Two- versus three-dimensional connectivity testing of first-order queries to semi-algebraic sets

TL;DR

This paper investigates whether the connectivity of three-dimensional semi-algebraic sets can be inferred from two-dimensional samples using first-order queries, concluding negatively for certain classes of queries.

Contribution

It provides a negative answer to the possibility of determining 3D connectivity from 2D samples for specific classes of first-order queries.

Findings

Connectivity cannot be inferred from 2D samples for cartesian-product-free queries.

Connectivity cannot be inferred from 2D samples for positive one-pass queries.

The results highlight limitations in sampling-based connectivity testing.

Abstract

This paper addresses the question whether one can determine the connectivity of a semi-algebraic set in three dimensions by testing the connectivity of a finite number of two-dimensional ``samples'' of the set, where these samples are defined by first-order queries. The question is answered negatively for two classes of first-order queries: cartesian-product-free, and positive one-pass.