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[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
3. Beyond Traditional Computation
Jerry Fodor is an example of a classicist who argues that Turing machines suffice to model cognition. Others, for example, Van Gelder, contest that we need a more computationally powerful model for human cognition. For, cognitive behavior might be "much more subtle and complex than the standard concept of representation [and therefore computation] can handle." (qt. 60b)
Horgan explains that the alternative dynamical systems approach "'involves a potentially more powerful kind of mathematics' that can deal with a cognition system that realizes 'a function so complex and subtle that it is not tractably computable.'" (qt60bc)
It is not entirely clear what "more powerful" means in these claims. We could use Schonbein's "computational hierarchy." We list which computers are capable of computing which functions. The more powerful ones are those that can compute more functions than the others. This is probably because they have greater resources to do so.
Those who argue for non-classical models also advocate continuous rather than discrete values for defining cognitive models. Van Gelder, for example, notes that "differential equations utilize continuous values." (60c) And Horgan explains that "the mathematics of dynamical systems is fundamentally continuous mathematics rather than discrete mathematics." (60c)
Turing required that computational states be discrete. But the values in continuous systems make use of "infinitely precise values" that can "differ by an arbitrarily small degree." Hence analog systems differ fundamentally from traditional digital automata. (60d)
Continuous systems, then, are "non-computational" in the sense that they can compute functions that are not Turing-computable. Furthermore, Horgan believes that "suitably modified neural networks are the sorts of things that could realize such 'non-computational' systems." (60d)
He writes,
dynamical systems whose transitions are computable are actually a relative rarity, and it is certainly possible for noncomputable dynamical systems to be subserved by neural networks – at least if the networks are made more analog in nature by letting the nodes take on a continuous range of activation values, and/or letting them update themselves instantaneously rather than by discrete time steps. (qt61a, emphasis mine)
In fact, it has already been shown that analog artificial neural networks (AANNs) are more computationally powerful than Turing machines. They are super-Turing-computable.
these networks are relatively simple: They are first-order, recurrent, and synchronously updated; they use saturated-linear activation functions; and have a finite number of nodes. (61b)
If such a system used just numerical quantities that were rational numbers (and hence are specifiable as ratios between integers), then they would be Turing equivalent. "However, if one allows for continuous weights and activations, AANNs are capable of computing functions not computable by TMs" (61c)
Dynamicists claim that we need computational models more powerful than Turing machines in order to model cognition. "Part of this claim revolves around the use of formalizations that make use of continuous values, which implies that these systems cannot be understood in terms of classical computation." (61c) Because certain neural networks can handle continuous variables, it is "at least logically possible" to produce super-Turing-equivalent systems.
However, a number of arguments have been offered to the conclusion that AANNs (and other continuously valued automata) are nomologically impossible, i.e., not realizable by physical systems in our world. (61d)
Schonbein will now examine three such arguments.
Schonbein, Whit. "Cognition and the Power of Continuous Dynamical Systems." Mind and Machines, Springer, (2005) 15: pp. 57-71.
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