Progress in Theoretical Computer Science

Beyond the Turing Limit

By: Hava Siegelmann

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Paperback | 21 October 2012

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204 Pages

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23.39 x 15.6 x 1.09

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Humanity's most basic intellectual quest to decipher nature and master it has led to numerous efforts to build machines that simulate the world or communi- cate with it [Bus70, Tur36, MP43, Sha48, vN56, Sha41, Rub89, NK91, Nyc92]. The computational power and dynamic behavior of such machines is a central question for mathematicians, computer scientists, and occasionally, physicists. Our interest is in computers called artificial neural networks. In their most general framework, neural networks consist of assemblies of simple processors, or "neurons," each of which computes a scalar activation function of its input. This activation function is nonlinear, and is typically a monotonic function with bounded range, much like neural responses to input stimuli. The scalar value produced by a neuron affects other neurons, which then calculate a new scalar value of their own. This describes the dynamical behavior of parallel updates. Some of the signals originate from outside the network and act as inputs to the system, while other signals are communicated back to the environment and are thus used to encode the end result of the computation.

Industry Reviews

"All of the three primary questions are considered: What computational models can the net simulate (within polynomial bounds)? What are the computational complexity classes that are relevant to the net? How does the net (which, after all, is an analog device) relate to Church's thesis? Moreover the power of the basic model is also analyzed when the domain of reals is replaced by the rationals and the integers."

-Mathematical Reviews

"Siegelmann's book focuses on the computational complexities of neural networks and making this research accessible...the book accomplishes the said task nicely."

---SIAM Review, Vol. 42, No 3.

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