Related papers: Universal Computation Is 'Almost Surely' Chaotic
One of the principal goals of controlling classical chaotic dynamical systems is known as targeting, which is the very weakly perturbative process of using the system's extreme sensitivity to initial conditions in order to arrive at a…
Stable chaos is a generalization of the chaotic behaviour exhibited by cellular automata to continuous-variable systems and it owes its name to an underlying irregular and yet linearly stable dynamics. In this review we discuss analogies…
Consider a universal Turing machine that produces a partial or total function (or a binary stream), based on the answers to the binary queries that it makes during the computation. We study the probability that the machine will produce a…
The striking fractal geometry of strange attractors underscores the generative nature of chaos: like probability distributions, chaotic systems can be repeatedly measured to produce arbitrarily-detailed information about the underlying…
A cell dynamical system model for deterministic chaos enables precise quantification of the round-off error growth,i.e., deterministic chaos in digital computer realizations of mathematical models of continuum dynamical systems. The model…
Chaotic iterations have been introduced on the one hand by Chazan, Mi- ranker [6] and Miellou [10] in a numerical analysis context, and on the other hand by Robert [12] and Pellegrin [11] in the discrete dynamical systems frame- work. In…
Chaotic iterations have been introduced on the one hand by Chazan, Miranker [5] and Miellou [9] in a numerical analysis context, and on the other hand by Robert [11] and Pellegrin [10] in the discrete dynamical systems framework. In both…
Chaotic neural networks have received a great deal of attention these last years. In this paper we establish a precise correspondence between the so-called chaotic iterations and a particular class of artificial neural networks: global…
We consider computations of a Turing machine subjected to noise. In every step, the action (the new state and the new content of the observed cell, the direction of the head movement) can differ from that prescribed by the transition…
It is known that the set of all nonnegative integers may be equipped with a total order that is chaotic in the sense that there is no monotone three-term arithmetic progressions. Such chaotic order must be so complicated that the resulting…
It is an increasingly important problem to study conditions on the structure of a network that guarantee a given behavior for its underlying dynamical system. In this paper we report that a Boolean network may fall within the chaotic…
Using nonstandard analysis, we will extend the classical Turing machines into the internal Turing machines. The internal Turing machines have the capability to work with infinite ($*$-finite) number of bits while keeping the finite…
We outline the construction of a molecular system that could, in principle, implement a thermodynamically reversible Universal Turing Machine (UTM). By proposing a concrete-albeit idealised-design and operational protocol, we reveal…
Computational problems are classified into computable and uncomputable problems. If there exists an effective procedure (algorithm) to compute a problem then the problem is computable otherwise it is uncomputable. Turing machines can…
The main theorem of this paper establishes conditions under which the "chaos game" algorithm almost surely yields the attractor of an iterated function system. The theorem holds in a very general setting, even for non contractive iterated…
Chaotic dynamics can be quite heterogeneous in the sense that in some regions the dynamics are unstable in more directions than in other regions. When trajectories wander between these regions, the dynamics is complicated. We say a chaotic…
Predicting chaotic dynamical systems is critical in many scientific fields, such as weather forecasting, but challenging due to the characteristic sensitive dependence on initial conditions. Traditional modeling approaches require extensive…
In this paper, we introduce the notion of distributional chaos and the measure of chaos for random dynamical systems generated by two interval maps. We give some sufficient conditions for a zero measure of chaos and examples of chaotic…
The Turing machine is one of the simple abstract computational devices that can be used to investigate the limits of computability. In this paper, they are considered from several points of view that emphasize the importance and the…
We show on the example of the Arnold cat map that classical chaotic systems can be simulated with exponential efficiency on a quantum computer. Although classical computer errors grow exponentially with time, the quantum algorithm with…