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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., Comment: 43 pages, 29 figures. arXiv admin note: substantial text overlap with arXiv:2306.06728, arXiv:2304.13575, arXiv:2104.01561, arXiv:2108.13094

In this paper, we prove that there is a strongly universal cellular automaton on the heptagrid with six states which is rotation invariant. This improves a previous paper of the author with 7 states. Here, the structures are modified and the number of rules is much less., Comment: 35 pages, 28 figures

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., Comment: 37 pages, 21 figures. arXiv admin note: substantial text overlap with arXiv:2108.13094, arXiv:2104.01561, arXiv:2107.04843

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., Comment: 55 pages, 38 figures

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., Comment: 35 pages, 5 tables

In this paper, we prove that there is an outer totalistic weakly universal cellular automaton in the dodecagrid, the tessellation {5,3,4} of the hyperbolic 3D space, with four states. It is the first result in such a context., Comment: 32 pages, 19 figures. arXiv admin note: substantial text overlap with arXiv:2104.01561, arXiv:2107.04843

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., Comment: 49 pages, 23 figures. arXiv admin note: substantial text overlap with arXiv:2104.01561

In this paper, we prove that there is a strongly universal cellular automaton in the dodecagrid, the tessellation {5,3,4} of the hyperbolic 3D-space, with five states which is rotation invariant. This improves a previous paper of the author where the automaton required ten states., Comment: 62p, 25 figures

In this paper, we prove that there is a strongly universal cellular automaton on the heptagrid with seven states which is rotation invariant. This improves a previous paper of the author where the automaton required ten states., Comment: 43 pages, 44 figures

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., Comment: 48 pages, 15 figures

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., Comment: 35 pages, 6 figures

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., Comment: 17 pages, 9 figures

In this paper, we investigate the halting problem for deterministic cellula automata in the pentagrid. We prove that the problem is decidable when the cellular automaton starts its computation from a finite configuration and when it has at most two states, one of them being a quiescent state., Comment: 18 pages, 10 figures

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 $\{p,3\}$., Comment: 32 pages, 16 figures. arXiv admin note: substantial text overlap with arXiv:1606.03938; text overlap with arXiv:1605.09518

In this paper, we construct a weakly universal cellular automaton on the tessellation $\{8,3\}$ which is not rotation invariant but which is truly planar., Comment: 32 pages, 17 figures. arXiv admin note: text overlap with arXiv:1605.09518. substantial text overlap with arXiv:1606.09488

In this paper, we indicate a new way to define coordinates for the tiles of the tilings $\{p,3\}$ and $\{p$$-$$2,4\}$ where the natural number $p$ satisfies $p\geq 7$., Comment: 33 pages, 9 figures; this second version improves one figure by inserting a new picture; it also improves two algorithms

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{\`a 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., Comment: 38 pages 24 figures. arXiv admin note: text overlap with arXiv:1510.09129

In this paper, we prove two results. First, there is a family of sequences of embedded quarters of the hyperbolic plane such that any sequence converges to a limit which is an end of the hyperbolic plane. Second, there is no algorithm which would allow us to check whether two given ends are equal or not., Comment: 17 pages, 5 figures

In this paper, we study the contribution of the theory of grossone to the study of infinigons in the hyperbolic plane. We can see that the theory of grossone can help us to obtain much more classification for these objects than in the traditional setting., Comment: 13 pages, 2 figures, correction of typos

In this paper, we construct a weakly universal cellular automaton with two states only on the tiling {11,3}. The cellular automaton is rotation invariant and it is a true planar one., Comment: 31 pages, 22 figures, 19 tables

The twin primes conjecture is a very old problem. Tacitly it is supposed that the primes it deals with are finite. In the present paper we consider three problems that are not related to finite primes but deal with infinite integers. The main tool of our investigation is a numeral system proposed recently that allows one to express various infinities and infinitesimals easily and by a finite number of symbols. The problems under consideration are the following and for all of them we give affermative answers: (i) do infinite primes exist? (ii) do infinite twin primes exist? (ii) is the set of infinite twin primes infinite? Examples of these three kinds of objects are given., Comment: 11 pages

In this paper, we construct a cellular automaton on the heptagrid which is planar, weakly universal and which have three states only. This result improves the best result which was with four states., Comment: 27 pages, 21 figures. arXiv admin note: substantial text overlap with arXiv:1403.2373

In this paper, we construct a cellular automaton on the pentagrid which is planar, weakly universal and which have five states only. This result much improves the best result which was with nine states, Comment: 23 pages, 21 figures

In this paper we study the word problem of groups corresponding to tessellations of the hyperbolic plane. In particular using the Fibonacci technology developed by the second author we show that groups corresponding to the pentagrid or the heptagrid are not automatic., Comment: 31 pages, 9 figures

In this paper, we try to give the appropriate class of languages to which belong various objects associated with tessellations in the hyperbolic plane., Comment: In Proceedings MCU 2013, arXiv:1309.1043

In this paper, we investigate the possibility to use two tilings of the hyperbolic plane as basic frame for devising a way to input texts in Chinese characters into messages of cellphones, smartphones, ipads and tablets., Comment: 28 pages, 40 figures

In this paper, we construct a strongly universal cellular automaton on the line with 11 states and the standard neighbourhood. We embed this construction into several tilings of the hyperbolic plane and of the hyperbolic 3D space giving rise to strongly universal cellular automata with 10 states., Comment: In Proceedings MCU 2013, arXiv:1309.1043

