Trace Complexity of Chaotic Reversible Cellular Automata

Delvenne, K\r{u}rka and Blondel have defined new notions of computational complexity for arbitrary symbolic systems, and shown examples of effective systems that are computationally universal in this sense. The notion is defined in terms of the trace function of the system, and aims to capture its dynamics. We present a Devaney-chaotic reversible cellular automaton that is universal in their sense, answering a question that they explicitly left open. We also discuss some implications and limitations of the construction.Comment: 12 pages + 1 page appendix, 4 figures. Accepted to Reversible Computation 2014 (proceedings published by Springer

Decidability and Universality in Symbolic Dynamical Systems

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 model-checking 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.Comment: 23 pages; a shorter version is submitted to conference MCU 2004 v2: minor orthographic changes v3: section 5.2 (collatz functions) mathematically improved v4: orthographic corrections, one reference added v5:27 pages. Important modifications. The formalism is strengthened: temporal logic replaced by finite automata. New results. Submitte

Recent advances in the evolution of interfaces: thermodynamics, upscaling, and universality

We consider the evolution of interfaces in binary mixtures permeating strongly heterogeneous systems such as porous media. To this end, we first review available thermodynamic formulations for binary mixtures based on \emph{general reversible-irreversible couplings} and the associated mathematical attempts to formulate a \emph{non-equilibrium variational principle} in which these non-equilibrium couplings can be identified as minimizers. Based on this, we investigate two microscopic binary mixture formulations fully resolving heterogeneous/perforated domains: (a) a flux-driven immiscible fluid formulation without fluid flow; (b) a momentum-driven formulation for quasi-static and incompressible velocity fields. In both cases we state two novel, reliably upscaled equations for binary mixtures/multiphase fluids in strongly heterogeneous systems by systematically taking thermodynamic features such as free energies into account as well as the system's heterogeneity defined on the microscale such as geometry and materials (e.g. wetting properties). In the context of (a), we unravel a \emph{universality} with respect to the coarsening rate due to its independence of the system's heterogeneity, i.e. the well-known ${\cal O}(t^{1/3})$-behaviour for homogeneous systems holds also for perforated domains. Finally, the versatility of phase field equations and their \emph{thermodynamic foundation} relying on free energies, make the collected recent developments here highly promising for scientific, engineering and industrial applications for which we provide an example for lithium batteries