Results 1  10
of
178
A Survey of Computational Complexity Results in Systems and Control
, 2000
"... The purpose of this paper is twofold: (a) to provide a tutorial introduction to some key concepts from the theory of computational complexity, highlighting their relevance to systems and control theory, and (b) to survey the relatively recent research activity lying at the interface between these fi ..."
Abstract

Cited by 187 (18 self)
 Add to MetaCart
The purpose of this paper is twofold: (a) to provide a tutorial introduction to some key concepts from the theory of computational complexity, highlighting their relevance to systems and control theory, and (b) to survey the relatively recent research activity lying at the interface between these fields. We begin with a brief introduction to models of computation, the concepts of undecidability, polynomial time algorithms, NPcompleteness, and the implications of intractability results. We then survey a number of problems that arise in systems and control theory, some of them classical, some of them related to current research. We discuss them from the point of view of computational complexity and also point out many open problems. In particular, we consider problems related to stability or stabilizability of linear systems with parametric uncertainty, robust control, timevarying linear systems, nonlinear and hybrid systems, and stochastic optimal control.
A General Framework for Adaptive Processing of Data Structures
 IEEE TRANSACTIONS ON NEURAL NETWORKS
, 1998
"... A structured organization of information is typically required by symbolic processing. On the other hand, most connectionist models assume that data are organized according to relatively poor structures, like arrays or sequences. The framework described in this paper is an attempt to unify adaptive ..."
Abstract

Cited by 150 (61 self)
 Add to MetaCart
A structured organization of information is typically required by symbolic processing. On the other hand, most connectionist models assume that data are organized according to relatively poor structures, like arrays or sequences. The framework described in this paper is an attempt to unify adaptive models like artificial neural nets and belief nets for the problem of processing structured information. In particular, relations between data variables are expressed by directed acyclic graphs, where both numerical and categorical values coexist. The general framework proposed in this paper can be regarded as an extension of both recurrent neural networks and hidden Markov models to the case of acyclic graphs. In particular we study the supervised learning problem as the problem of learning transductions from an input structured space to an output structured space, where transductions are assumed to admit a recursive hidden statespace representation. We introduce a graphical formalism for r...
Analog Computation via Neural Networks
 THEORETICAL COMPUTER SCIENCE
, 1994
"... 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 ..."
Abstract

Cited by 96 (10 self)
 Add to MetaCart
(Show Context)
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 [20].) 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 ...
A DNA and restriction enzyme implementation of Turing Machines.
 DIMACS SERIES IN DISCRETE MATHEMATICS AND THEORETICAL COMPUTER SCIENCE
"... Bacteria employ restriction enzymes to cut or restrict DNA at or near specific words in a unique way. Many restriction enzymes cut the two strands of doublestranded DNA at different positions leaving overhangs of singlestranded DNA. Two pieces of DNA may be rejoined or ligated if their terminal ov ..."
Abstract

Cited by 84 (1 self)
 Add to MetaCart
Bacteria employ restriction enzymes to cut or restrict DNA at or near specific words in a unique way. Many restriction enzymes cut the two strands of doublestranded DNA at different positions leaving overhangs of singlestranded DNA. Two pieces of DNA may be rejoined or ligated if their terminal overhangs are complementary. Using these operations fragments of DNA, or oligonucleotides, may be inserted and deleted from a circular piece of plasmid DNA. We propose an encoding for the transition table of a Turing machine in DNA oligonucleotides and a corresponding series of restrictions and ligations of those oligonucleotides that, when performed on circular DNA encoding an instantaneous description of a Turing machine, simulate the operation of the Turing machine encoded in those oligonucleotides. DNA based Turing machines have been proposed by Charles Bennett but they invoke imaginary enzymes to perform the statesymbol transitions. Our approach differs in that every operation can be pe...
Extraction of Rules from Discretetime Recurrent Neural Networks
, 1996
"... The extraction of symbolic knowledge from trained neural networks and the direct encoding of (partial) knowledge into networks prior to training are important issues. They allow the exchange of information between symbolic and connectionist knowledge representations. The focas of this paper is on t ..."
Abstract

Cited by 69 (15 self)
 Add to MetaCart
The extraction of symbolic knowledge from trained neural networks and the direct encoding of (partial) knowledge into networks prior to training are important issues. They allow the exchange of information between symbolic and connectionist knowledge representations. The focas of this paper is on the quality of the rules that are extracted from recurrent neural networks. Discretetime recurrent neural networks can be trained to correctly classify strings of a regular language. Rules defining the learned grammar can be extracted from networks in the form of deterministic finitestate automata (DFAs) by applying clustering algorithms in the output space of recurrent state neurons. Our algorithm can extract different finitestate automata that are consistent with a training set from the same network. We compare the generalization performances of these different models and the trained network and we introduce a heuristic that permits us to choose among the consistent DFAs the model which best approximates the learned regular grammar.
Lower Bounds for the Computational Power of Networks of Spiking Neurons
 NEURAL COMPUTATION
, 1995
"... We investigate the computational power of a formal model for networks of spiking neurons. It is shown that simple operations on phasedifferences between spiketrains provide a very powerful computational tool that can in principle be used to carry out highly complex computations on a small network o ..."
Abstract