In this paper, we construct a family of weakly universal rotation invariant cellular automaton for all grids $\{p,3\}$ of the hyperbolic plane for $p\geq 13$. The scheme is general for $p\geq 17$ and for $13\leq p<17$, we give such a cellular automaton for $p=13$, which is enough. Also, an important property of this family is that the set of cells of the cellular automaton which are subject to changes is actually a planar set. The problem for $p<13$ for a truly planar construction is still open. The best result, for $p=7$, is four states and was obtained by the same author., Comment: 83 pages, 38 figures, 53 tables. arXiv admin note: text overlap with arXiv:0903.2108

We study a Life-like cellular automaton rule $B2/S2345$ where a cell in state `0' takes state `1' if it has exactly two neighbors in state `1' and the cell remains in the state `1' if it has between two and five neighbors in state `1.' This automaton is a discrete analog spatially extended chemical media, combining both properties of sub-excitable and precipitating chemical media. When started from random initial configuration B2/S2345 automaton exhibits chaotic behavior. Configurations with low density of state `1' show emergence of localized propagating patterns and stationary localizations. We construct basic logical gates and elementary arithmetical circuits by simulating logical signals with mobile localizations reaction propagating geometrically restricted by stationary non-destructible localizations. Values of Boolean variables are encoded into two types of patterns --- symmetric (False) and asymmetric (True) patterns --- which compete for the `empty' space when propagate in the channels. Implementations of logical gates and binary adders are illustrated explicitly., Comment: Chapter in In: A. Adamatzky (Ed.), ISBN 9781849962162

In this paper, we look at how to count the number of elements of a set within the frame of Sergeyev's numeral system. We also look at the connection between the number of elements of a set and the notion of bijection in this new setting. We also show the difference between this new numeral system and the results of the traditional naive set theory., Comment: 13 pages, 2 figures, a slightly different variant is accepted by the journal "p-Adic Numbers, Ultrametric Analysis and Applications"

In this paper, we look at the improvement of our knowledge on a family of tilings of the hyperbolic plane which is brought in by the use of Sergeyev's numeral system based on grossone. It appears that the information we can get by using this new numeral system depends on the way we look at the tilings. The ways are significantly different but they confirm some results which were obtained in the traditional but constructive frame and allow us to obtain an additional precision with respect to this information., Comment: 25 pages, 13 figures. The paper will appear in the journal Applied Mathematics and Computations, see below at DOI

In this paper, we first describe the isothermal constrained loop extension DNA amplification (ICLEDA), which is a new variant of amplification combining the advantages of rolling circle amplification (RCA) and of strand displacement amplification (SDA). Then, we formalize this process in terms of the theory of formal languages and show, on the basis of this formulation, how to manage OR and AND gates. We then explain how to introduce negation, which allows us to prove that, in principle, it is possible to implement the computation of any boolean function on DNA strands using ICLEDA., Comment: 14 pages, 6 figures

This paper introduces a communication system for the tiles of the heptagrid, a tiling of the hyperbolic plane. The method can be extended to other tilings of this plane. The paper focuses on an actual implementation at the programming stage with a short account of two experiments., Comment: 30 pages, 5 figures

In this paper, we look at the possibility to implement the algorithm to construct a discrete line devised by the first author in cellular automata. It turns out that such an implementation is feasible., Comment: 44 pages, 28 figures

In this paper we define an infinite family of triangular tilings of the hyperbolic plane defined by two parameters ranging in the natural nummbers and we give a uniform way to define coordinates for locating the triangles of the tiling., Comment: 15 pages, 8 figures, 3 tables

In this paper we give three examples of expending patterns defined by hyperbolic cellular automata whose growth seems to be very similar to the growth of colonies of bacteria., Comment: 16 pages, 13 figures

In this paper, following the way opened by a previous paper deposited on arXiv, we give an upper bound to the number of states for a hyperbolic cellular automaton in the pentagrid. Indeed, we prove that there is a hyperbolic cellular automaton which is rotation invariant and whose halting problem is undecidable and which has 9~states., Comment: 17 pages, 9 figures

In this paper, we show a construction of a weakly universal cellular automaton in the 3D hyperbolic space with two states. The cellular automaton is rotation invariant and, moreover, based on a new implementation of a railway circuit in the dodecagrid,the construction is a truly 3D-one., Comment: 38p., 25 figures, 8 tables

In this paper, we look at two ways to implement determinisitic one dimensional cellular automata into hyperbolic cellular automata in three contexts: the pentagrid, the heptagrid and the dodecagrid, these tilings being classically denoted by $\{5,4\}$, $\{7,3\}$ and $\{5,3,4\}$ respectively., Comment: 19 pages, 9 figures

This paper introduces a method of navigation in a large family of tilings of the hyperbolic plane and looks at the question of possible applications in the light of the few ones which were already obtained., Comment: 7 pages, 10 figures

In this paper, we give an application of iterated pushdown automata to contour words of balls and two other domains in infinitely many tilings of the hyperbolic plane. We also give a similar application for the tiling {5,3,4} of the hyperbolic 3D space and for the tiling {5,3,3,4} of the hyperbolic 4D space as well., Comment: 15 pages, 6 figures

In this paper, we look at the following question. We consider cellular automata in the hyperbolic plane, (see Margenstern, 2000, 2007 and Margenstern, Morita, 2001) and we consider the global function defined on all possible configurations. Is the injectivity of this function undecidable? The problem was answered positively in the case of the Euclidean plane by Jarkko Kari, in 1994. In the present paper, we show that the answer is also positive for the hyperbolic plane: the problem is undecidable.

In this paper, we construct a new weakly universal cellular automaton on the ternary heptagrid. The previous result, obtained by the same author and Y. Song required six states only. This time, the number of states is four. This is the best result up to date for cellular automata in the hyperbolic plane., Comment: 35 pages, 33 figures