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The surjectivity problem for 2D cellular automata

Bruno Durand
1994 Journal of computer and system sciences (Print)  
The surjectivity problem for 2D cellular automata was proved undecidable in 1989 by Jarkko Kari. The proof consists in a reduction of a problem concerning finite tilings into the previous one.  ...  This reduction uses a special and very sophisticated tile set. In this article, we present a much more simple tile set which can play the same role.  ...  A finite version of the tiling problem (see Theorem 2) is given and reduced to the surjectivity problem for two-dimensional cellular automata (2D CA for short).  ... 
doi:10.1016/s0022-0000(05)80077-7 fatcat:amqchrlsqnbn3luawh3vdmr27u

Inversion of 2D cellular automata: some complexity results

B. Durand
1994 Theoretical Computer Science  
Then, we present a transformation of problems concerning tilings into problems concerning cellular automata.  ...  Inversion of 2D cellular automata: some complexity results, Theoretical Computer Science 134 (1994) 387401.  ...  We have already used this method in order to provide a simple proof for the undecidability of the surjectivity problem in [4] (improved version in [S] ).  ... 
doi:10.1016/0304-3975(94)90244-5 fatcat:26flq6gxnbegfbskeecy5heaka

Reversibility and surjectivity problems of cellular automata

Jarkko Kari
1994 Journal of computer and system sciences (Print)  
We also prove that the corresponding surjectivity problem--the problem of deciding if the global function is surjective--is undecidable for two-dimensional CA.  ...  The problem of deciding if a given cellular automaton (CA) is reversible (or, equivalently, if its global transition function is injective) is called the reversibility problem of CA.  ...  The tiling problem was proved undecidable by . A simplified proof was given later by Robinson [13] .  ... 
doi:10.1016/s0022-0000(05)80025-x fatcat:nytnexobrnb6nolhwud2bsk66a

2D cellular automata: dynamics and undecidability [article]

Enrico Formenti, Alberto Dennunzio, Michael Weiss
2009 arXiv   pre-print
Moreover, we show a tight relation between closingness and openness for 2D CA. Finally, the undecidability of closingness property for 2D CA is proved.  ...  In this paper we introduce the notion of quasi-expansivity for 2D CA and we show that it shares many properties with expansivity (that holds only for 1D CA).  ...  If the K ν,µ -tiling is valid along all macro-tiles of this infinite path, it means that the τ -tiling of c and c is valid in arbitrary large squares (since a plane-pattern-filling path visits all the  ... 
arXiv:0906.0857v2 fatcat:ulusbesmzjg25ovs7bhx4xnnxq

Number conserving cellular automata: new results on decidability and dynamics

Bruno Durand, Enrico Formenti, Aristide Grange, Zsuzsanna Róka
2003 Discrete Mathematics & Theoretical Computer Science  
International audience This paper is a survey on our recent results about number conserving cellular automata. First, we prove the linear time decidability of the property of number conservation.  ...  The sequel focuses on dynamical evolutions of number conserving cellular automata.  ...  For example, in [7] , the authors used this algorithm to find out a special NCA in order to prove the main result of their paper, namely, the undecidability of the surjectivity problem for NCA in dimension  ... 
doi:10.46298/dmtcs.2301 fatcat:ftprar4r7je4nk6g6mruvnz25a

A random NP-complete problem for inversion of 2D cellular automata [chapter]

Bruno Durand
1995 Lecture Notes in Computer Science  
In this paper, we prove the co-RNP-completeness (RNP = Random NP) of the following decision problem: "Given a 2-dimensional cellular automaton &, is & reversible when restricted to finite configurations  ...  In order to prove this result, we introduce a polynomial reduction from problems concerning finite tilings into problems concerning cellular automata.  ...  The key point of our proof is that Our complexity result for 2D cellular automata A transformation between tilings and 2-dimensional cellular automata was first presented by Jarkko Kari in [14] (a  ... 
doi:10.1007/3-540-59042-0_65 fatcat:s3ochhl2cnfjlgxinez7ej4jmi

A Random NP-complete problem for inversion of 2D cellular automata

Bruno Durand
1995 Theoretical Computer Science  
In this paper, we prove the co-RNP-completeness (RNP = Random NP) of the following decision problem: "Given a 2-dimensional cellular automaton &, is & reversible when restricted to finite configurations  ...  In order to prove this result, we introduce a polynomial reduction from problems concerning finite tilings into problems concerning cellular automata.  ...  The key point of our proof is that Our complexity result for 2D cellular automata A transformation between tilings and 2-dimensional cellular automata was first presented by Jarkko Kari in [14] (a  ... 
doi:10.1016/0304-3975(94)00293-r fatcat:r7iga3hcejcmnej3uiutj4pzxe

