Multi-track Turing machine

A Multitrack Turing machine is a specific type of multi-tape Turing machine.

In a standard n-tape Turing machine, n heads move independently along n tracks. In a n-track Turing machine, one head reads and writes on all tracks simultaneously. A tape position in an n-track Turing Machine contains n symbols from the tape alphabet. It is equivalent to the standard Turing machine and therefore accepts precisely the recursively enumerable languages.

Formal definition

A multitrack Turing machine with $n$-tapes can be formally defined as a 6-tuple$M=\langle Q,\Sigma ,\Gamma ,\delta ,q_{0},F\rangle$, where

• $Q$ is a finite set of states;
• $\Sigma \subseteq \Gamma \setminus \{b\}$ is a finite set of input symbols, that is, the set of symbols allowed to appear in the initial tape contents;
• $\Gamma$ is a finite set of tape alphabet symbols;
• $q_{0}\in Q$ is the initial state;
• $F\subseteq Q$ is the set of final or accepting states;
• $\delta :\left(Q\backslash F\times \Gamma ^{n}\right)\rightarrow \left(Q\times \Gamma ^{n}\times \{L,R\}\right)$ is a partial function called the transition function.

Sometimes also denoted as $\delta \left(Q_{i},[x_{1},x_{2}...x_{n}]\right)=(Q_{j},[y_{1},y_{2}...y_{n}],d)$, where $d\in \{L,R\}$.

A non-deterministic variant can be defined by replacing the transition function $\delta$ by a transition relation $\delta \subseteq \left(Q\backslash F\times \Gamma ^{n}\right)\times \left(Q\times \Gamma ^{n}\times \{L,R\}\right)$.

Proof of equivalency to standard Turing machine

This will prove that a two-track Turing machine is equivalent to a standard Turing machine. This can be generalized to a n-track Turing machine. Let L be a recursively enumerable language. Let $M=\langle Q,\Sigma ,\Gamma ,\delta ,q_{0},F\rangle$ be standard Turing machine that accepts L. Let M' is a two-track Turing machine. To prove ⁠$M=M'$⁠ it must be shown that $M\subseteq M'$ and $M'\subseteq M$.

• $M\subseteq M'$

If the second track is ignored then M and M' are clearly equivalent.

• $M'\subseteq M$

The tape alphabet of a one-track Turing machine equivalent to a two-track Turing machine consists of an ordered pair. The input symbol a of a Turing machine M' can be identified as an ordered pair ⁠$[x,y]$⁠ of Turing machine M. The one-track Turing machine is:

$M=\langle Q,\Sigma \times {B},\Gamma \times \Gamma ,\delta ',q_{0},F\rangle$ with the transition function $\delta \left(q_{i},[x_{1},x_{2}]\right)=\delta '\left(q_{i},[x_{1},x_{2}]\right)$

This machine also accepts L.

References

• Thomas A. Sudkamp (2006). Languages and Machines, Third edition. Addison-Wesley. ISBN 0-321-32221-5. Chapter 8.6: Multitape Machines: pp 269–271