20 Apr 2009

Argument from Noise, 4. Three Arguments, 4.3 The Argument from Noise, in Schonbein, Cognition and the Power of Continuous Dynamical Systems

[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

4.3 The Argument from Noise

Previously we saw Fields argue that measurement in analog systems causes disruptions in operation. Schonbein disagreed. However, noise will cause this problem. (65c)

Neuronal information transmission involves an element of intrinsic noise. In fact, analog artificial neural networks (AANNs) can still function normally even without absolute precision.
If we conceive of the ideal case of inter-node communication in an AANN as involving infinite-precision weights, then successful communication in real-world contexts (i.e., despite noise) indicates that not all the precision provided by the posited real values is required for the system to function normally. This is because noise renders the lesser-significant bits useless (i.e., unreliable) for the purposes of carrying out the relevant computation, and therefore these bits can be ignored. (65d)
Hence analog noise limits its computational power.
If the presence of noise removes the utility of lesser-significant digits for carrying information, we should expect the computational power of systems that rely on those digits to be reduced in the presence of noise. Indeed, AANNs subjected to noise are reduced in computational power to finite automata, and often to a power less than that of finite automata. (65-66)
When subjected to typical noise, AANNs do not exhibit computational power greater than TMs, and will probably be reduced to levels below that of TMs. In short, AANNs only enjoy super-Turing-computability under ideal circumstances, and such circumstances are not what we find in actual cognitive systems. (66)

Schonbein addresses two possible responses to this attack on analog.

1) The Noise does not Matter

Say we want to reduce an AANN's computational power to the level of a Turing machine. To do so, we need to designate points along its operation. At these points the machine is in a certain determinate state at a determinate time. All the information in between we disrupt with noise. The Turing machine will be able to compute certain functions. The AANN will be able to compute no more than the Turing machine if the noise renders useless the variations between points.
However, noise is random. Therefore, we cannot determine a priori which bits will be ineffectual: At one time it may be at the nth bit. So (the response goes) we cannot segment the space of the network in such a way as to yield a set of discrete states, as required by computation, traditionally defined, since the boundaries of the desired state are constantly shifting. (66b)
[We cannot reduce AANNs to Turing machines, because there is no formula to determine how random noise can disqualify certain bits of information. Such a method might always inevitably make the AANN less computationally powerful than Turing machines, because the noise eliminates needed data.]

However, a system will not often use reliable information. So if noise makes some information unusable, it will not use it. So consider the method above when we disrupt the AANN system with noise at certain points. The argument above was that this will make the system less powerful than a Turing machine, rather than equal. For, the noise might disrupt vital information rather than inconsequential information. However, Schonbein's point is that if the information proves unreliable, the system will not use it anyway. So the system will not be any more disrupted. We could in fact render it equally powerful as the Turing machine using this method.

2) The Noise is Necessary

There is a second defense that AANN systems are super-Turing computable despite noise. We might argue that noise is a necessary part of cognition. In fact, perhaps the very reason we need analog is because it has noise.

The problem with this response is that it treats random noise interference and the relevant information as equally important. However, when we explain the relation between beliefs and desires, we need to maintain this distinction. Bob desires milk. He believes it is at the store. So Bob goes to the store and returns with milk. No part of his decision was decided by noise.

Thus this response « fails to honor a basic commitment of belief/desire psychology. » (67a) This field of study « explains behavior by showing how particular events fall under generalizations. » Consider Bob. He operates according to a relevant generalization : « anyone who desires something and believes it can be obtained at a certain location will take steps to reach that location, » with all things being equal. But if noise is necessary to how we make decisions, then we cannot articulate a standard generalization for what guides our decisions. We might as well just say, « if someone knows where to get what they want, only God knows what they will do about it. »

So we cannot argue that noise is irrelevant or necessary. But analog is only superior to digital when noise is either irrelevant or necessary. So there are no grounds to say that analog is superior to digital. (67c-d)

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