20 Apr 2009

Argument from Noise, 4. Three Arguments, 4.2 Argument from Measure, in Schonbein, Cognition and the Power of Continuous Dynamical Systems

by Corry Shores
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Schonbein's Cognition and the Power of Continuous Dynamical Systems, Entry Directory][Nick Bostrom & Anders Sandberg argue for digital computation instead of analog for simulating human cognition. They base their contention in part on the "argument from noise." The entries in this series summarize Schonbein's defense of that argument.]

Whit Schonbein

Cognition and the Power
of Continuous Dynamical Systems

4. Three Arguments against AANNs

4.2 The Argument from Measure

Fields argues that it does not matter if analog systems are more precise. On account of certain physical limitations, our model can only correspond one-to-one with the system in a digital fashion. Schonbein formulates the argument thus:

1. If a virtual machine M (e.g. a Turing machine) is to be realized in a physical system S, it must be possible to put states of M into correspondence with states of S. [We have a modeled design for a pipe-joint that doubles the amount of incoming water.]
2. In order to construct this mapping, the states of S must be measured. [Our pipe-joint needs to measure incoming water-volume in order to know what amount to add to it.]
3. Measuring a state of S involves adding energy to the system. [We will have to use an electrical device to measure the incoming water amounts.]
4. The more precise our measurements are – the « smaller » the state being measured is – the more probable it is that our measurements will influence the behavior of the system. Thus measuring a state will result in S not making the transition to the state it would have entered had it not been perturbed. [If we have to measure the water to absolute precision, that will take a computer an enormous amount of time, maybe even infinite time. So such an analog measurement will hinder the operation of the pipe-fixture.]
5. Therefore, there is an upper bound on the precision we can achieve in measuring the states of S. [Our device can only be so precise without disrupting its own operation.]
6. Therefore, the only correspondences that can be constructed are those that possess a finite number of states. [Our model or design for the pipe-fixture will potentially be able to handle infinitely many variations in water volume. However, we will only be able to correlate these infinite possibilities to a finite number of possible states in our physical device.]
7. Therefore, a Turing machine (TM) can compute any machine realized by S. [Thus, some digital pipe device of sufficient computational/measuring ability can be just as precise as an analog one, even if our design/model itself can theoretically deal with continuous variables.]

According to Fields, so long as we accept premises 3 and 4, « a continuous dynamical system cannot, even in principle, exhibit behavior that cannot be simulated by a universal Turing machine. » (65a)

Schonbein finds two problems with Fields' argument.

1) We cannot follow Fields to his conclusion that a TM machine can compute anything that a continuous system can compute. For, we know already that analog artificial neural networks (AANNs) are super-Turing-computable. They can compute functions that Turing machines cannot.

2) We should not think that our inability to measure the computed variations poses any limit on the machine's ability to compute them. [Two rivers merge. The juncture adds the volumes of both tributaries. We cannot measure the inputs and outputs exactly. But that does not stop the juncture from combining every smallest bit from both.]
it is fallacious to infer from our lack of ability to measure the states of a system to the conclusion that those unmeasured (or unmeasurable) states are not relevant to the behavior of the system – to do so is to confuse our metaphysics with our epistemology. (65b)
We are not interested so much in measuring the systems. We firstly want just to model it.

Schonbein, Whit. "Cognition and the Power of Continuous Dynamical Systems." Mind and Machines, Springer, (2005) 15: pp. 57-71.
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