Two heads are better than two tapes

TL;DR

This paper investigates the computational power of two-head Turing machines with single-dimensional tapes, showing limitations in recognizing certain languages in real time and comparing capabilities with other machine configurations.

Contribution

It proves that two-head one-dimensional tape Turing machines cannot recognize a specific language in real time, settling a longstanding conjecture about their computational limitations.

Findings

Two-head one-dimensional tape Turing machines cannot recognize certain languages in real time.

Machines with three tapes or two-dimensional tapes can recognize the same language in real time.

The paper confirms that head configuration affects real-time recognition capabilities.

Abstract

We show that a Turing machine with two single-head one-dimensional tapes cannot recognize the set {x2x'| x \in {0,1}^* and x' is a prefix of x} in real time, although it can do so with three tapes, two two-dimensional tapes, or one two-head tape, or in linear time with just one tape. In particular, this settles the longstanding conjecture that a two-head Turing machine can recognize more languages in real time if its heads are on the same one-dimensional tape than if they are on separate one-dimensional tapes.