Cited by 67 (16 self)
 Add to MetaCart
We investigate the computational power of a formal model for networks of spiking neurons. It is shown that simple operations on phasedifferences between spiketrains provide a very powerful computational tool that can in principle be used to carry out highly complex computations on a small network of spiking neurons. We construct networks of spiking neurons that simulate arbitrary threshold circuits, Turing machines, and a certain type of random access machines with real valued inputs. We also show that relatively weak basic assumptions about the response and thresholdfunctions of the spiking neurons are sufficient in order to employ them for such computations.
Interconnected Automata and Linear Systems: A Theoretical Framework in DiscreteTime
 in Hybrid Systems III: Verification and Control
, 1996
"... . This paper summarizes the definitions and several of the main results of an approach to hybrid systems, which combines finite automata and linear systems, developed by the author in the early 1980s. Some related more recent results are briefly mentioned as well. 1 Introduction  The Need for Hybri ..."
Abstract

Cited by 60 (0 self)
 Add to MetaCart
(Show Context)
. This paper summarizes the definitions and several of the main results of an approach to hybrid systems, which combines finite automata and linear systems, developed by the author in the early 1980s. Some related more recent results are briefly mentioned as well. 1 Introduction  The Need for Hybrid Systems Linear control theory is welldeveloped and highly sophisticated, and is widely applied in areas ranging from aerospace to automotive control. Linear systems provide highly accurate models of many physical systems; furthermore, the use of linear systems as "robust" controllers often allows the tolerance of even severe model nonlinearities and uncertainties. However, it remains a fact that many continuous physical processes cannot be satisfactorily modeled linearly, nor can be adequately regulated by means of linear controllers alone. It has long been recognized that the control of more complex systems than those handled by the linear theory will require switching mechanisms (disco...
Noisy Time Series Prediction using a Recurrent Neural Network and Grammatical Inference
 Machine Learning
, 2001
"... Financial forecasting is an example of a signal processing problem which is challenging due to small sample sizes, high noise, nonstationarity, and nonlinearity. Neural networks have been very successful in a number of signal processing applications. We discuss fundamental limitations and inherent ..."
Abstract

Cited by 58 (0 self)
 Add to MetaCart
(Show Context)
Financial forecasting is an example of a signal processing problem which is challenging due to small sample sizes, high noise, nonstationarity, and nonlinearity. Neural networks have been very successful in a number of signal processing applications. We discuss fundamental limitations and inherent difficulties when using neural networks for the processing of high noise, small sample size signals. We introduce a new intelligent signal processing method which addresses the difficulties. The method proposed uses conversion into a symbolic representation with a selforganizing map, and grammatical inference with recurrent neural networks. We apply the method to the prediction of daily foreign exchange rates, addressing difficulties with nonstationarity, overfitting, and unequal a priori class probabilities, and we find significant predictability in comprehensive experiments covering 5 different foreign exchange rates. The method correctly predicts the direction of change for th...
Complexity of Stability and Controllability of Elementary Hybrid Systems
, 1997
"... this paper, weconsider simple classes of nonlinear systems and provethatbasic questions related to their stabilityandcontrollabilityare either undecidable or computationally intractable (NPhard). As a special case, weconsider a class of hybrid systems in which the state space is partitioned into tw ..."
Abstract

Cited by 57 (9 self)
 Add to MetaCart
this paper, weconsider simple classes of nonlinear systems and provethatbasic questions related to their stabilityandcontrollabilityare either undecidable or computationally intractable (NPhard). As a special case, weconsider a class of hybrid systems in which the state space is partitioned into two halfspaces, and the dynamics in eachhalfspace correspond to a differentlinear system
Finiteness Results for Sigmoidal "Neural" Networks
 In Proceedings of 25th Annual ACM Symposium on the Theory of Computing
, 1993
"... ) Angus Macintyre Mathematical Inst., University of Oxford Oxford OX1 3LB, England, UK Email: ajm@maths.ox.ac.uk Eduardo D. Sontag 3 Dept. of Mathematics, Rutgers University New Brunswick, NJ 08903 Email: sontag@hilbert.rutgers.edu Abstract Proc. 25th Annual Symp. Theory Computing , San Diego, ..."
Abstract

Cited by 47 (12 self)
 Add to MetaCart
) Angus Macintyre Mathematical Inst., University of Oxford Oxford OX1 3LB, England, UK Email: ajm@maths.ox.ac.uk Eduardo D. Sontag 3 Dept. of Mathematics, Rutgers University New Brunswick, NJ 08903 Email: sontag@hilbert.rutgers.edu Abstract Proc. 25th Annual Symp. Theory Computing , San Diego, May 1993 This paper deals with analog circuits. It establishes the finiteness of VC dimension, teaching dimension, and several other measures of sample complexity which arise in learning theory. It also shows that the equivalence of behaviors, and the loading problem, are effectively decidable, modulo a widely believed conjecture in number theory. The results, the first ones that are independent of weight size, apply when the gate function is the "standard sigmoid" commonly used in neural networks research. The proofs rely on very recent developments in the elementary theory of real numbers with exponentiation. (Some weaker conclusions are also given for more general analytic gate functions...