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**1 - 3**of**3**### Universality, Turing Incompleteness and Observers

"... The development of the mathematical theory of computability was motivated in large part by the foundational crisis in mathematics. D. Hilbert suggested an antidote to all the foundational problems that were discovered in the late 19th century: his proposal, in essence, was to formalize mathematics a ..."

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The development of the mathematical theory of computability was motivated in large part by the foundational crisis in mathematics. D. Hilbert suggested an antidote to all the foundational problems that were discovered in the late 19th century: his proposal, in essence, was to formalize mathematics and construct a finite set of axioms that are strong enough to prove all proper theorems, but no more. Thus a proof of consistency and a proof of completeness were required. These proofs should be carried only by strictly finitary means so as to be beyond any reasonable criticism. As Hilbert pointed out [19], to carry out this project one needs to develop a better understanding of proofs as objects of mathematical discourse: To reach our goal, we must make the proofs as such the object of our investigation; we are thus compelled to a sort of proof theory which studies operations with the proofs themselves. Furthermore, Hilbert hoped to find a single, mechanical procedure that would, at least in principle, provide correct answers to all well-defined questions

### RESEARCH ARTICLE Computational Classification of Cellular Automata

"... We discuss attempts at the classification of cellular automata, in particular with a view towards decidability. We will see that a large variety of properties relating to the short-term evolution of configurations is decidable in principle, but questions relating to the long-term evolution are typic ..."

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We discuss attempts at the classification of cellular automata, in particular with a view towards decidability. We will see that a large variety of properties relating to the short-term evolution of configurations is decidable in principle, but questions relating to the long-term evolution are typically undecidable. Even in the decidable case, computational hardness poses a major obstacle for the automatic analysis of cellular automata.

### IOS Press Stochastic Cellular Automata: Correlations, Decidability and Simulations

, 2013

"... Abstract. This paper introduces a simple formalism for dealing with deterministic, nondeterministic and stochastic cellular automata in an unified and composable manner. This formalism allows for local probabilistic correlations, a feature which is not present in usual definitions. We show that this ..."

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Abstract. This paper introduces a simple formalism for dealing with deterministic, nondeterministic and stochastic cellular automata in an unified and composable manner. This formalism allows for local probabilistic correlations, a feature which is not present in usual definitions. We show that this feature allows for strictly more behaviors (for instance, number conserving stochastic cellular automata require these local probabilistic correlations). We also show that several problems which are deceptively simple in the usual definitions, become undecidable when we allow for local probabilistic correlations, even in dimension one. Armed with this formalism, we extend the notion of intrinsic simulation between deterministic cellular automata, to the non-deterministic and stochastic settings. Although the intrinsic simulation relation is shown to become undecidable in dimension two and higher, we provide explicit tools to prove or disprove the existence of such a simulation between any two given stochastic cellular automata. Those tools rely upon a characterization of equality of stochastic global maps, shown to be equivalent to the existence of a stochastic coupling between the random sources. We apply them to prove that there is no universal stochastic cellular automaton. Yet we provide stochastic cellular automata achieving optimal partial universality, as well as a universal non-deterministic cellular automaton. This work has been partially funded by the ANR-10-JCJC-0208 CausaQ grant.