Efficient Computation by Three Counter Machines

TL;DR

This paper demonstrates that multiplication and certain functions of two variables can be computed efficiently in polynomial time on a simple three counter machine, extending to functions computable in linear space.

Contribution

It introduces a method for efficient computation of multiplication and linear-space functions on a minimal three counter machine.

Findings

Multiplication is computable in polynomial time on a three counter machine.

Functions of two variables computable in linear space are also efficiently computable.

The technique generalizes to a class of functions beyond basic multiplication.

Abstract

We show that multiplication can be done in polynomial time on a three counter machine that receives its input as the contents of two counters. The technique is generalized to functions of two variables computable by deterministic Turing machines in linear space.