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Computational universality in symbolic dynamical systems
 Fundamenta Informaticae
"... Abstract. Many different definitions of computational universality for various types of systems have flourished since Turing’s work. In this paper, we propose a general definition of universality that applies to arbitrary discrete time symbolic dynamical systems. For Turing machines and tag systems, ..."
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Abstract. Many different definitions of computational universality for various types of systems have flourished since Turing’s work. In this paper, we propose a general definition of universality that applies to arbitrary discrete time symbolic dynamical systems. For Turing machines and tag systems, our definition coincides with the usual notion of universality. It however yields a new definition for cellular automata and subshifts. Our definition is robust with respect to noise on the initial condition, which is a desirable feature for physical realizability. We derive necessary conditions for universality. For instance, a universal system must have a sensitive point and a proper subsystem. We conjecture that universal systems have an infinite number of subsystems. We also discuss the thesis that computation should occur at the ‘edge of chaos ’ and we exhibit a universal chaotic system. 1
On the computability of the topological entropy of subshifts
 Discrete Math. Theor. Comput. Sci
"... We prove that the topological entropy of subshifts having decidable language is uncomputable in the following sense: For no error bound less than 1/4 does there exists a program that, given a decision procedure for the language of a subshift as input, will approximate the entropy of the subshift wit ..."
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We prove that the topological entropy of subshifts having decidable language is uncomputable in the following sense: For no error bound less than 1/4 does there exists a program that, given a decision procedure for the language of a subshift as input, will approximate the entropy of the subshift within the error bound. In addition, we prove that not only is the topological entropy of sofic shifts computable to arbitary precision (a wellknown fact), but all standard comparisons of the topological entropy with rational numbers are decidable.
On BetaShifts Having Arithmetical Languages
"... Abstract. Let β be a real number with 1 < β < 2. We prove that the language of the βshift is ∆ 0 n iff β is a ∆nreal. The special case where n is 1 is the independently interesting result that the language of the βshift is decidable iff β is a computable real. The “if ” part of the proof is ..."
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Abstract. Let β be a real number with 1 < β < 2. We prove that the language of the βshift is ∆ 0 n iff β is a ∆nreal. The special case where n is 1 is the independently interesting result that the language of the βshift is decidable iff β is a computable real. The “if ” part of the proof is nonconstructive; we show that for Walters ’ version of the βshift, no constructive proof exists. 1
REVISITING THE RICE THEOREM OF CELLULAR AUTOMATA
"... A cellular automaton is a parallel synchronous computing model, which consists in a juxtaposition of finite automata whose state evolves according to that of their neighbors. It induces a dynamical system on the set of configurations, i.e. the infinite sequences of cell states. The limit set of the ..."
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A cellular automaton is a parallel synchronous computing model, which consists in a juxtaposition of finite automata whose state evolves according to that of their neighbors. It induces a dynamical system on the set of configurations, i.e. the infinite sequences of cell states. The limit set of the cellular automaton is the set of configurations which can be reached arbitrarily late in the evolution. In this paper, we prove that all properties of limit sets of cellular automata with binarystate cells are undecidable, except surjectivity. This is a refinement of the classical “Rice Theorem” that Kari proved on cellular automata with arbitrary state sets.
IOS Press Decidability and Universality in Symbolic Dynamical Systems
"... 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 un ..."
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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 modelchecking 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. 1.
On Immortal Configurations in Turing Machines
, 2012
"... Abstract We investigate the immortality problem for Turing machines and prove that there exists a Turing Machine that is immortal but halts on every recursive configuration. The result is obtained by combining a new proof of Hooper’s theorem [11] with recent results on effective symbolic dynamics. ..."
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Abstract We investigate the immortality problem for Turing machines and prove that there exists a Turing Machine that is immortal but halts on every recursive configuration. The result is obtained by combining a new proof of Hooper’s theorem [11] with recent results on effective symbolic dynamics.