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The Computational Power of Continuous Time Neural Networks
 In Proc. SOFSEM'97, the 24th Seminar on Current Trends in Theory and Practice of Informatics, Lecture Notes in Computer Science
, 1995
"... We investigate the computational power of continuoustime neural networks with Hopfieldtype units. We prove that polynomialsize networks with saturatedlinear response functions are at least as powerful as polynomially spacebounded Turing machines. 1 Introduction In a paper published in 1984 [11 ..."
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We investigate the computational power of continuoustime neural networks with Hopfieldtype units. We prove that polynomialsize networks with saturatedlinear response functions are at least as powerful as polynomially spacebounded Turing machines. 1 Introduction In a paper published in 1984 [11], John Hopfield introduced a continuoustime version of the neural network model whose discretetime variant he had discussed in his seminal 1982 paper [10]. The 1984 paper also contains an electronic implementation scheme for the continuoustime networks, and an argument showing that for sufficiently largegain nonlinearities, these behave similarly to the discretetime ones, at least when used as associative memories. The power of Hopfield's discretetime networks as generalpurpose computational devices was analyzed in [17, 18]. In this paper we conduct a similar analysis for networks consisting of Hopfield's continuoustime units; however we are at this stage able to analyze only the gen...
Neural Networks with Real Weights: Analog Computational Complexity
, 1992
"... We pursue a particular approach to analog computation, based on dynamical systems of the type used in neural networks research. Our systems have a fixed structure, invariant in time, corresponding to an unchanging number of "neurons". If allowed exponential time for computation, they turn ..."
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We pursue a particular approach to analog computation, based on dynamical systems of the type used in neural networks research. Our systems have a fixed structure, invariant in time, corresponding to an unchanging number of "neurons". If allowed exponential time for computation, they turn out to have unbounded power. However, under polynomialtime constraints there are limits on their capabilities, though being more powerful than Turing Machines. (A similar but more restricted model was shown to be polynomialtime equivalent to classical digital computation in the previous work [17].) Moreover, there is a precise correspondence between nets and standard nonuniform circuits with equivalent resources, and as a consequence one has lower bound constraints on what they can compute. This relationship is perhaps surprising since our analog devices do not change in any manner with input size. We note that these networks are not likely to solve polynomially NPhard problems, as the equality "p...
The Broad Conception Of Computation
 American Behavioral Scientist
, 1997
"... A myth has arisen concerning Turing's paper of 1936, namely that Turing set forth a fundamental principle concerning the limits of what can be computed by machine  a myth that has passed into cognitive science and the philosophy of mind, to wide and pernicious effect. This supposed principle, ..."
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A myth has arisen concerning Turing's paper of 1936, namely that Turing set forth a fundamental principle concerning the limits of what can be computed by machine  a myth that has passed into cognitive science and the philosophy of mind, to wide and pernicious effect. This supposed principle, sometimes incorrectly termed the 'ChurchTuring thesis', is the claim that the class of functions that can be computed by machines is identical to the class of functions that can be computed by Turing machines. In point of fact Turing himself nowhere endorses, nor even states, this claim (nor does Church). I describe a number of notional machines, both analogue and digital, that can compute more than a universal Turing machine. These machines are exemplars of the class of nonclassical computing machines. Nothing known at present rules out the possibility that machines in this class will one day be built, nor that the brain itself is such a machine. These theoretical considerations undercut a numb...
Upper and Lower Bounds on ContinuousTime Computation
"... We consider various extensions and modifications of Shannon's General Purpose Analog Computer, which is a model of computation by differential equations in continuous time. We show that several classical computation classes have natural analog counterparts, including the primitive recursive fun ..."
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We consider various extensions and modifications of Shannon's General Purpose Analog Computer, which is a model of computation by differential equations in continuous time. We show that several classical computation classes have natural analog counterparts, including the primitive recursive functions, the elementary functions, the levels of the Grzegorczyk hierarchy, and the arithmetical and analytical hierarchies.
The Quantum Fourier Transform and Extensions of the Abelian Hidden Subgroup Problem
, 2002
"... The quantum Fourier transform (QFT) has emerged as the primary tool in quantum algorithms which achieve exponential advantage over classical computation and lies at the heart of the solution to the abelian hidden subgroup problem, of which Shor’s celebrated factoring and discrete log algorithms are ..."
