Random Access Machines with only addition, multiplication, equality The literature is fairly clear that unit-cost RAMs with primitive multiplication are unreasonable, in that they


cannot be simulated by Turing machines in polynomial time
can solve PSPACE-complete problems in polynomial time


However, all of the references I can find on this topic (Simon 1974, Schonhage 1979) also involve boolean operations, integer division, etc.

Do there exist any results for the "reasonableness" of RAMs that only have addition, multiplication, and equality? In other words, which do not have boolean operations, truncated integer division, truncated subtraction, etc?

One would think that such RAMs are still quite "unreasonable." The main red flag is that they enable the generation of exponentially large integers in linear time, and due to the convolution-ish effects of multiplication, this can get particularly complex. However, I cannot actually find any results showing that this allows for any sort of "unreasonable" result (exponential speedup of Turing machine, unreasonable relationship to PSPACE, etc).

Does the literature have any results on this topic?
The other day I was reading a paper on parallelized random access machines without bit operations, which sounded very much like what you are describing. For these models NC is known not to equal P. See here: https://epubs.siam.org/doi/10.1137/S0097539794282930

The paper can also be found on Professor Mulmuley’s website.