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(nitpick) Are metabolic systems and neuron networks really discrete (as opposed to analog) systems?

by rntz 10y ago

pron 10y ago
I think that the common belief is that they are (or, at least, that a discrete model can capture their behavior), although I don't think it matters, as the common belief is that analog computers and digital computers have the same computational power.
- rntz 10y ago
  What does it mean to say that analog and digital computers have the same computational power? My intuition would be something like: for any function calculable by an analog computer, for any desired degree of precision, there is a Turing-computable function which approximates it to that degree of precision. But you have to be a bit finnicky, since you can't actually feed an analogue input to a digital computer, either. So it has to be some relation between the precision of the input and the precision of the output. And of course there's the other direction, that analogue computers can simulate digital ones. Do you know whether anything like this been formalized and proven?
  - pron 10y ago
    Turing himself addressed this issue, and actually his famous 1936 paper talks about computing real numbers, not natural numbers. See http://cs.stackexchange.com/questions/35343/analog-computers-and-the-church-turing-thesis http://cs.stackexchange.com/questions/35343/analog-computers... for a brief discussion. In short, we don't know of anything that an analog computer can do that a digital computer can't. If you think a bit about your point on precision, you'll see that "desired digital precision" actually covers a problem that is essential to the analog computation itself. In other words, try to think what "infinite precision" actually means. But let me give an example that would make it more concrete. Suppose you want to build a computer that simulates the behavior of n bodies in gravitational interaction in space. It's obvious that with a digital computer, there would eventually be discrepancies between the actual system and the simulation (which would only grow). But now consider this: suppose that instead you want to do the simulation by building a "computer" which is nothing more than a copy of the original system (i.e., the "simulation" is composed of another set of n bodies in space). Is it possible to have this analog "simulation" more accurate than a digital simulation? The answer is no, because even supposing a true continuum of positions and velocities exists in the first place, in order to reconstruct the system precisely, you would need to pass an infinite amount of information to the "simulation", which is impossible.

(nitpick) Are metabolic systems and neuron networks really discrete (as opposed to analog) systems?

by rntz 10y ago

(nitpick) Are metabolic systems and neuron networks really discrete (as opposed to analog) systems?

pron 10y ago

I think that the common belief is that they are (or, at least, that a discrete model can capture their behavior), although I don't think it matters, as the common belief is that analog computers and digital computers have the same computational power.

rntz 10y ago
What does it mean to say that analog and digital computers have the same computational power? My intuition would be something like: for any function calculable by an analog computer, for any desired degree of precision, there is a Turing-computable function which approximates it to that degree of precision. But you have to be a bit finnicky, since you can't actually feed an analogue input to a digital computer, either. So it has to be some relation between the precision of the input and the precision of the output. And of course there's the other direction, that analogue computers can simulate digital ones. Do you know whether anything like this been formalized and proven?
- pron 10y ago
  Turing himself addressed this issue, and actually his famous 1936 paper talks about computing real numbers, not natural numbers. See http://cs.stackexchange.com/questions/35343/analog-computers-and-the-church-turing-thesis http://cs.stackexchange.com/questions/35343/analog-computers... for a brief discussion. In short, we don't know of anything that an analog computer can do that a digital computer can't. If you think a bit about your point on precision, you'll see that "desired digital precision" actually covers a problem that is essential to the analog computation itself. In other words, try to think what "infinite precision" actually means. But let me give an example that would make it more concrete. Suppose you want to build a computer that simulates the behavior of n bodies in gravitational interaction in space. It's obvious that with a digital computer, there would eventually be discrepancies between the actual system and the simulation (which would only grow). But now consider this: suppose that instead you want to do the simulation by building a "computer" which is nothing more than a copy of the original system (i.e., the "simulation" is composed of another set of n bodies in space). Is it possible to have this analog "simulation" more accurate than a digital simulation? The answer is no, because even supposing a true continuum of positions and velocities exists in the first place, in order to reconstruct the system precisely, you would need to pass an infinite amount of information to the "simulation", which is impossible.