# Multi Head universal Turing machine

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?

• 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

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

• 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 1 '16 at 2:33
• A single tape with multiple heads is the same as multiple tapes, up to linear factors. – Emil Jeřábek supports Monica Mar 1 '16 at 8:55
• A nice place to look for information about different versions of TMs is the first chapter of handbook of theoretical computer science. – Kaveh Mar 1 '16 at 14:22