"Equivalent device" for testing for NP and coNP? I'm trying to understand some aspects of the $P=^?NP$ and $NP=^?coNP$ problems.

I am engineer and not mathematician nor computer scientist so I do not completely understand what a turing machine is.

I'm currently thinking about a variant of the "subset sum problem" known to be in $NP$ and suspected to be not in $coNP$. However I'm thinking of a modified variant of the problem.

It is easy to prove that a computer based on a "real" CPU can prove the answer "yes, a subset exists" (when the subset is known) in less than $(n+a)^2$ operations (while $a$ depends on the type of CPU used) when the original set contains $n$ elements.

From my understanding of $NP$ this implies that the problem belongs to $NP$. My first question is if this understanding is correct.

(I write "belongs to $NP$" and not "$\in NP$" because I know I mix up terms like "problem", "formal language", ...)

I suspect the minimum number of operations (on the real CPU) to prove the answer "no, no matching subset exists" to be more than $\frac{2^n}b$ (while $b$ depends on the type of CPU used).

So my second question is if this would mean that the problem does not belong to $coNP$.

As far as I know that there are differences between Turing Machines and CPU-based computers:


A Turing Machine has unlimited memory while a computer has limited memory
A Turing Machine can't generate true (undeterministic) random numbers while on a computer you may use the noise of the microphone input...
...


So I'm not sure if "cannot be calculated by a computer in polynomal time" implies "cannot be calculated by a Turing Machine in polynomal time".

-- Edit --

Having read the answers I think I have to say something about the background of the question:

I think I had already understood what P, NP or coNP is before asking the question but I was not 100% sure because I was not sure about the differences between a Turing Machine and a computer.

I have an idea how it could be possible to prove the impossibility to build a digital computer that can solve a certain problem in less than $a^nb$ steps while $n$ is the number of inputs and $a$ and $b$ depends on the computer and the problem.

I was just wondering if such a proof was sufficient to prove $P \neq NP$ if it could be done for a problem that is known to be in NP.

Having read your answers I think that such a proof would be sufficient. Thanks.

(Needless to say that I am not able to carry the proof I'm thinking of for any problem.)
I will not directly answer your questions, as this would result in explaining the definitions of $\mathsf{NP}$ and $\mathsf{coNP}$, which has been done before. However, using the following hint combined with the definition of the complexity classes indirectly answers your questions.



A modern computer with unlimited memory is equivalent to the concept of a register machine, which in turn is turing equivalent. Thus everything that can be computed on a modern computer with infinite memory can also be computed by a turing machine and vice-versa.


  A Turing Machine can't generate true (undeterministic) random numbers
  while on a computer you may use the noise of the microphone input...


A microphone, or any other hardware-based true random number generator, is an input of the computer. That means the values delivered by it are not computed by the computer itself. Note that you can also model random input by a two-tape turing machine, where one tape contains the random numbers.