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Maurice Margenstern

Publications and source records attributed to Maurice Margenstern.

At least 19 recordsLinked to original sources
arXiv2023

A strongly universal cellular automaton on the pentagrid with six states and Moore neighbourhood

In this paper, we prove that there is a strongly universal cellular automaton on the pentagrid with six states. For each cell c, Moore neighbourhood consists of the cells which share a vertex with c. Moreover, the rules are rotation invariant. There are 1072 of them. The result is different from arXiv:2306.06728. The present paper deals with the pentagrid and not with the heptagrid, it is not at all the same context although the implementated model is basically the same. The implementation of that model is different from the quoted arXiv paper.

nlin.CG↗
arXiv2023

A weakly universal weighted cellular automaton in the heptagrid with 6 states

In this paper we prove that there is a weakly universal weighted cellular automaton in the heptagrid, the tessellation {7,3} of the hyperbolic plane, with 6 states. The present paper improves the same result deposited on arXiv:2301.10691v1 and also arXiv:2301.10691v2. In the deposited papers, the result is proved with 7 states. In the present replacement the number of states is reduced to 6. Such a reducing is not trivial and requires substantial changes in the implementation. The maximal weight is now 34, a very strong reduction with the best result with 7 states. Also, the table has 137 entries, signifcantly less than the 160 entries of the paper with 7 states. The reduction is obtained by a new implementation of the tracks which play a key role as far as without tracks there is no computational universality result.

cs.FL↗
arXiv2022

The domino problem of the hyperbolic plane is undecidable, new proof

The present paper is a new version of the arXiv paper revisiting the proof given in a previous paper of the author published in 2008 proving that the general tiling problem of the hyperbolic plane is undecidable by proving a slightly stronger version using only a regular polygon as the basic shape of the tiles. The problem was raised by a paper of Raphael Robinson in 1971, in his famous simplified proof that the general tiling problem is undecidable for the Euclidean plane, initially proved by Robert Berger in 1966. The present construction improves that of the recent arXiv paper. It also strongly reduces the number of prototiles.

cs.DM↗
arXiv2021

What can we learn from universal Turing machines?

In the present paper, we construct what we call a pedagogical universal Turing machine. We try to understand which comparisons with biological phenomena can be deduced from its encoding and from its working.

cs.FL↗
arXiv2021

A strongly universal cellular automaton with four states

In this paper, we prove that there is a strongly universal cellular automaton in the dodecagrid, the tesselllation {5,3,4} of the hyperbolic 3D space, with four states but, it is not rotation invariant as the automaton of arXiv:2104.01561 is with five states. The present paper is not an improvement of arXiv:2104.01561. The reduction to four states of the automaton of the present paper results from a relaxation of the condition of rotation invariance. However, the relaxation is not complete: the rules are invariant with respect to rotations of the dodecahedron which leave it globally invariant which also leave invariant a couple of opposite faces of the dodecahedron. As there are twenty five such rotations, it can be considered that the reduction to four states is significant although the relaxation it implied.

nlin.CG↗
arXiv2019

About Fibonacci trees III: multiple Fibonacci trees

In this third paper, we revisit the question to which extent the properties of the trees associated to the tilings $\{p,4\}$ of the hyperbolic plane are still true if we consider a finitely generated tree by the same rules but rooted at a black node? What happens if, considering the same distinction between black and white nodes but changing the place of the black son in the rules. What happens if we change the representation of the numbers by another set of digits? We tackle all of these questions in the paper. The present paper is an extension of the previous papers arXiv:1904.12135 and arXiv:1907.04677.

cs.DM↗
arXiv2019

About Fibonacci trees. II -- generalized Fibonacci trees

In this second paper, we look at the following question: are the properties of the trees associated to the tilings $\{p,4\}$ and $\{p$+$2,3\}$ of the hyperbolic plane still true if we consider a finitely generated tree by the same rules but rooted at a black node? The direct answer is no, but new properties arise, no more complex than in the case of a tree rooted at a white node, and worth of interest. The present paper is an extension of the previous paper: arXiv:1904.12135.

cs.DM↗
arXiv2019

About Fibonacci trees. I

In this first paper, we look at the following question: are the properties of the Fibonacci tree still true if we consider a finitely generated tree by the same rules but rooted at a black node? The direct answer is no, but new properties arise, a bit more complex than in the case of a tree rooted at a white node, but still of interest.

cs.FL↗
arXiv2016

A weakly universal cellular automaton in the heptagrid

In this paper, we construct a weakly universal cellular automaton in the heptagrid, the tessellation $\{7,3\}$ which is not rotation invariant but which is truly planar. This result, under these conditions, cannot be improved for the tessellations $\{p,3\}$.

cs.DM↗
arXiv2016

A weakly universal cellular automaton on the pentagrid with two states

In this paper, we prove that there is a weakly universal cellular automaton on the pentagrid with two states. This paper improves in some sense a previous result with three states. Both results make use of \textit{à la Moore} neighbourhood. However, the result with three states is rotation invariant while that with two states is not. In both cases, at each step of the computation, the set of non quiescent states has always infinitely many cycles.

cs.DM↗