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A theory of complexity for continuous time systems
 Journal of Complexity
, 2002
"... We present a model of computation with ordinary differential equations (ODEs) which converge to attractors that are interpreted as the output of a computation. We introduce a measure of complexity for exponentially convergent ODEs, enabling an algorithmic analysis of continuous time flows and their ..."
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We present a model of computation with ordinary differential equations (ODEs) which converge to attractors that are interpreted as the output of a computation. We introduce a measure of complexity for exponentially convergent ODEs, enabling an algorithmic analysis of continuous time flows and their comparison with discrete algorithms. We define polynomial and logarithmic continuous time complexity classes and show that an ODE which solves the maximum network flow problem has polynomial time complexity. We also analyze a simple flow that solves the Maximum problem in logarithmic time. We conjecture that a subclass of the continuous P is equivalent to the classical P. 2001 Elsevier Science (USA) Key Words: theory of analog computation; dynamical systems.
Grounding Analog Computers
 Think
, 1993
"... Although analog computation was eclipsed by digital computation in the second half of the twentieth century, it is returning as an important alternative computing technology. Indeed, as explained in this report, theoretical results imply that analog computation can escape from the limitations of dig ..."
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Cited by 12 (7 self)
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Although analog computation was eclipsed by digital computation in the second half of the twentieth century, it is returning as an important alternative computing technology. Indeed, as explained in this report, theoretical results imply that analog computation can escape from the limitations of digital computation. Furthermore, analog computation has emerged as an important theoretical framework for discussing computation in the brain and other natural systems. The report (1) summarizes the fundamentals of analog computing, starting with the continuous state space and the various processes by which analog computation can be organized in time; (2) discusses analog computation in nature, which provides models and inspiration for many contemporary uses of analog computation, such as neural networks; (3) considers generalpurpose analog computing, both from a theoretical perspective and in terms of practical generalpurpose analog computers; (4) discusses the theoretical power of
Neural and SuperTuring Computing
 Minds and Machines
, 2003
"... Abstract. “Neural computing ” is a research field based on perceiving the human brain as an information system. This system reads its input continuously via the different senses, encodes data into various biophysical variables such as membrane potentials or neural firing rates, stores information us ..."
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Cited by 11 (0 self)
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Abstract. “Neural computing ” is a research field based on perceiving the human brain as an information system. This system reads its input continuously via the different senses, encodes data into various biophysical variables such as membrane potentials or neural firing rates, stores information using different kinds of memories (e.g., shortterm memory, longterm memory, associative memory), performs some operations called “computation”, and outputs onto various channels, including motor control commands, decisions, thoughts, and feelings. We show a natural model of neural computing that gives rise to hypercomputation. Rigorous mathematical analysis is applied, explicating our model’s exact computational power and how it changes with the change of parameters. Our analog neural network allows for supraTuring power while keeping track of computational constraints, and thus embeds a possible answer to the superiority of the biological intelligence within the framework of classical computer science. We further propose it as standard in the field of analog computation, functioning in a role similar to that of the universal Turing machine in digital computation. In particular an analog of the ChurchTuring thesis of digital computation is stated where the neural network takes place of the Turing machine. Key words: analog computation, computational theory, chaos, dynamical systems, neuron
SuperTuring or NonTuring? Extending the Concept of Computation
"... “Hypercomputation ” is often defined as transcending Turing computation in the sense of computing a larger class of functions than can Turing machines. While this possibility is important and interesting, this paper argues that there are many other important senses in which we may “transcend Turing ..."
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Cited by 8 (8 self)
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“Hypercomputation ” is often defined as transcending Turing computation in the sense of computing a larger class of functions than can Turing machines. While this possibility is important and interesting, this paper argues that there are many other important senses in which we may “transcend Turing computation. ” Turing computation, like all models, exists in a frame of relevance, which underlies the assumptions on which it rests and the questions that it is suited to answer. Although appropriate in many circumstances, there are other important applications of the idea of computation for which this model is not relevant. Therefore we should supplement it with new models based on different assumptions and suited to answering different questions. In alternative frames of relevance, including natural computation and nanocomputation, the central issues include realtime response, continuity, indeterminacy, and parallelism. Once we understand computation in a broader sense, we can see new possibilities for using physical processes to achieve computational goals, which will increase in importance as we approach the limits of electronic binary logic. Key words: hypercomputation, ChurchTuring thesis, natural computation, theory of computation, model of computation, Turing computation,
The general purpose analog computer and computable analysis are two equivalent paradigms of analog computation
 Theory and Applications of Models of Computation, Third International Conference, TAMC 2006
, 2006
"... Abstract. In this paper we revisit one of the first models of analog computation, Shannon’s General Purpose Analog Computer (GPAC). The GPAC has often been argued to be weaker than computable analysis. As main contribution, we show that if we change the notion of GPACcomputability in a natural way, ..."
