Results 1  10
of
30
Computational mechanics: Pattern and prediction, structure and simplicity
 Journal of Statistical Physics
, 1999
"... Computational mechanics, an approach to structural complexity, defines a process’s causal states and gives a procedure for finding them. We show that the causalstate representation—an Emachine—is the minimal one consistent with ..."
Abstract

Cited by 48 (8 self)
 Add to MetaCart
Computational mechanics, an approach to structural complexity, defines a process’s causal states and gives a procedure for finding them. We show that the causalstate representation—an Emachine—is the minimal one consistent with
An analog characterization of the Grzegorczyk hierarchy
 Journal of Complexity
, 2002
"... We study a restricted version of Shannon's General . . . ..."
Abstract

Cited by 34 (16 self)
 Add to MetaCart
We study a restricted version of Shannon's General . . .
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 ..."
Abstract

Cited by 34 (16 self)
 Add to MetaCart
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.
Computational aspects of feedback in neural circuits
 PLOS Computational Biology
, 2007
"... It has previously been shown that generic cortical microcircuit models can perform complex realtime computations on continuous input streams, provided that these computations can be carried out with a rapidly fading memory. We investigate the computational capability of such circuits in the more re ..."
Abstract

Cited by 21 (5 self)
 Add to MetaCart
(Show Context)
It has previously been shown that generic cortical microcircuit models can perform complex realtime computations on continuous input streams, provided that these computations can be carried out with a rapidly fading memory. We investigate the computational capability of such circuits in the more realistic case where not only readout neurons, but in addition a few neurons within the circuit, have been trained for specific tasks. This is essentially equivalent to the case where the output of trained readout neurons is fed back into the circuit. We show that this new model overcomes the limitation of a rapidly fading memory. In fact, we prove that in the idealized case without noise it can carry out any conceivable digital or analog computation on timevarying inputs. But even with noise, the resulting computational model can perform a large class of biologically relevant realtime computations that require a nonfading memory. We demonstrate these computational implications of feedback both theoretically, and through computer simulations of detailed cortical microcircuit models that are subject to noise and have complex inherent dynamics. We show that the application of simple learning procedures (such as linear regression or perceptron learning) to a few neurons enables such circuits to represent time over behaviorally relevant long time spans, to integrate evidence from incoming spike trains over longer periods of time, and to process new information contained in such spike trains in diverse ways according to the current internal state of the circuit. In particular we show that such generic cortical microcircuits with feedback provide a new model for working memory that is consistent with a large set of biological constraints.
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 ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
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
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 fun ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
(Show Context)
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.
Abstract geometrical computation: Turingcomputing ability and undecidability
, 2004
"... In the Cellular Automata (CA) literature, discrete lines inside (discrete) spacetime diagrams are often idealized as Euclidean lines in order to analyze a dynamics or to design CA for special purposes. In this article, we present a parallel analog model of computation corresponding to this ideali ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
In the Cellular Automata (CA) literature, discrete lines inside (discrete) spacetime diagrams are often idealized as Euclidean lines in order to analyze a dynamics or to design CA for special purposes. In this article, we present a parallel analog model of computation corresponding to this idealization: dimensionless signals are moving on a continuous space in continuous time generating Euclidean lines on (continuous) spacetime diagrams. Like CA, this model is parallel, synchronous, uniform in space and time, and uses local updating. The main difference is that space and time are continuous and not discrete (i.e. R instead of Z). In this article, the model is restricted to Q in order to remain inside Turingcomputation theory. We prove that our model can carry out any Turingcomputation through twocounter automata simulation and provide some undecidability results.
A ContinuousTime Hopfield Net Simulation of Discrete Neural Networks
, 2000
"... We investigate the computational power of continuoustime symmetric Hopfield nets. As is well known, such networks have very constrained, Liapunovfunction controlled dynamics. Nevertheless, we show that they are universal and efficient computational devices, in the sense that any convergent fully p ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
We investigate the computational power of continuoustime symmetric Hopfield nets. As is well known, such networks have very constrained, Liapunovfunction controlled dynamics. Nevertheless, we show that they are universal and efficient computational devices, in the sense that any convergent fully parallel computation by a network of n discretetime binary neurons, with in general asymmetric interconnections, can be simulated by a symmetric continuoustime Hopfield net containing only 14n + 6 units using the saturatedlinear sigmoid activation function. In terms of standard discrete computation models this result implies that any polynomially spacebounded Turing machine can be simulated by a polynomially sizeincreasing sequence of continuoustime Hopfield nets.