Theory of cellular automata: A survey

Jarkko Kari
2005 Theoretical Computer Science  
This article surveys some theoretical aspects of cellular automata CA research.  ...  The main goal is to provide a tutorial of CA theory to researchers in other branches of natural computing, to give a compact collection of known results with references to their proofs, and to suggest  ...  Introduction Cellular automata (CA) are among the oldest models of natural computing, dating back over half a century.  ... 
doi:10.1016/j.tcs.2004.11.021 fatcat:bmxech4lfra6rjew5kd2biywhq

Topological Dynamics of Cellular Automata: Dimension Matters [article]

Mathieu Sablik
2009 arXiv   pre-print
Topological dynamics of cellular automata (CA), inherited from classical dynamical systems theory, has been essentially studied in dimension 1.  ...  Finally, we show that the set of sensitive CA is only Pi_2 in dimension 1, but becomes Sigma_3-hard for dimension 3.  ...  -what happens when we restrict to reversible cellular automata? more generally to surjective ones?  ... 
arXiv:0811.2731v2 fatcat:tkjdlk2o5bhk5im5ihbqowlqo4

The Mirage of Universality in Cellular Automata [article]

Guillaume Theyssier
2021 arXiv   pre-print
This note is a survey of examples and results about cellular automata with the purpose of recalling that there is no 'universal' way of being computationally universal.  ...  Finally we show how strong forms of universality can be hidden inside some seemingly simple cellular automata according to some classical dynamical parameters.  ...  We are now going to define a notion of universality for cellular automata.  ... 
arXiv:2112.01090v1 fatcat:tr4ett74xjenjjghomozmdf5pi

Topological Dynamics of Cellular Automata: Dimension Matters

Mathieu Sablik, Guillaume Theyssier
2010 Theory of Computing Systems  
Topological dynamics of cellular automata (CA), inherited from classical dynamical systems theory, has been essentially studied in dimension 1.  ...  Finally, we show that the set of sensitive CA is only Π 0 2 in dimension 1, but becomes Σ 0 3 -hard for dimension 3.  ...  -what happens when we restrict to reversible cellular automata? more generally to surjective ones?  ... 
doi:10.1007/s00224-010-9255-x fatcat:2ov3rpwjzbdwvpvw6jzfkyw2cm

A brief history of cellular automata

Palash Sarkar
2000 ACM Computing Surveys  
Here we trace a history of cellular automata from their beginnings with von Neumann to the present day.  ...  The work should be of interest to both new entrants into the field as well as researchers working on particular aspects of cellular automata.  ...  ACKNOWLEDGMENTS The author is greatly indebted to Rana Barua for reading the manuscript and, more importantly, for all the discussions on CA that made this work possible.  ... 
doi:10.1145/349194.349202 fatcat:llv74746vnbwxcslsvadfiv4xi

Characterisation of limit measures of higher-dimensional cellular automata [article]

Martin Delacourt, Benjamin Hellouin de Menibus
2017 arXiv   pre-print
The main tool is the implementation of arbitrary computation in the time evolution of a cellular automata in such a way that it emerges and self-organises from a random configuration.  ...  We consider the typical asymptotic behaviour of cellular automata of higher dimension (greater than 2).  ...  A longer-term research direction concerns surjective cellular automata.  ... 
arXiv:1512.03696v2 fatcat:v5k7iqhjwnhaznweq2h3wepx4m

μ-Limit Sets of Cellular Automata from a Computational Complexity Perspective [article]

Laurent Boyer , Mathieu Sablik
2015 arXiv   pre-print
More precisely, we investigate the computational complexity of these sets and of related decision problems.  ...  This paper concerns μ-limit sets of cellular automata: sets of configurations made of words whose probability to appear does not vanish with time, starting from an initial μ-random configuration.  ...  Acknowledgment We are grateful for the time spent by the anonymous referees on the first version of this paper and for the incitative to write a better version through their numerous comments.  ... 
arXiv:1309.6730v2 fatcat:ms64o75wkjbufis76my24fb2se

Characterisation of Limit Measures of Higher-Dimensional Cellular Automata

Martin Delacourt, Benjamin Hellouin de Menibus
2017 Theory of Computing Systems  
The main tool is the implementation of arbitrary computation in the time evolution of a cellular automata in such a way that it emerges and self-organises from a random conguration.  ...  We consider the typical asymptotic behaviour of cellular automata of higher dimension (≥ 2).  ...  A longer-term research direction concerns surjective cellular automata.  ... 
doi:10.1007/s00224-017-9753-1 fatcat:oaxbzggu6vcyllj66l2djn3cm4
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