Addition Machines, Automatic Functions and Open Problems of Floyd and Knuth

TL;DR

This paper proves positive answers to open problems posed by Floyd and Knuth regarding register machine bounds for multiplication, division, and power sum computations, and explores the automaticity of functions within register machines.

Contribution

It improves bounds on register counts for fast multiplication/division and characterizes automatic functions in register machines, solving longstanding open problems.

Findings

Fast multiplication and division bounds are improved.

Automatic functions with one input can be computed with four registers in linear time.

A nonautomatic function with one input can be computed with two registers in linear time.

Abstract

Floyd and Knuth investigated in 1990 register machines which can add, subtract and compare integers as primitive operations. They asked whether their current bound on the number of registers for multiplying and dividing fast (running in time linear in the size of the input) can be improved and whether one can output fast the powers of two summing up to a positive integer in subquadratic time. Both questions are answered positively. Furthermore, it is shown that every function computed by only one register is automatic and that automatic functions with one input can be computed with four registers in linear time; automatic functions with a larger number of inputs can be computed with 5 registers in linear time. There is a nonautomatic function with one input which can be computed with two registers in linear time.