Arithmetic Branching Programs with Memory

TL;DR

This paper explores how adding memory to arithmetic branching programs enhances their computational power, providing new characterizations for complexity classes like VP and VNP.

Contribution

It introduces memory into branching programs, broadening their expressive capacity and offering natural characterizations of VP and VNP.

Findings

Memory increases branching program power even at constant width

New characterizations of VP and VNP using memory-enhanced branching programs

Memory types influence computational capabilities significantly

Abstract

We extend the well known characterization of $\vpws$ as the class of polynomials computed by polynomial size arithmetic branching programs to other complexity classes. In order to do so we add additional memory to the computation of branching programs to make them more expressive. We show that allowing different types of memory in branching programs increases the computational power even for constant width programs. In particular, this leads to very natural and robust characterizations of $\vp$ and $\vnp$ by branching programs with memory.