Related papers: Trace Complexity of Chaotic Reversible Cellular Au…
A universal map is derived for all deterministic 1D cellular automata (CA) containing no freely adjustable parameters. The map can be extended to an arbitrary number of dimensions and topologies and its invariances allow to classify all CA…
The TTE approach to Computable Analysis is the study of so-called representations (encodings for continuous objects such as reals, functions, and sets) with respect to the notions of computability they induce. A rich variety of such…
Cellular automata have long been celebrated for their ability to generate complex behaviors from simple, local rules, with well-known discrete models like Conway's Game of Life proven capable of universal computation. Recent advancements…
Reversible computation is key in developing new, energy-efficient paradigms, but also in providing forward-only concepts with broader definitions and finer frames of study.Among other fields, the algebraic specification and representation…
In the present paper, an essential generalization of the symbolic dynamics is considered. We apply the notions of abstract self-similar sets and the similarity map for a chaos introduction, which orbits are expanded among infinitely many…
This work redefines the framework of chaos in dynamical systems by extending Devaney's definition to multiple mappings, emphasizing the pivotal role of nonlinearity. We propose a novel theorem demonstrating how nonlinear dynamics within a…
The search for symmetry as an unusual yet profoundly appealing phenomenon, and the origin of regular, repeating configuration patterns have long been a central focus of complexity science and physics. To better grasp and understand symmetry…
We define quantum cellular automata as infinite quantum lattice systems with discrete time dynamics, such that the time step commutes with lattice translations and has strictly finite propagation speed. In contrast to earlier definitions…
Cellular automata with memory (CAM) are widely used in fields such as image processing, pattern recognition, simulation, and cryptography. The invertibility of CAM is generally considered to be chaotic. Paper [Invertible behavior in…
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 define a notion of complexity, which quantifies the nonlinearity of the computation of a neural network, as well as a complementary measure of the effective dimension of feature representations. We investigate these observables both for…
Cellular automata represent physical systems where both space and time are discrete, and the associated physical quantities assume a limited set of values. While previous research has applied cellular automata in modeling chemical,…
We introduce a formal definition of Wolfram's notion of computational process based on cellular automata, a physics-like model of computation. There is a natural classification of these processes into decidable, intermediate and complete.…
We give exact formulae for a wide family of complexity measures that capture the organization of hidden nonlinear processes. The spectral decomposition of operator-valued functions leads to closed-form expressions involving the full…
This talk advocates intrinsic universality as a notion to identify simple cellular automata with complex computational behavior. After an historical introduction and proper definitions of intrinsic universality, which is discussed with…
Since their inception at {\it Macy conferences} in later 1940s complex systems remain the most controversial topic of inter-disciplinary sciences. The term `complex system' is the most vague and liberally used scientific term. Using…
Cellular automata (CA) are discrete-time dynamical systems with local update rules on a lattice. Despite their elementary definition, CA support a wide spectrum of macroscopic phenomena central to statistical physics: equilibrium and…
Describing complex phenomena by means of cellular automata (CA) has shown to be a very effective approach in pure and applied sciences. In fact, the number of published papers concerning this topic has tremendously increased over the last…
Computational complexity is a core theory of computer science, which dictates the degree of difficulty of computation. There are many problems with high complexity that we have to deal, which is especially true for AI. This raises a big…
Building on the seminal work of Gromov on endomorphisms of symbolic algebraic varieties [10], we introduce a notion of cellular automata over schemes which generalize affine algebraic cellular automata in [7]. We extend known results to…