Stack and register complexity of radix conversions

TL;DR

This paper explores the computational complexity of radix conversions, demonstrating limitations of pushdown automata and capabilities of two-counter machines, with insights into input order effects and structural properties.

Contribution

It extends existing results on radix conversion complexity by analyzing two-counter machines and input order effects, providing new structural insights.

Findings

No PDA can compute the significand of certain floating point approximations.

Two counter machines can perform radix conversions with input.

Input order affects the ability of two counter machines to decode online.

Abstract

We investigate the question of computational resources (such as stacks and counters) necessary to perform radix conversions. To this end it is shown that no PDA can compute the significand of the best $n$-digit floating point approximation of a power of an incommensurable radix. This extends the results of W.~Clinger. We also prove that a two counter machine with input is capable of such conversions. On the other hand we note a curious asymmetry with respect to the order in which the digits are input by showing that a two counter machine can decode its input online if the digits are presented in the most-to-least significant order while no such machine can decode its input in this manner if the digits are presented in the least-to-most significant order. Some structural results about two counter machines (with input) are also established.