PolylogSpace, respectively. 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 a structure directly. We investigate whether an explicit predicate for the ordering of the domain is needed in our PolylogTime logic. Finally, we present the open problem of finding an exact characterization of order-invariant queries in PolylogTime.