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On the Complexity of Limit Sets of Cellular Automata Associated with Probability Measures [article]

Laurent Boyer, Guillaume Theyssier
2006 arXiv   pre-print
It is a refinement of the classical notion of omega-limit sets dealing with the typical long term behavior of cellular automata.  ...  We study the notion of limit sets of cellular automata associated with probability measures (mu-limit sets). This notion was introduced by P. Kurka and A. Maass.  ...  ) = { ω (01) ω , ω (10) ω }. ⊓ ⊔ We will now give a characterization of the persistent language of non sensitive cellular automata.  ... 
arXiv:cs/0604007v2 fatcat:asbf3p4xnvbwzi5xgztjnvlymm

On the limit set of some universal cellular automata

Eric Goles, Alejandro Maass, Servet Martinez
1993 Theoretical Computer Science  
For this class of cellular automata we prove that the associated limit language is regular.  ...  B- (4) Limi[ set of unitlersal cellular automata In block IB the size of the blocks of states E+ vary from 2 to 7.  ... 
doi:10.1016/0304-3975(93)90350-3 fatcat:ejgjojpmrff6bjp5kq54ptwjvy

Limit Sets and the Rate of Convergence for One-Dimensional Cellular Automata Traffic Models [chapter]

T. Namiki
2003 Traffic and Granular Flow'01  
A critical exponent of phase transition for one-dimensional cellular automata with conservative law is proposed and it is shown that the exponent represents the asymptotic behavior of the dynamics.  ...  After here we denote the shift map by σ. For one-dimensional cellular automata τ : S Z → S Z , the limit set X ∞ = n≥0 τ n X 0 , X 0 = S Z is always well defined.  ...  We estimate such phenomena, that is the asymptotic behavior of one-dimensional cellular automata traffic model from initial distribution to the limit distribution with respect to Gibbs measure [1] on the  ... 
doi:10.1007/978-3-662-10583-2_9 fatcat:s2dw6ftskngozpwyfjuwbwg6im

On the Complexity of Limit Sets of Cellular Automata Associated with Probability Measures [chapter]

Laurent Boyer, Victor Poupet, Guillaume Theyssier
2006 Lecture Notes in Computer Science  
It is a refinement of the classical notion of ω-limit sets dealing with the typical long term behavior of cellular automata.  ...  We study the notion of limit sets of cellular automata associated with probability measures (µ-limit sets). This notion was introduced by P. Kůrka and A. Maass in [1].  ...  { ω (01) ω , ω (10) ω }. ⊓ ⊔ We will now give a characterisation of the persistent language of non sensible cellular automata.  ... 
doi:10.1007/11821069_17 fatcat:4ky3yqqspbgwxkdhaqw3snqhbm

Revisiting the Rice Theorem of Cellular Automata [article]

Pierre Guillon, Gaétan Richard
2011 arXiv   pre-print
This is a refinement of the classical "Rice Theorem" that Kari proved on cellular automata with arbitrary state sets.  ...  The limit set of the cellular automaton is the set of configurations which can be reached arbitrarily late in the evolution.  ...  For any nontrivial property P over the limit sets of nonsurjective cellular automata on ¾, there exist two cellular automata G 0 , G 1 on alphabet ¾ sharing the same quiescent state q ∈ ¾, and such that  ... 
arXiv:1001.0253v4 fatcat:gt63drfonzcv3euzkgogglzzsm

Revisiting the Rice Theorem of Cellular Automata

Pierre Guillon, Gaétan Richard, Marc Herbstritt
2010 Symposium on Theoretical Aspects of Computer Science  
This is a refinement of the classical "Rice Theorem" that Kari proved on cellular automata with arbitrary state sets.  ...  The limit set of the cellular automaton is the set of configurations which can be reached arbitrarily late in the evolution.  ...  For any nontrivial property P over the limit sets of nonsurjective cellular automata on ¾, there exist two cellular automata G 0 , G 1 on alphabet ¾ sharing the same quiescent state q ∈ ¾, and such that  ... 
doi:10.4230/lipics.stacs.2010.2474 dblp:conf/stacs/GuillonR10 fatcat:mwmnqjguzbfk5mlbcenzmkaoqy

Computation theory of cellular automata

Stephen Wolfram
1984 Communications in Mathematical Physics  
The limit sets generated by some classes of cellular automata correspond to regular languages. For other classes of cellular automata they appear to correspond to more complicated languages.  ...  The sets of configurations generated after a finite number of time steps of cellular automaton evolution are shown to form regular languages. Many examples are given.  ...  Franks, and many of the other participants for their comments. Some of the results given above were obtained using the computer mathematics system SMP [50] .  ... 
doi:10.1007/bf01217347 fatcat:yrtu66o6xjbkbivnnm3mh2xsqy

