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Computational Complexity Of Neural Networks: A Survey
, 1994
"... . We survey some of the central results in the complexity theory of discrete neural networks, with pointers to the literature. Our main emphasis is on the computational power of various acyclic and cyclic network models, but we also discuss briefly the complexity aspects of synthesizing networks fr ..."
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Cited by 22 (6 self)
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. We survey some of the central results in the complexity theory of discrete neural networks, with pointers to the literature. Our main emphasis is on the computational power of various acyclic and cyclic network models, but we also discuss briefly the complexity aspects of synthesizing networks from examples of their behavior. CR Classification: F.1.1 [Computation by Abstract Devices]: Models of Computationneural networks, circuits; F.1.3 [Computation by Abstract Devices ]: Complexity Classescomplexity hierarchies Key words: Neural networks, computational complexity, threshold circuits, associative memory 1. Introduction The currently again very active field of computation by "neural" networks has opened up a wealth of fascinating research topics in the computational complexity analysis of the models considered. While much of the general appeal of the field stems not so much from new computational possibilities, but from the possibility of "learning", or synthesizing networks...
Complexity Issues in Discrete Hopfield Networks
, 1994
"... We survey some aspects of the computational complexity theory of discretetime and discretestate Hopfield networks. The emphasis is on topics that are not adequately covered by the existing survey literature, most significantly: 1. the known upper and lower bounds for the convergence times of Hopfi ..."
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Cited by 18 (4 self)
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We survey some aspects of the computational complexity theory of discretetime and discretestate Hopfield networks. The emphasis is on topics that are not adequately covered by the existing survey literature, most significantly: 1. the known upper and lower bounds for the convergence times of Hopfield nets (here we consider mainly worstcase results); 2. the power of Hopfield nets as general computing devices (as opposed to their applications to associative memory and optimization); 3. the complexity of the synthesis ("learning") and analysis problems related to Hopfield nets as associative memories. Draft chapter for the forthcoming book The Computational and Learning Complexity of Neural Networks: Advanced Topics (ed. Ian Parberry).
Neural Networks and Complexity Theory
 In Proc. 17th International Symposium on Mathematical Foundations of Computer Science
, 1992
"... . We survey some of the central results in the complexity theory of discrete neural networks, with pointers to the literature. 1 Introduction The recently revived field of computation by "neural" networks provides the complexity theorist with a wealth of fascinating research topics. While much of ..."
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Cited by 14 (4 self)
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. We survey some of the central results in the complexity theory of discrete neural networks, with pointers to the literature. 1 Introduction The recently revived field of computation by "neural" networks provides the complexity theorist with a wealth of fascinating research topics. While much of the general appeal of the field stems not so much from new computational possibilities, but from the possibility of "learning", or synthesizing networks directly from examples of their desired inputoutput behavior, it is nevertheless important to pay attention also to the complexity issues: firstly, what kinds of functions are computable by networks of a given type and size, and secondly, what is the complexity of the synthesis problems considered. In fact, inattention to these issues was a significant factor in the demise of the first stage of neural networks research in the late 60's, under the criticism of Minsky and Papert [51]. The intent of this paper is to survey some of the centra...
The Computational Power of Discrete Hopfield Nets with Hidden Units
 Neural Computation
, 1996
"... We prove that polynomial size discrete Hopfield networks with hidden units compute exactly the class of Boolean functions PSPACE/poly, i.e., the same functions as are computed by polynomial spacebounded nonuniform Turing machines. As a corollary to the construction, we observe also that networks wi ..."
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Cited by 11 (6 self)
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We prove that polynomial size discrete Hopfield networks with hidden units compute exactly the class of Boolean functions PSPACE/poly, i.e., the same functions as are computed by polynomial spacebounded nonuniform Turing machines. As a corollary to the construction, we observe also that networks with polynomially bounded interconnection weights compute exactly the class of functions P/poly, i.e., the class computed by polynomial timebounded nonuniform Turing machines.
An Overview Of The Computational Power Of Recurrent Neural Networks
 Proceedings of the 9th Finnish AI Conference STeP 2000{Millennium of AI, Espoo, Finland (Vol. 3: "AI of Tomorrow": Symposium on Theory, Finnish AI Society
, 2000
"... INTRODUCTION The two main streams of neural networks research consider neural networks either as a powerful family of nonlinear statistical models, to be used in for example pattern recognition applications [6], or as formal models to help develop a computational understanding of the brain [10]. His ..."
