Jérôme Durand-Lose

Intrinsic Universality of a 1-Dimensional Reversible Cellular Automaton

This paper deals with simulation and reversibility in the context of Cellular Automata (ca). We recall the definitions of ca and of the Block (bca) and Partitioned (pca) subclasses. We note that pca simulate ca. A simulation of reversible ca (r-ca) with reversible pca is built contradicting the intuition of known undecidability results. We build a 1-r-ca which is intrinsic universal, i.e., able to simulate any 1-r-ca.

Abstract geometrical computation 3: black holes for classical and analog computing

The so-called Black Hole model of computation involves a non Euclidean space-time where one device is infinitely “accelerated” on one world-line but can send some limited information to an observer working at “normal pace”. The key stone is that after a finite duration, the observer has received the information or knows that no information was ever sent by the device which had an infinite time to complete its computation. This allows to decide semi-decidable problems and clearly falls out of classical computability. A setting in a continuous Euclidean space-time that mimics this is presented. Not only is Zeno effect possible but it is used to unleash the black hole power. Both discrete (classical) computation and analog computation (in the understanding of Blum, Shub and Smale) are considered. Moreover, using nested singularities (which are built), it is shown how to decide higher levels of the corresponding arithmetical hierarchies.

Token-Based Self-Stabilizing Uniform Algorithms

This work focuses on self-stabilizing algorithms for mutual exclusion and leader election?two fundamental tasks for distributed systems. Self-stabilizing systems are able to recover by themselves, regaining their consistency from any initial or intermediary faulty configuration. The proposed algorithms are designed for any directed, anonymous network and stabilize under any distributed scheduler. The keystones of the algorithms are the token management and routing policies. In order to break the network symmetry, randomization is used. The space complexity is O((D++D?)(log(snd(n))+2)) where n is the network size, snd(n) is the smallest integer that does not divide n and D+ and D? are the maximal out and in degree, respectively. It should be noted that snd(n) is constant on the average and equals 2 on odd-size networks.

Reversible Cellular Automaton Able to Simulate Any Other Reversible One Using Partitioning Automata

Partitioning automata (PA) are defined. They are equivalent to cellular automata (CA). Reversible sub-classes are also equivalent. A simple, reversible and universal partitioning automaton is described. Finally, it is shown that there are reversible PA and CA that are able to simulate any reversible PA or CA on any configuration.

Abstract geometrical computation for black hole computation

The Black hole model of computation provides super-Turing computing power since it offers the possibility to decide in finite (observers) time any recursively enumerable (

Abstract geometrical computation: turing-computing ability and undecidability

\mathcal{R.E.}

Abstract Geometrical Computation and Computable Analysis

) problem. In this paper, we provide a geometric model of computation, conservative abstract geometrical computation, that, although being based on rational numbers, has the same property: it can simulate any Turing machine and can decide any

Abstract Geometrical Computation and the Linear Blum, Shub and Smale Model

\mathcal{R.E.}

Parallel transient time of one-dimensional sand pile

problem through the creation of an accumulation. Finitely many signals can leave any accumulation, and it can be known whether anything leaves. This corresponds to a black hole effect.

Reversible space-time simulation of cellular automata

In the Cellular Automata (CA) literature, discrete lines inside (discrete) space-time diagrams are often idealized as Euclidean lines in order to analyze a dynamics or to design CA for special purposes. In this article, we present a parallel analog model of computation corresponding to this idealization: dimensionless signals are moving on a continuous space in continuous time generating Euclidean lines on (continuous) space-time diagrams. Like CA, this model is parallel, synchronous, uniform in space and time, and uses local updating. The main difference is that space and time are continuous and not discrete (ie ℝ instead of ℤ). In this article, the model is restricted to ℚ in order to remain inside Turing-computation theory. We prove that our model can carry out any Turing-computation through two-counter automata simulation and provide some undecidability results.