Parallel Identity Testing for Skew Circuits with Big Powers and Applications

TL;DR

This paper introduces powerful skew arithmetic circuits with big powers, proving polynomial identity testing for them is in coRNC^2, and applies this to string program equivalence and group word problems.

Contribution

It extends polynomial identity testing to powerful skew circuits with big powers and demonstrates new applications in string program equivalence and group theory.

Findings

Polynomial identity testing for powerful skew circuits is in coRNC^2.

Equivalence of higher-dimensional straight-line programs is in coRNC^2.

The compressed word problem for certain wreath products is in coRNC^2.

Abstract

Powerful skew arithmetic circuits are introduced. These are skew arithmetic circuits with variables, where input gates can be labelled with powers $x^n$ for binary encoded numbers $n$. It is shown that polynomial identity testing for powerful skew arithmetic circuits belongs to $\mathsf{coRNC}^2$, which generalizes a corresponding result for (standard) skew circuits. Two applications of this result are presented: (i) Equivalence of higher-dimensional straight-line programs can be tested in $\mathsf{coRNC}^2$; this result is even new in the one-dimensional case, where the straight-line programs produce strings. (ii) The compressed word problem (or circuit evaluation problem) for certain wreath products of finitely generated abelian groups belongs to $\mathsf{coRNC}^2$.