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 $$M' \subseteq M$$.

If the second track is ignored then $M$ and $M'$ are clearly equivalent. 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:
 * $$ M \subseteq M' $$
 * $$ M' \subseteq M $$


 * $$M = \langle Q, \Sigma \times {B}, \Gamma \times \Gamma, \delta ', q_0, F \rangle $$ 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.