A Measure of Space for Computing over the Reals

TL;DR

This paper introduces a new space complexity measure for real computation, defining classes LOGSPACE_W and PSPACE_W, and demonstrates their relationships with known complexity classes and problems.

Contribution

It defines and analyzes new real-space complexity classes LOGSPACE_W and PSPACE_W, establishing their inclusions in existing classes and their relevance for natural algorithms.

Findings

LOGSPACE_W is included in NC^2_R and P_W

Real Circuit Decision Problem is P_R-complete under LOGSPACE_W reductions

PSPACE_W is included in PAR_R

Abstract

We propose a new complexity measure of space for the BSS model of computation. We define LOGSPACE\_W and PSPACE\_W complexity classes over the reals. We prove that LOGSPACE\_W is included in NC^2\_R and in P\_W, i.e. is small enough for being relevant. We prove that the Real Circuit Decision Problem is P\_R-complete under LOGSPACE\_W reductions, i.e. that LOGSPACE\_W is large enough for containing natural algorithms. We also prove that PSPACE\_W is included in PAR\_R.