On topological lower bounds for algebraic computation trees

TL;DR

This paper establishes lower bounds on the height of algebraic computation trees for semialgebraic set membership, linking it to Betti numbers, thus providing fundamental complexity limits for algebraic decision processes.

Contribution

It introduces new topological lower bounds for algebraic computation tree complexity based on Betti numbers, extending prior bounds and applying to projections of sets.

Findings

Lower bounds proportional to Betti numbers for decision tree height.

Bounds depend on the m-th Betti number of the set and its projections.

Examples demonstrate the bounds' applicability to specific problems.

Abstract

We prove that the height of any algebraic computation tree for deciding membership in a semialgebraic set is bounded from below (up to a multiplicative constant) by the logarithm of m-th Betti number (with respect to singular homology) of the set, divided by m+1. This result complements the well known lower bound by Yao for locally closed semialgebraic sets in terms of the total Borel-Moore Betti number. We also prove that the height is bounded from below by the logarithm of m-th Betti number of a projection of the set onto a coordinate subspace, divided by (m+1)^2. We illustrate these general results by examples of lower complexity bounds for some specific computational problems.