Reducibility and Artificial Neural Networks
Not really.  I respect what you're trying to achieve, but I don't think it's possible to achieve what you want, given our current level of knowledge of neural networks.

We already know that convolutional neural networks perform better for some problems, and fully-connected neural networks (what you call MP) work better for other problems.  So you can't expect to find that one is always better than the other.

It has been proven that any computable function can be approximated arbitrarily well by some fully-connected neural network.  However, the catch in this theorem is that the theorem doesn't tell us how large the neural network needs to be.  We already know that sometimes a larger network is more accurate, but is also slower to train, so that's a pretty big caveat in the theoretical result.

And if you want to compare two different network architectures -- say, convolutional vs fully-connected -- then such a comparison won't be useful if it doesn't take into account the size of the network.  If you need 1 billion parameters to make a fully-connected network work well, or 1 million parameters to make a convolutional network work well, are they equally good?  No, you'll probably prefer the convolutional network.  It's not at all clear how to get useful reductions between the two architectures that tells us anything about the size of the neural network.

So, no, you're probably not going to get some useful theory this way that says "you can throw away all other architectures other than type X".