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 an 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 -tapes can be formally defined as a 6-tuple, where
  • is a finite set of states;
  • is a finite set of input symbols, that is, the set of symbols allowed to appear in the initial tape contents;
  • is a finite set of tape alphabet symbols;
  • is the initial state;
  • is the set of final or accepting states;
  • is a partial function called the transition function.
A non-deterministic variant can be defined by replacing the transition function by a transition relation.

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 be standard Turing machine that accepts L. Let is a two-track Turing machine. To prove it must be shown that and.
If the second track is ignored then and 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 can be identified as an ordered pair of Turing machine. The one-track Turing machine is:
This machine also accepts L.