# Can all programs reducible to ones with only arithmetic operations on inputs be simulated with polynomial overhead by arithmetic machine?

In Can all programs be modeled as operations of elementary arithmetic operations on inputs? and computabiltiy theory, I asked:

we treat all inputs and intermediate results and final outputs as natural number. While algorithms/programs themselves are considered natural numbers, here we treat these programs/functions/algorithms as just computable functions. The question is, when the function operates on an input to produce an output, can we consider the operation of function as using only a number of arithmetic operations (addition, subtraction, multiplication and division) on an input? Or does the use of if/else make the aforementioned not true? If this is true, is the number of arithmetic operations polynomially proportional to the lowest time complexity bound possible for solving a problem? (That is, if the lowest time complexity is $\text{O(whatever)}$, then the number of arithmetic operations is $\text{O(whatever)}^k$ where $k$ is some rational number.)

I learned an answer to this, and now I would like to present variation: If we limit our scope to programs that can be modeled as operations of arithmetic operations on inputs, can these program be simulated by a machine that can only do basic arithmetic processes on inputs (multiplciation, division, subtraction, addition) with polynomial overhead (That is, if the lowest time complexity is $\text{O(whatever)}$, then the number of arithmetic operations is $\text{O(whatever}^k)$ where $k$ is some rational number.) to the lowest possible time complexity for solving a problem?

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still no comments? Is this a hard question? –  logica Apr 22 '13 at 6:31
Cross-posted at mathoverflow.net/questions/128343 –  logica Apr 22 '13 at 12:03