Universal Neural Field Computation

Peter beim Graben () and Roland Potthast
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Peter beim Graben: Humboldt-Universität zu Berlin, Department of German Studies and Linguistics
Roland Potthast: University of Reading, Department of Mathematics and Statistics

Chapter Chapter 11 in Neural Fields, 2014, pp 299-318 from Springer

Abstract: Abstract Turing machines and Gödel numbers are important pillars of the theory of computation. Thus, any computational architecture needs to show how it could relate to Turing machines and how stable implementations of Turing computation are possible. In this chapter, we implement universal Turing computation in a neural field environment. To this end, we employ the canonical symbologram representation of a Turing machine obtained from a Gödel encoding of its symbolic repertoire and generalized shifts. The resulting nonlinear dynamical automaton (NDA) is a piecewise affine-linear map acting on the unit square that is partitioned into rectangular domains. Instead of looking at point dynamics in phase space, we then consider functional dynamics of probability distribution functions (p.d.f.s) over phase space. This is generally described by a Frobenius-Perron integral transformation that can be regarded as a neural field equation over the unit square as feature space Feature space of a Dynamic Field Theory Dynamic field theory (DFT) (DFT). Solving the Frobenius-Perron equation Frobenius-Perron equation yields that uniform p.d.f.s with rectangular support are mapped onto uniform p.d.f.s with rectangular support, again. We call the resulting representation dynamic field automaton.

Keywords: Turing Machine; Phrase Structure Tree; Denotational Semantic; Neural Field; Complex Data Type (search for similar items in EconPapers)
Date: 2014
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-54593-1_11

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DOI: 10.1007/978-3-642-54593-1_11

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