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Decidability and Universality in Symbolic Dynamical Systems | | Jean-Charles Delvenne
; Petr Kurka
; Vincent Blondel
; | | Date: |
8 Apr 2004 | | Subject: | Computational Complexity; Logic in Computer Science ACM-class: F.1.1; F.4.1 | cs.CC cs.LO | | 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. | | Source: | arXiv, cs.CC/0404021 | | Services: | Forum | Review | PDF | Favorites | |
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