# 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 a 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 multitape Turing machine 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
• $\Sigma$ is a finite set of symbols called the tape alphabet
• $\Gamma \in Q$
• $q_0 \in Q$ is the initial state
• $F \subseteq Q$ is the set of final or accepting states.
• $\delta \subseteq \left(Q \backslash A \times \Sigma\right) \times \left( Q \times \Sigma \times d \right)$ is a relation on states and symbols called the transition relation.
• $\delta \left(Q_i,[x_1,x_2...x_n]\right)=(Q_j,[y_1,y_2...y_n],d)$

where $d \in \{L,R\}$

## 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 all but the first 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. Adison Wesley. ISBN 0-321-32221-5. Chapter 8.6: Multitape Machines: pp 269-271