# 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. :) Feb 29, 2016 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? Mar 1, 2016 at 2:33
• A single tape with multiple heads is the same as multiple tapes, up to linear factors. Mar 1, 2016 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. Mar 1, 2016 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. Mar 17, 2022 at 12:58