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The quantum Fourier transform (QFT) has emerged as the primary tool in quantum algorithms which achieve exponential advantage over classical computation and lies at the heart of the solution to the abelian hidden subgroup problem, of which Shor’s celebrated factoring and discrete log algorithms are a special case. We begin by addressing various computational issues surrounding the QFT and give improved parallel circuits for both the QFT over a power of 2 and the QFT over an arbitrary cyclic group. These circuits are based on new insight into the relationship between the discrete Fourier transform over different cyclic groups. We then exploit this insight to extend the class of hidden subgroup problems with efficient quantum solutions. First we relax the condition that the underlying hidden subgroup function be distinct on distinct cosets of the subgroup in question and show that this relaxation can be solved whenever G is a finitelygenerated abelian group. We then extend this reasoning to the hidden cyclic subgroup problem over the reals, showing how to efficiently generate the bits of the period of any sufficiently piecewisecontinuous function on ℜ. Finally, we show that this problem of periodfinding over ℜ, viewed as an oracle promise problem, is strictly harder than its integral counterpart. In particular, periodfinding
Lipschitz continuous ordinary differential equations are polynomialspace complete
 Comput. Complexity
, 2010
"... ABSTRACT. In answer to Ko’s question raised in 1983, we show that an initial value problem given by a polynomialtime computable, Lipschitz continuous function can have a polynomialspace complete solution. The key insight is simple: the Lipschitz condition means that the feedback in the differentia ..."
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ABSTRACT. In answer to Ko’s question raised in 1983, we show that an initial value problem given by a polynomialtime computable, Lipschitz continuous function can have a polynomialspace complete solution. The key insight is simple: the Lipschitz condition means that the feedback in the differential equation is weak. We define a class of polynomialspace computation tableaux with equally restricted feedback, and show that they are still polynomialspace complete. The same technique also settles Ko’s two later questions on Volterra integral equations.
How much can analog and hybrid systems be proved (super)Turing
 Applied Mathematics and Computation
, 2006
"... Church thesis and its variants say roughly that all reasonable models of computation do not have more power than Turing Machines. In a contrapositive way, they say that any model with superTuring power must have something unreasonable. Our aim is to discuss how much theoretical computer science can ..."
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Church thesis and its variants say roughly that all reasonable models of computation do not have more power than Turing Machines. In a contrapositive way, they say that any model with superTuring power must have something unreasonable. Our aim is to discuss how much theoretical computer science can quantify this, by considering several classes of continuous time dynamical systems, and by studying how much they can be proved Turing or superTuring. 1
History of "Church's theses" and a manifesto on converting physics into a rigorous algorithmic discipline
, 1999
"... Church's thesis claims that any "reasonable computer" may be simulated by a Turing machine. The "strong" thesis says that the simulation may be performed with only polynomial slowdown. This document is both a history of "Church's thesis"  and particularly of ..."
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Church's thesis claims that any "reasonable computer" may be simulated by a Turing machine. The "strong" thesis says that the simulation may be performed with only polynomial slowdown. This document is both a history of "Church's thesis"  and particularly of the notion that it is a statement about physics  and an opinionated philosophical statement.
Accurate and Precise Computation using Analog VLSI, with Applications to Computer Graphics and Neural Networks
, 1993
"... This thesis develops an engineering practice and design methodology to enable us to use CMOS analog VLSI chips to perform more accurate and precise computation. These techniques form the basis of an approach that permits us to build computer graphics and neural network applications using analog VLSI ..."
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This thesis develops an engineering practice and design methodology to enable us to use CMOS analog VLSI chips to perform more accurate and precise computation. These techniques form the basis of an approach that permits us to build computer graphics and neural network applications using analog VLSI. The nature of the design methodology focuses on defining goals for circuit behavior to be met as part of the design process. To increase the accuracy of analog computation, we develop techniques for creating compensated circuit building blocks, where compensation implies the cancellation of device variations, offsets, and nonlinearities. These compensated building blocks can be used as components in larger and more complex circuits, which can then also be compensated. To this end, we develop techniques for automatically determining appropriate parameters for circuits, using constrained optimization. We also fabricate circuits that implement multidimensional gradient estimation for a grad...
Exact Cover with Light
 New Generation Computing, SpringerVerlag
, 2008
"... www.cs.ubbcluj.ro/∼moltean We suggest a new optical solution for solving the YES/NO version of the Exact Cover problem by using the massive parallelism of light. The idea is to build an optical device which can generate all possible solutions of the problem and then to pick the correct one. In our c ..."
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www.cs.ubbcluj.ro/∼moltean We suggest a new optical solution for solving the YES/NO version of the Exact Cover problem by using the massive parallelism of light. The idea is to build an optical device which can generate all possible solutions of the problem and then to pick the correct one. In our case the device has a graphlike representation and the light is traversing it by following the routes given by the connections between nodes. The nodes are connected by arcs in a special way which lets us to generate all possible covers (exact or not) of the given set. For selecting the correct solution we assign to each item, from the set to be covered, a special integer number. These numbers will actually represent delays induced to light when it passes through arcs. The solution is represented as a subray arriving at a certain moment in the destination node. This will tell us if an exact cover does exist or not.