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It's easy to see how a multi-track Turing machine can simulate a single-track Turing machine; it does so by ignoring all but the first track. But how does it work the other way? I need a specification of a transition function that does the job. If there are $k$ tracks, then we can think of symbols as being vectors and arrange them one after another in the tape; but again, what's the transition function like in the equivalent single-track machine?

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    $\begingroup$ This is a fairly standard exercise. What are your specific problems with the transition function? (I seem to remember that there are some related questions around, discussion the performance impact of such a simulation.) $\endgroup$ – Raphael Sep 13 '12 at 12:38
  • $\begingroup$ @Raphael If $\delta_k(q, [x_1,\dots,x_k]) = [y_1,\dots,y_k]$ where $\delta_k$ is the multi-track transition function, how is $\delta$, the single-track transition function, defined as to guarantee equivalence? $\endgroup$ – saadtaame Sep 13 '12 at 12:46
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    $\begingroup$ I think you need a more general idea before looking at individual transitions. You may need to do some additional bookkeeping/tape management not easily seen/expressed in single transitions. $\endgroup$ – Raphael Sep 13 '12 at 13:23
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If $\Sigma = (x_1,...,x_n)$ is the alphabet of the $m$-tracks $TM$, just use an expanded alphabet $\Sigma' = \Sigma \times ... \times \Sigma$ for the single-track $TM'$ ($|\Sigma'| = n^m$).

Every vector $\bar{x}_i$ of $m$ symbols from $\Sigma$ can be mapped to a unique alphabet symbol $u_i$ in $\Sigma'$: $\bar{x}_i = (x_{i_1},x_{i_2},...,x_{i_m}) \rightarrow u_i \in \Sigma'$

Hence every transition of $TM$ $(q_h,(x_{i_1},x_{i_2},...,x_{i_m}))\rightarrow (q_k,(x_{j_1},x_{j_2},...,x_{j_m}),dir)$ can be mapped to an equivalent transition in $TM'$ where the "read vector" $\bar{x_i}$ and "write vector" $\bar{x_j}$ are replaced with the corresponding alphabet symbols in $\Sigma'$: $(q_h,u_i)\rightarrow (q_k,u_j,dir)$

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  • $\begingroup$ Clear, but one question. Do the symbols in $\Sigma'$ have length $=1$? That's bugging me. $\endgroup$ – saadtaame Sep 16 '12 at 20:08
  • $\begingroup$ @saadtaame, yes all the $u_{i}$s are single symbols, there's just $\Sigma^{m}$ of them. $\endgroup$ – Luke Mathieson Dec 1 '12 at 0:18
  • $\begingroup$ @Vor I accepted this answer but now as I'm reading I have new questions. Why is there only one $dir$ in the transition function of the m-tracks TM? Isn't it that every track has a head that can move left, right, or not move at all? $\endgroup$ – saadtaame Dec 12 '13 at 22:33
  • $\begingroup$ @saadtaame: no, if the heads can move independently (in different directions), then the name of the model is multitape Turing machine, not multitrack Turing machine as in your question (and in my answer :) $\endgroup$ – Vor Dec 13 '13 at 9:09
  • $\begingroup$ I imagine that the simulation is much harder then. $\endgroup$ – saadtaame Dec 13 '13 at 14:10

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