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Abstract. In this paper we revisit one of the first models of analog computation, Shannon’s General Purpose Analog Computer (GPAC). The GPAC has often been argued to be weaker than computable analysis. As main contribution, we show that if we change the notion of GPACcomputability in a natural way, we compute exactly all real computable functions (in the sense of computable analysis). Moreover, since GPACs are equivalent to systems of polynomial differential equations then we show that all real computable functions can be defined by such models. 1
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.
Will the robot do the right thing
 In Proc. Artificial Intelligence 94, 255
, 1994
"... Constraint Nets have been developed as an algebraic online computational model of robotic systems. A robotic system consists of a robot and its environment. A robot consists of a plant and a controller. A constraint net is used to model the dynamics of each component and the complete system. The ov ..."
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Cited by 5 (4 self)
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Constraint Nets have been developed as an algebraic online computational model of robotic systems. A robotic system consists of a robot and its environment. A robot consists of a plant and a controller. A constraint net is used to model the dynamics of each component and the complete system. The overall behavior of the system emerges from the coupling of each of its components. The question posed in the title is decomposed into two questions: rst, what is the right thing? second, how does one guarantee the robot will do it? We answer these questions by establishing a formal approach to the speci cation and veri cation of robotic behaviors. In particular, we develop a realtime temporal logic for the speci cation of behaviors and a new veri cation method, based on timed 8automata, for showing that the constraint net model of a robotic system satis es the specication of a desired global behavior of the system. Since the constraint net model of the controller can also serve as the online controller of the real plant, this is a practical way of building wellbehaved robots. Running examples of a coordinator for a twohanded robot performing an assembly task and a reactive maze traveler illustrate the approach. Shell Canada Fellow, Canadian Institute for Advanced Research 1
C.: Empty space computes: The evolution of an unconventional supercomputer
 ACM International Conference on Computing Frontiers
, 2006
"... Lee A. Rubel defined the extended analog computer to remove the limitations of Shannon’s general purpose analog computer. Partial differential equation solvers were a “quintessential ” part of Rubel’s theoretical machine. These components have been implemented with “empty space ” (VLSI circuits with ..."
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Lee A. Rubel defined the extended analog computer to remove the limitations of Shannon’s general purpose analog computer. Partial differential equation solvers were a “quintessential ” part of Rubel’s theoretical machine. These components have been implemented with “empty space ” (VLSI circuits without transistors), as well as conductive plastic. For the past decade research at Indiana University has explored the design and applications of extended analog computers. The machines have become increasingly sophisticated and flexible. The “empty ” computational area solves partial differential equations. The rest of the space includes fuzzy logic elements, configuration memory and input/output channels. This paper describes the theoretical definition, architecture and implementation of these unconventional computers. Two applications are described in detail. Rubel’s model can be viewed as an abstract specification for a distributed supercomputer. We close with a description of this inexpensive 64node supercomputer that is based on our current single processor, and which has been placed into the public domain. The next step is to implement the improved architecture in VLSI, and seek computation speeds approaching trillions of partial differential equations per second.
Computational bounds on polynomial differential equations
, 2008
"... In this paper we study from a computational perspective some properties of the solutions of polynomial ordinary differential equations. We consider elementary (in the sense of Analysis) discretetime dynamical systems satisfying certain criteria of robustness. We show that those systems can be simul ..."
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Cited by 4 (3 self)
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In this paper we study from a computational perspective some properties of the solutions of polynomial ordinary differential equations. We consider elementary (in the sense of Analysis) discretetime dynamical systems satisfying certain criteria of robustness. We show that those systems can be simulated with elementary and robust continuoustime dynamical systems which can be expanded into fully polynomial ordinary differential equations in Q[π]. This sets a computational lower bound on polynomial ODEs since the former class is large enough to include the dynamics of arbitrary Turing machines. We also apply the previous methods to show that the problem of determining whether the maximal interval of definition of an initialvalue problem defined with polynomial ODEs is bounded or not is in general undecidable, even if the parameters of the system are computable and comparable and if the degree of the corresponding polynomial is at most 56. Combined with earlier results on the computability of solutions of polynomial ODEs, one can conclude that there is from a computational point of view a close connection between these systems and Turing machines.