Counting is Easy

TL;DR

The paper presents a simple real-time simulation method for multiple counters using a single-tape Turing machine, with efficient space usage proportional to the number of counters and logarithmic in the number of commands.

Contribution

It introduces a novel, straightforward simulation technique for multiple counters on a single-tape Turing machine, using only five rewriting rules.

Findings

Simulation runs in real time for any fixed number of counters.

Space complexity is approximately (k+ε) log n bits for n commands.

The method uses only five simple rewriting rules.

Abstract

For any fixed $k$, a remarkably simple single-tape Turing machine can simulate $k$ independent counters in real time. Informally, a counter is a storage unit that maintains a single integer (initially 0), incrementing it, decrementing it, or reporting its sign (positive, negative, or zero) on command. Any automaton that responds to each successive command as a counter would is said to simulate a counter. (Only for a sign inquiry is the response of interest, of course. And zeroness is the only real issue, since a simulator can readily use zero detection to keep track of positivity and negativity in finite-state control. In this paper we describe a remarkably simple real-time simulation, based on just five simple rewriting rules, of any fixed number $k$ of independent counters. On a Turing machine with a single, binary work tape, the simulation runs in real time, handling an arbitrary counter command at each step. The space used by the simulation can be held to $(k+\epsilon) \log_2 n$ bits for the first $n$ commands, for any specified $\epsilon > 0$.