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160
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 ..."
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Cited by 133 (20 self)
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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.
Analog computers and recursive functions over the reals
 Journal of Complexity
, 2003
"... In this paper we show that Shannon’s General Purpose Analog Computer (GPAC) is equivalent to a particular class of recursive functions over the reals with the flavour of Kleene’s classical recursive function theory. We first consider the GPAC and several of its extensions to show that all these mode ..."
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Cited by 40 (19 self)
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In this paper we show that Shannon’s General Purpose Analog Computer (GPAC) is equivalent to a particular class of recursive functions over the reals with the flavour of Kleene’s classical recursive function theory. We first consider the GPAC and several of its extensions to show that all these models have drawbacks and we introduce an alternative continuoustime model of computation that solve these problems. We also show that this new model preserve all the significant relations involving the previous models (namely, the equivalence with the differentially algebraic functions). We then continue with the topic of recursive functions over the reals, and we show full connections between functions generated by the model introduced so far and a particular class of recursive functions over the reals. 1
Beyond Turing Machines
"... In this paper we describe and analyze models of problem solving and computation going beyond Turing Machines. Three principles of extending the Turing Machine's expressiveness are identified, namely, by interaction, evolution and infinity. Several models utilizing the above principles are pr ..."
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Cited by 35 (4 self)
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In this paper we describe and analyze models of problem solving and computation going beyond Turing Machines. Three principles of extending the Turing Machine's expressiveness are identified, namely, by interaction, evolution and infinity. Several models utilizing the above principles are presented. Other
Iteration, Inequalities, and Differentiability in Analog Computers
, 1999
"... Shannon's General Purpose Analog Computer (GPAC) is an elegant model of analog computation in continuous time. In this paper, we consider whether the set G of GPACcomputable functions is closed under iteration, that is, whether for any function f(x) 2 G there is a function F (x; t) 2 G s ..."
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Cited by 34 (16 self)
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Shannon's General Purpose Analog Computer (GPAC) is an elegant model of analog computation in continuous time. In this paper, we consider whether the set G of GPACcomputable functions is closed under iteration, that is, whether for any function f(x) 2 G there is a function F (x; t) 2 G such that F (x; t) = f t (x) for nonnegative integers t. We show that G is not closed under iteration, but a simple extension of it is. In particular, if we relax the definition of the GPAC slightly to include unique solutions to boundary value problems, or equivalently if we allow functions x k (x) that sense inequalities in a dierentiable way, the resulting class, which we call G + k , is closed under iteration. Furthermore, G + k includes all primitive recursive functions, and has the additional closure property that if T (x) is in G+k , then any function of x computable by a Turing machine in T (x) time is also.
ContextFree and ContextSensitive Dynamics in Recurrent Neural Networks
, 2000
"... Continuousvalued recurrent neural networks can learn mechanisms for processing contextfree languages. The dynamics of such networks is usually based on damped oscillation around fixed points in state space and requires that the dynamical components are arranged in certain ways. It is shown tha ..."
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Cited by 29 (6 self)
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Continuousvalued recurrent neural networks can learn mechanisms for processing contextfree languages. The dynamics of such networks is usually based on damped oscillation around fixed points in state space and requires that the dynamical components are arranged in certain ways. It is shown that qualitatively similar dynamics with similar constraints hold for a n b n c n , a contextsensitive language. The additional difficulty with a n b n c n , compared with the contextfree language a n b n , consists of "counting up" and "counting down" letters simultaneously. The network solution is to oscillate in two principal dimensions, one for counting up and one for counting down. This study focuses on the dynamics employed by the Sequential Cascaded Network, in contrast with the Simple Recurrent Network, and the use of Backpropagation Through Time. Found solutions generalize well beyond training data, however, learning is not reliable. The contribution of this ...
Principles for an Integrated Connectionist/Symbolic Theory of Higher Cognition
, 1992
"... The main claim of this paper is that connectionism offers cognitive science a number of excellent opportunities for turning methodological, theoretical. and metatheoretica! schisms into powerfnl integrationsopportunities for forging constructive synergy out of the destructive interference whic ..."
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Cited by 23 (4 self)
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The main claim of this paper is that connectionism offers cognitive science a number of excellent opportunities for turning methodological, theoretical. and metatheoretica! schisms into powerfnl integrationsopportunities for forging constructive synergy out of the destructive interference which plagues the field. The paper begins with an analysis of the rifts in tile field and what it would take to overcome them. We argue that while connectionism ha,s often contributed to the deepexLing of these schisms, ]t is nonetheless possible to turn this trend aroundpossible for connectionism to play a central role in a unification of cognitive science. Essential o this process is the development of strong theoretical principles founded (in part) on connectionist computation; a main goal of this paper is to demonstrate that such principles are indeed within the reach of a connectionistgrounded theory of cognition. The enterprise rests on a willingness to entertain, analyze, and extend characterizations of cognitive problems, and hypothesized solutions, which are deliberately overly simple and generalin order to discover the insights they can offer through mathematical analyses which this simplicity and generality are makes possible. In this
Polynomial differential equations compute all real computable functions on computable compact intervals
, 2007
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Computability with Polynomial Differential Equations
, 2007
"... In this paper, we show that there are Initial Value Problems defined with polynomial ordinary differential equations that can simulate universal Turing machines in the presence of bounded noise. The polynomial ODE defining the IVP is explicitly obtained and the simulation is performed in real time. ..."
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Cited by 22 (14 self)
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In this paper, we show that there are Initial Value Problems defined with polynomial ordinary differential equations that can simulate universal Turing machines in the presence of bounded noise. The polynomial ODE defining the IVP is explicitly obtained and the simulation is performed in real time.
Some recent developments on Shannon’s general purpose analog computer
 Mathematical Logic Quarterly
"... This paper revisits one of the first models of analog computation, the General Purpose Analog Computer (GPAC). In particular, we restrict our attention to the improved model presented in [11] and we show that it can be further refined. With this we prove the following: (i) the previous model can be ..."
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Cited by 20 (7 self)
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This paper revisits one of the first models of analog computation, the General Purpose Analog Computer (GPAC). In particular, we restrict our attention to the improved model presented in [11] and we show that it can be further refined. With this we prove the following: (i) the previous model can be simplified; (ii) it admits extensions having close connections with the class of smooth continuous time dynamical systems. As a consequence, we conclude that some of these extensions achieve Turing universality. Finally, it is shown that if we introduce a new notion of computability for the GPAC, based on ideas from computable analysis, then one can compute transcendentally transcendental functions such as the Gamma function or Riemann’s Zeta function. 1
Neuromodulation of Reactive Sensorimotor Mappings as a ShortTerm Memory Mechanism
 in Delayed Response Tasks,” Adaptive Behavior
, 2002
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