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16
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 such t ..."
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Cited by 29 (15 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.
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 function ..."
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Cited by 8 (2 self)
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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.
An analog characterization of the subrecursive functions
 PROC. 4TH CONFERENCE ON REAL NUMBERS AND COMPUTERS
, 2000
"... We study a restricted version of Shannon’s General Purpose Analog Computer in which we only allow the machine to solve linear differential equations. This corresponds to only allowing local feedback in the machine’s variables. We show that if this computer is allowed to sense inequalities in a dif ..."
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Cited by 6 (1 self)
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We study a restricted version of Shannon’s General Purpose Analog Computer in which we only allow the machine to solve linear differential equations. This corresponds to only allowing local feedback in the machine’s variables. We show that if this computer is allowed to sense inequalities in a differentiable way, then it can compute exactly the elementary functions. Furthermore, we show that if the machine has access to an oracle which computes a function f(x) with a suitable growth as x goes to infinity, then it can compute functions on any given level of the Grzegorczyk hierarchy. More precisely, we show that the model contains exactly the nth level of the Grzegorczyk hierarchy if it is allowed to solve n − 3 nonlinear differential equations of a certain kind. Therefore, we claim that there is a close connection between analog complexity classes, and the dynamical systems that compute them, and classical sets of subrecursive functions.
Decidability and universality in symbolic dynamical systems
 Fund. Inform
"... Abstract. Many different definitions of computational universality for various types of dynamical systems have flourished since Turing’s work. We propose a general definition of universality that applies to arbitrary discrete time symbolic dynamical systems. Universality of a system is defined as un ..."
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Cited by 5 (0 self)
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Abstract. Many different definitions of computational universality for various types of dynamical systems have flourished since Turing’s work. We propose a general definition of universality that applies to arbitrary discrete time symbolic dynamical systems. Universality of a system is defined as undecidability of a modelchecking problem. For Turing machines, counter machines and tag systems, our definition coincides with the classical one. It yields, however, a new definition for cellular automata and subshifts. Our definition is robust with respect to initial condition, which is a desirable feature for physical realizability. We derive necessary conditions for undecidability and universality. For instance, a universal system must have a sensitive point and a proper subsystem. We conjecture that universal systems have infinite number of subsystems. We also discuss the thesis according to which computation should occur at the ‘edge of chaos ’ and we exhibit a universal chaotic system. 1.
Iteration, Inequalities, and Dierentiability in Analog Computers
"... . 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 t ..."
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Cited by 4 (3 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 denition 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. Key words: Analog computation, recursion theory, iteration, die...
Computational universality in symbolic dynamical systems
 Fundamenta Informaticae
"... Abstract. Many different definitions of computational universality for various types of systems have flourished since Turing’s work. In this paper, we propose a general definition of universality that applies to arbitrary discrete time symbolic dynamical systems. For Turing machines and tag systems, ..."
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Cited by 3 (0 self)
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Abstract. Many different definitions of computational universality for various types of systems have flourished since Turing’s work. In this paper, we propose a general definition of universality that applies to arbitrary discrete time symbolic dynamical systems. For Turing machines and tag systems, our definition coincides with the usual notion of universality. It however yields a new definition for cellular automata and subshifts. Our definition is robust with respect to noise on the initial condition, which is a desirable feature for physical realizability. We derive necessary conditions for universality. For instance, a universal system must have a sensitive point and a proper subsystem. We conjecture that universal systems have an infinite number of subsystems. We also discuss the thesis that computation should occur at the ‘edge of chaos ’ and we exhibit a universal chaotic system. 1
Solving SAT with bilateral computing
 Romanian Journal of Information Science and Technology
"... We solve a simple instance of the SAT problem using a natural physical computing system based on fluid mechanics. The natural system functions in a way that avoids the combinatorial explosion which generally arises from the exponential number of assignments to be examined. The solution may be viewed ..."
