Evaluating Matrix Circuits

TL;DR

This paper investigates the computational complexity of the circuit evaluation problem for various classes of finitely generated linear groups, establishing bounds and equivalences with polynomial identity testing.

Contribution

It demonstrates that the compressed word problem for finitely generated nilpotent groups is in , and identifies cases within polycyclic groups where the problem matches the complexity of polynomial identity testing.

Findings

The problem is in  for finitely generated nilpotent groups.

For , the problem is equivalent to polynomial identity testing.

Some polycyclic groups have problems as hard as polynomial identity testing.

Abstract

The circuit evaluation problem (also known as the compressed word problem) for finitely generated linear groups is studied. The best upper bound for this problem is $\mathsf{coRP}$, which is shown by a reduction to polynomial identity testing. Conversely, the compressed word problem for the linear group $\mathsf{SL}_3(\mathbb{Z})$ is equivalent to polynomial identity testing. In the paper, it is shown that the compressed word problem for every finitely generated nilpotent group is in $\mathsf{DET} \subseteq \mathsf{NC}^2$. Within the larger class of polycyclic groups we find examples where the compressed word problem is at least as hard as polynomial identity testing for skew arithmetic circuits.