Arithmetic Circuits with Division

TL;DR

This paper explores the computational complexity of membership problems for arithmetic circuits over natural numbers that include division, revealing that certain division-including subsets are PSPACE-complete, with some cases being NP-hard.

Contribution

It provides new complexity classifications for arithmetic circuits with division, including PSPACE-completeness results and contrasts between different operation subsets.

Findings

{intersection,union,complement,+,/} is PSPACE-complete

{intersection,union,complement,x,/} is PSPACE-complete

{union,/} is NP-hard

Abstract

We study the computational complexity of the membership problem for arithmetic circuits over natural numbers with division. We consider different subsets of the operations {intersection,union,complement,+,x,/}, where / is the element-wise integer division (without remainder and without rounding). Results for the subsets without division have been studied before, in particular by McKenzie and Wagner and Yang. The division is expressive because it makes it possible to describe the set of factors of a given number as a circuit. Surprisingly, the cases {intersection,union,complement,+,/} and {intersection,union,complement,x,/} are PSPACE-complete and therefore equivalent to the corresponding cases without division. The case {union,/} is NP-hard in contrast to the case {union} which is NL-complete. Further upper bounds, lower bounds and completeness results are given.