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It is common knowledge that a universal Turing machine can simulate any Turing machine with logarithmic overhead. Is it possible to make this overhead constant by constructing an analogous "Universal" multi head Turing machine?

Cpt Wobbles
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  • I believe that there are machine models where the time-complexity overhead factor is a constant. Maybe someone out there can provide some examples. :) – Michael Wehar Feb 29 '16 at 14:48

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If we have a fixed number of tapes then yes we can simulate them without the logarithmic overhead. E.g simulation of two-tape (and in general $k$-tape) TMs on a two-tape machine can be done without the logarithmic factor increase.

If we want to simulate an arbitrary constant number of tapes then AFAIK we don't know any such simulation.

See these questions and the references there:

Justification of log f in DTIME hierarchy theorem

Universal simulation of Turing machines

Community
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Kaveh
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  • Hmm interesting. What about a single tape with fixed input alphabet (ie the input output colors are restricted to ${0,1,#}$ (blank space) and still multil head? – Cpt Wobbles Mar 01 '16 at 02:33
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    A single tape with multiple heads is the same as multiple tapes, up to linear factors. – Emil Jeřábek Mar 01 '16 at 08:55
  • A nice place to look for information about different versions of TMs is the first chapter of handbook of theoretical computer science. – Kaveh Mar 01 '16 at 14:22
  • Would you have a reference for constant overhead simulation of k tapes by 2 tapes ? I've yet to find it in the links you posted. – ULechine Mar 17 '22 at 12:58