Computing with and without arbitrary large numbers

TL;DR

This paper characterizes how the availability of large arbitrary numbers as input affects the computational power of random access machines, correcting and strengthening classical results in the field.

Contribution

It provides a general characterization of the computational impact of large arbitrary inputs and corrects and enhances foundational results by Simon and Szegedy (1992) and Simon (1981).

Findings

Corrected classical results with new constructions

Strengthened the theoretical understanding of large number inputs

Provided tools for analyzing arbitrary large number power

Abstract

In the study of random access machines (RAMs) it has been shown that the availability of an extra input integer, having no special properties other than being sufficiently large, is enough to reduce the computational complexity of some problems. However, this has only been shown so far for specific problems. We provide a characterization of the power of such extra inputs for general problems. To do so, we first correct a classical result by Simon and Szegedy (1992) as well as one by Simon (1981). In the former we show mistakes in the proof and correct these by an entirely new construction, with no great change to the results. In the latter, the original proof direction stands with only minor modifications, but the new results are far stronger than those of Simon (1981). In both cases, the new constructions provide the theoretical tools required to characterize the power of arbitrary large numbers.