Counting is Easy

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 $\epsilon > 0$.