The evaluation of geometric queries: constraint databases and quantifier elimination

TL;DR

This paper models geometric elimination tasks using constraint database queries, establishing exponential lower bounds on their evaluation complexity, and explores the limits of implementing such algorithms based on advanced geometric elimination methods.

Contribution

It introduces a formal framework for geometric elimination via constraint database queries and proves exponential complexity bounds for their evaluation.

Findings

Exponential lower bounds for evaluating geometric queries.

Modeling geometric elimination as constraint database queries.

Analysis of the complexity limits of geometric elimination algorithms.

Abstract

We model the algorithmic task of geometric elimination (e.g., quantifier elimination in the elementary field theories of real and complex numbers) by means of certain constraint database queries, called geometric queries. As a particular case of such a geometric elimination task, we consider sample point queries. We show exponential lower complexity bounds for evaluating geometric queries in the general and in the particular case of sample point queries. Although this paper is of theoretical nature, its aim is to explore the possibilities and (complexity-)limits of computer implemented query evaluation algorithms for Constraint Databases, based on the principles of the most advanced geometric elimination procedures and their implementations, like, e.g., the software package "Kronecker".