Related papers: Dynamic Level Sets
In this paper we present an introduction to the area of computability in dynamical systems. This is a fairly new field which has received quite some attention in recent years. One of the central questions in this area is if relevant…
Cellular automata are discrete dynamical systems and a model of computation. The limit set of a cellular automaton consists of the configurations having an infinite sequence of preimages. It is well known that these always contain a…
Dynamical Systems theory generally deals with fixed point iterations of continuous functions. Computation by Turing machine although is a fixed point iteration but is not continuous. This specific category of fixed point iterations can only…
We study the computational complexity theory of smooth, finite-dimensional dynamical systems. Building off of previous work, we give definitions for what it means for a smooth dynamical system to simulate a Turing machine. We then show that…
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…
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…
Many branches of theoretical and applied mathematics require a quantifiable notion of complexity. One such circumstance is a topological dynamical system - which involves a continuous self-map on a metric space. There are many notions of…
Self-replication is central to all life, and yet how it dynamically emerges in physical, non-equilibrium systems remains poorly understood. Von Neumann's pioneering work in the 1940s and subsequent developments suggest a natural hypothesis:…
In a recent article, we introduced and studied a precise class of dynamical systems called solvable systems. These systems present a dynamic ruled by discontinuous ordinary differential equations with solvable right-hand terms and unique…
The present paper introduces a novel notion of `(effective) computability', called viability, of strategies in game semantics in an intrinsic (i.e., without recourse to the standard Church-Turing computability), non-inductive and…
The study of automorphisms of computable and other structures connects computability theory with classical group theory. Among the noncomputable countable structures, computably enumerable structures are one of the most important objects of…
Generic computability has been studied in group theory and we now study it in the context of classical computability theory. A set A of natural numbers is generically computable if there is a partial computable function f whose domain has…
Turing's (1936) paper on computable numbers has played its role in underpinning different perspectives on the world of information. On the one hand, it encourages a digital ontology, with a perceived flatness of computational structure…
We wish to investigate some elementary problems concerning topological dynamics revolving around our proposed definition of escaping set. We also discuss the notion of escaping set in the induced dynamics of the hyperspace. Moreover, we…
Dynamic Topological Logic ($\mathcal{DTL}$) is a combination of $\mathcal{S}${\em 4}, under its topological interpretation, and the temporal logic $\mathcal{LTL}$ interpreted over the natural numbers. $\mathcal{DTL}$ is used to reason about…
Although there is a somewhat standard formalization of computability on countable sets given by Turing machines, the same cannot be said about uncountable sets. Among the approaches to define computability in these sets, order-theoretic…
There are several forms of irreducibility in computing systems, ranging from undecidability to intractability to nonlinearity. This paper is an exploration of the conceptual issues that have arisen in the course of investigating speed-up…
Many real-world dynamic systems, both natural and artificial, are understood to be performing computations. For artificial dynamic systems, explicitly designed to perform computation - such as digital computers - by construction, we can…
Approaching limitations of digital computing technologies have spurred research in neuromorphic and other unconventional approaches to computing. Here we argue that if we want to systematically engineer computing systems that are based on…
The concept of random dynamical system is a comparatively recent development combining ideas and methods from the well developed areas of probability theory and dynamical systems. Due to our inaccurate knowledge of the particular physical…