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In Turing Machine, we know that there's (fine-grained complexity) difference between one-tape, 2-taped and multi-taped TM, even though they could be simulated efficiently.

(Well, actually I'm not quite sure they are indeed seperated: i.e. $TIME(t)\neq TIME_{2-taped}(t)\neq TIME_{multi-taped}(t)$ )

But today I'd like to focus on RAM model, is there any complexity seperation between one-taped RAM model and multi-taped model? Though we might expect it be quite small because of random accessibility.

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    $\begingroup$ The RAM model doesn’t have any tapes. $\endgroup$ – Yuval Filmus Jun 14 at 8:05
  • $\begingroup$ It seems RAM model is to use a $log(n)$-bit index to access certain cell on a tape. $\endgroup$ – Taylor Huang Jun 14 at 8:20
  • $\begingroup$ Having more than one “tape” gives you no advantage at all. That’s a nice exercise for you. $\endgroup$ – Yuval Filmus Jun 14 at 8:29
  • $\begingroup$ I'm not sure about that. A "tape-manner" turing machine would require zigzagging to simulate its multi-taped version. In RAM model, it seems we still need zigzagging except that we only need log time to jump to that specific location. $\endgroup$ – Taylor Huang Jun 14 at 15:36
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Random-access machines support the following operation in constant time: $$ x \gets M[y], $$ where $M$ is the memory array, and $y$ is an index whose allowable size depends on the exact model. Whether $M$ is an array of bits or an array of words depends on your exact model.

If you had several different memory arrays, say $M_1[y],\ldots,M_r[y]$, then you could simulate the command $$ x \gets M_i[y] $$ with the command $$ x \gets M[ry + i] $$ which also takes constant time.

Therefore there is nothing to be gained by allowing several "memories".

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    $\begingroup$ The integer arithmetic you performed before memory access, i.e. $ry+i$, isn't that at least log time? $\endgroup$ – Taylor Huang Jun 14 at 16:27
  • $\begingroup$ It depends on the model. In the unit cost RAM it will take unit time. You could also transform your program so that for each variable $y$, it also computes $ry+i$ for $i=1,\ldots,r$. The resulting overhead will be $O(r) = O(1)$. $\endgroup$ – Yuval Filmus Jun 14 at 17:10

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