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Cited by 2 (2 self)
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We solve a simple instance of the SAT problem using a natural physical computing system based on fluid mechanics. The natural system functions in a way that avoids the combinatorial explosion which generally arises from the exponential number of assignments to be examined. The solution may be viewed as part of a more general type of natural computation called Bilateral Computing. The paper will also describe this new computing paradigm and will compare it with Reversible Computing. Our approach is informal: the emphasis is on motivation, ideas and implementation rather than a formal description. 1
Brain Organization and Computation
"... Abstract. Theories of how the brain computes can be differentiated in three general conceptions: the algorithmic approach, the neural information processing (neurocomputational) approach and the dynamical systems approach. The discussion of key features of brain organization (i.e. structure with fun ..."
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Cited by 2 (2 self)
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Abstract. Theories of how the brain computes can be differentiated in three general conceptions: the algorithmic approach, the neural information processing (neurocomputational) approach and the dynamical systems approach. The discussion of key features of brain organization (i.e. structure with function) demonstrates the selforganizing character of brain processes at the various spatiotemporal scales. It is argued that the features associated with the brain are in support of its description in terms of dynamical systems theory, and of a concept of computation to be developed further within this framework. 1
Exponential Transients in ContinuousTime Liapunov Systems
 In Proceedings of the 11th ICANN'2001 Conference on Arti Neural Networks, LNCS 2130
, 2003
"... We consider the convergence behavior of a class of continuoustime dynamical systems corresponding to so called symmetric Hopfield nets studied in neural networks theory. We prove that such systems may have transient times that are exponential in the system dimension (i.e. number of "neurons"), desp ..."
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We consider the convergence behavior of a class of continuoustime dynamical systems corresponding to so called symmetric Hopfield nets studied in neural networks theory. We prove that such systems may have transient times that are exponential in the system dimension (i.e. number of "neurons"), despite the fact that their dynamics are controlled by Liapunov functions. This result stands in contrast to many proposed uses of such systems in e.g. combinatorial optimization applications, in which it is often implicitly assumed that their convergence is rapid. An additional interesting observation is that our example of an exponentialtransient continuoustime system (a simulated binary counter) in fact converges more slowly than any discretetime Hopfield system of the same representation size. This suggests that continuoustime systems may be worth investigating for gains in descriptional efficiency as compared to their discretetime counterparts.
Inaccessibility and undecidability in computation, geometry and dynamical systems
 PHYSICA D
, 2001
"... Nonselfsimilar sets defined by a decision procedure are numerically investigated by introducing the notion of inaccessibility to (ideal) decision procedure, that is connected with undecidability. A halting set of a universal Turing machine (UTM), the Mandelbrot set and a riddled basin are mainly i ..."
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Nonselfsimilar sets defined by a decision procedure are numerically investigated by introducing the notion of inaccessibility to (ideal) decision procedure, that is connected with undecidability. A halting set of a universal Turing machine (UTM), the Mandelbrot set and a riddled basin are mainly investigated as nonselfsimilar sets with a decision procedure. By encoding a symbol sequence to a point in a Euclidean space, a halting set of a UTM is shown to be geometrically represented as a nonselfsimilar set, having different patterns and different fine structures on arbitrarily small scales. The boundary dimension of this set is shown to be equal to the space dimension, implying that the ideal decision procedure is inaccessible in the presence of error. This property is shown to be invariant under application of “fractal ” code transformations. Thus, a characterization of undecidability is given by the inaccessibility to the ideal decision procedure and its invariance against the code transformations. It is also shown that the distribution of halting time of the UTM, decays with a power law (or slower), and that this characteristic is also unchanged under code transformation. The Mandelbrot set is shown to have these features including the invariance against the code transformation, in common, and is connected with undecidable sets. In contrast, although a riddled basin, as a geometric representation of a certain contextfree language, has the boundary dimension equal to the space dimension and a power law halting time distribution, these properties are not invariant against the code transformation. Thus, the riddled basin is ranked as middle between an ordinary fractal and a halting set of a UTM or the Mandelbrot set. Last, we