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;
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 M' is a two-track Turing machine. To prove it must be shown that and .
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 of Turing machine M. The one-track Turing machine is:
with the transition function
This machine also accepts L.
References
Thomas A. Sudkamp (2006). Languages and Machines, Third edition. Addison-Wesley. ISBN0-321-32221-5. Chapter 8.6: Multitape Machines: pp 269–271
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