Read e-book online Neural Networks and Analog Computation: Beyond the Turing PDF

By Hava T. Siegelmann

ISBN-10: 146120707X

ISBN-13: 9781461207078

ISBN-10: 1461268753

ISBN-13: 9781461268758

The theoretical foundations of Neural Networks and Analog Computation conceptualize neural networks as a selected form of computing device which includes a number of assemblies of easy processors interconnected in an complicated constitution. analyzing those networks below quite a few source constraints unearths a continuum of computational units, a number of of which coincide with famous classical versions. What emerges is a Church-Turing-like thesis, utilized to the sphere of analog computation, which good points the neural community version rather than the electronic Turing laptop. This new thought can function some extent of departure for the improvement of other, supra-Turing, computational theories. On a mathematical point, the therapy of neural computations enriches the idea of computation but additionally explicated the computational complexity linked to organic networks, adaptive engineering instruments, and similar types from the fields of keep watch over concept and nonlinear dynamics.

The themes lined during this paintings will entice a large readership from numerous disciplines. distinctive care has been taken to provide an explanation for the speculation essentially and concisely. the 1st bankruptcy evaluate s the basic phrases of recent computational concept from the viewpoint of neural networks and serves as a reference for the rest of the ebook. all of the next chapters opens with introductory fabric and proceeds to give an explanation for the chapter’s connection to the advance of the idea. Thereafter, the concept that is outlined in mathematical terms.

Although the concept of a neural community basically arises from biology, many engineering purposes were discovered via hugely idealized and simplified versions of neuron habit. specific components of software were as diversified as explosives detection in airport protection, signature verification, monetary and scientific instances sequence prediction, imaginative and prescient, speech processing, robotics, nonlinear keep an eye on, and sign processing. the focal point in all of those versions is fullyyt at the habit of networks as computer.

The fabric during this e-book should be of curiosity to researchers in various engineering and technologies disciplines. furthermore, the paintings could provide the bottom of a graduate-level seminar in neural networks for machine technological know-how students.

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Get Neural Networks and Analog Computation: Beyond the Turing PDF

The theoretical foundations of Neural Networks and Analog Computation conceptualize neural networks as a specific form of laptop which include a number of assemblies of simple processors interconnected in an elaborate constitution. reading those networks below numerous source constraints finds a continuum of computational units, numerous of which coincide with recognized classical versions.

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1 Cantor-like Encoding of Stacks Let a = al a2 ... be the stack from top to bottom. Assume we were to encode a stack's binary stream a = ala2 ... with 82 into the number a12. This value CHAPTER 3. 34 o NETWORKS WITH RATIONAL WEIGHTS 1/4 1/2 3/4 1 • • • • • • ••• • •• . ft.. •. 1: 4-Cantor set could be held in a neuron since it ranges in [0,1]; however it cannot be decoded efficiently in the sense that the number of operations required to retrieve the first bit is not fixed. Two different values that are close cannot be clearly differentiated with continuous activation functions.

Dr), . , dP), . , d~r») E Ur, and for all i = 1, ... , t, at most one of the d~k) is positive }. We denote by I the set of multi-indices (il' ... ' it), with ij E {O, 1, ... , r}, for all j = 1, ... , t. For each function {3 : Sr,t -+ {O, I} that is signal invariant, there exist vectors {Vi E ~t+2 liE I} and scalars {Ci E ~ liE I} such that, for each (dil ), ... , d~r») E Sr,t and any x E {O, I}, we can write (3(di1), ... L(il, ... • , t , X , and where we are defining d~O) = o. " denotes the inner product in ~t+2.

04(Wp)). Then the machine M is said to output the string WI, and we let 'lj;M (w) = WI. This defines a partial map 'lj;M : {a, 1}+ ---- {a, 1}+, the I/O map of M. 3. 4 via the above encoding map 04. 4 Network with Four Layers To prove the recursive power of neural networks, we simulate a p-stack machine, M. Without loss of generality, we assume that the initial state qI differs from the halting state qH (otherwise the function computed is the identity, which can be easily implemented by a network).

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Neural Networks and Analog Computation: Beyond the Turing Limit by Hava T. Siegelmann

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