Decidability and Universality in Symbolic Dynamical Systems

Issue title: SPECIAL ISSUE MCU2004

Article type: Research Article

Authors: Delvenne, Jean-Charles | Kůrka, Petr | Blondel, Vincent

Affiliations: Université catholique de Louvain, Department of Mathematical Engineering, Avenue Georges Lemaitre 4, B-1348 Louvain-la-Neuve, Belgium. E-mail: {delvenne,blondel}@inma.ucl.ac.be | Center for Theoretical Study & Faculty of Mathematics and Physics, Charles University in Prague, Jilská 1, CZ-11000 Praha 1, Czech Republic. E-mail: kurka@ms.mff.cuni.cz

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 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.

Journal: Fundamenta Informaticae, vol. 74, no. 4, pp. 463-490, 2006