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Cited by 10 (3 self)
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INTRODUCTION The two main streams of neural networks research consider neural networks either as a powerful family of nonlinear statistical models, to be used in for example pattern recognition applications [6], or as formal models to help develop a computational understanding of the brain [10]. Historically, the brain theory interest was primary [32], but with the advances in computer technology, the application potential of the statistical modeling techniques has shifted the balance. 1 The study of neural networks as general computational devices does not strictly follow this division of interests: rather, it provides a general framework outlining the limitations and possibilities aecting both research domains. The prime historic example here is obviously Minsky's and Papert's 1969 study of the computational limitations of singlelayer perceptrons [34], which was a major inuence in turning away interest from neural network learning to symbolic AI techniques for more
On the Computational Power of Discrete Hopfield Nets
 In: Proc. 20th International Colloquium on Automata, Languages, and Programming
, 1993
"... . We prove that polynomial size discrete synchronous Hopfield networks with hidden units compute exactly the class of Boolean functions PSPACE/poly, i.e., the same functions as are computed by polynomial spacebounded nonuniform Turing machines. As a corollary to the construction, we observe also th ..."
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Cited by 7 (4 self)
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. We prove that polynomial size discrete synchronous Hopfield networks with hidden units compute exactly the class of Boolean functions PSPACE/poly, i.e., the same functions as are computed by polynomial spacebounded nonuniform Turing machines. As a corollary to the construction, we observe also that networks with polynomially bounded interconnection weights compute exactly the class of functions P/poly. 1 Background Recurrent, or cyclic, neural networks are an intriguing model of massively parallel computation. In the recent surge of research in neural computation, such networks have been considered mostly from the point of view of two types of applications: pattern classification and associative memory (e.g. [16, 18, 21, 24]), and combinatorial optimization (e.g. [1, 7, 20]). Nevertheless, recurrent networks are capable also of more general types of computation, and issues of what exactly such networks can compute, and how they should be programmed, are becoming increasingly topica...
Combinatorial Optimization: A Survey
, 1993
"... This paper is a chapter of the forthcoming Handbook of Combinatorics, to be published by NorthHolland. It surveys the basic techniques and methods in combinatorial optimization. We organize our material according to the fundamental algorithmic techniques and illustrate them on problems to which the ..."
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Cited by 3 (0 self)
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This paper is a chapter of the forthcoming Handbook of Combinatorics, to be published by NorthHolland. It surveys the basic techniques and methods in combinatorial optimization. We organize our material according to the fundamental algorithmic techniques and illustrate them on problems to which these methods have been applied successfully. Special attention is given to approximation algorithms and fast (primal and dual) heuristics.
EnergyBased Computation with Symmetric Hopfield Nets
"... We propose a unifying approach to the analysis of computational aspects of symmetric Hopfield nets which is based on the concept of "energy source". Within this framework we present different results concerning the computational power of various Hopfield model classes. It is shown that polynomial ..."
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Cited by 2 (0 self)
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We propose a unifying approach to the analysis of computational aspects of symmetric Hopfield nets which is based on the concept of "energy source". Within this framework we present different results concerning the computational power of various Hopfield model classes. It is shown that polynomialtime computations by nondeterministic Turing machines can be reduced to the process of minimizing the energy in Hopfield nets (the MIN ENERGY problem). Furthermore, external and internal sources of energy are distinguished. The external sources include e.g. energizing inputs from socalled Hopfield languages, and also certain external oscillators that prove finite analog Hopfield nets to be computationally Turing universal. On the other hand, the internal source of energy can be implemented by a symmetric clock subnetwork producing an exponential number of oscillations which are used to energize the simulation of convergent asymmetric networks by Hopfield nets. This shows that infinite families of polynomialsize Hopfield nets compute the complexity class PSPACE/poly. A special attention is paid to generalizing these results for analog states and continuous time to point out alternative sources of efficient computation. 1
General Purpose Computation with Neural Networks: A Survey of Complexity Theoretic Results
, 2003
"... We survey and summarize the existing literature on the computational aspects of neural network models, by presenting a detailed taxonomy of the various models according to their complexity theoretic characteristics. The criteria of classi cation include e.g. the architecture of the network (fee ..."
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Cited by 2 (0 self)
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We survey and summarize the existing literature on the computational aspects of neural network models, by presenting a detailed taxonomy of the various models according to their complexity theoretic characteristics. The criteria of classi cation include e.g. the architecture of the network (feedforward vs. recurrent), time model (discrete vs. continuous), state type (binary vs. analog), weight constraints (symmetric vs. asymmetric), network size ( nite nets vs. in  nite families), computation type (deterministic vs. probabilistic), etc.