Two-dimensional cellular automata

Norman H. Packard, Stephen Wolfram
1985 Journal of statistical physics  
A largely phenomenological study of two-dimensional cellular automata is reported. Qualitative classes of behavior similar to those in one-dimensional cellular automata are found.  ...  Some global properties of cellular automata can be described by entropies and Lyapunov exponents. Others are undecidahle.  ...  Shaw for writing the kernel of our software simulation system for two-dimensional cellular automata in Ridge assembly language. We thank  ... 
doi:10.1007/bf01010423 fatcat:4tdguoisl5fmraggit5gstkkne

Some applications of information theory to cellular automata

Michael S. Waterman
1984 Physica D : Non-linear phenomena  
In this paper general deterministic one-dimensional cellular automata are identified with mappings of the unit interval into itself.  ...  A body of related work by Russian authors on probabilistic automata is briefly examined. Important questions about the existence and properties of limiting distributions remain unresolved.  ...  The author is grateful to John R. Kinney for useful discussions.  ... 
doi:10.1016/0167-2789(84)90247-1 fatcat:o3x5ac2djzctddrbo7ufctrw3y

Page 2824 of Mathematical Reviews Vol. , Issue 92e [page]

1992 Mathematical Reviews  
Summary: “The problem of identification of cellular automata the set of configurations of which is specified on an n-dimensional integer lattice is examined.  ...  Summary: “Given a one-dimensional cellular automaton rule f, a block transform of f is a rule 7, f such that there exists between the limit sets of both rules a bijection that replaces each site value  ... 

Evolution of 2D apoptotic cellular automata

Jennifer Garner, Daniel Ashlock
2015 2015 IEEE Congress on Evolutionary Computation (CEC)  
A collection of examples of renderings of evolved cellular automata are provided and steps for additional work to improve the system are outlined.  ...  This study generalizes earlier work on evolving 1D apoptotic automata to evolving 2D automata, producing a type of evolved art.  ...  the value zero) or until the time limit expired. • If the automata still had living (nonzero) cells at the end of the time limit or if one of the cells at the edge of the cell array was alive, the automata  ... 
doi:10.1109/cec.2015.7257151 dblp:conf/cec/GarnerA15 fatcat:fljk4ezlzzbqnbe5juj6t2skky

Cellular automata as models of complexity

Stephen Wolfram
1984 Nature  
Cellular Automata Cellular automata may serve as suitable models for a wide range of natural systems such as: • The growth of dendritic crystals (snowflakes) • Pigmentation patterns on mollusk shells This  ...  ,k-1) that update according to a deterministic rule depending on a neighborhood Complex behavior can arise from simple rules Introduction • Cellular Automata • Applications • Types of Cellular Automata  ...  Computation theory • For class 3 and 4: • the occurrence of a particular length n site value sequence in infinite time is undecidable. • Decidable in finite time considering initial sequence length n0=  ... 
doi:10.1038/311419a0 fatcat:kbs6fi7g5vb6dhyuuzf6bdz6xq

On Stable and Unstable Limit Sets of Finite Families of Cellular Automata [chapter]

Ville Salo, Ilkka Törmä
2012 Lecture Notes in Computer Science  
In this paper, we define the notion of limit set for a finite family of cellular automata, which is a generalization of the limit set of a single automaton.  ...  We prove that the hierarchy formed by increasing the number of automata in the defining set is infinite, and study the boolean closure properties of different classes of limit sets.  ...  Acknowledgements We'd like to thank Pierre Guillon for Lemma 12 and for pointing us in the direction of [7] .  ... 
doi:10.1007/978-3-642-28332-1_43 fatcat:xfdxnzgsszd65ojx5c5n63dylu

Page 4276 of Mathematical Reviews Vol. , Issue 95g [page]

1995 Mathematical Reviews  
A property is called nontrivial if there are both automata whose limit sets have the property and automata whose limit sets do not have it.  ...  The limit set of a cellular automaton consists of all the config- urations that can occur after arbitrarily long computations.  ... 

Decidability in Group Shifts and Group Cellular Automata

Pierre Béaur, Jarkko Kari, Daniel Kráľ, Javier Esparza
2020 International Symposium on Mathematical Foundations of Computer Science  
Our decidability proofs are based on developing algorithms to manipulate arbitrary group shifts, and viewing the set of space-time diagrams of group cellular automata as multidimensional group shifts.  ...  It also turns out that the the trace shift and the limit set can be effectively constructed, that injectivity always implies surjectivity, and that jointly periodic points are dense in the limit set.  ...  Then in Section 4 we apply the projections on space-time diagrams of cellular automata to effectively construct their traces and limit sets.  ... 
doi:10.4230/lipics.mfcs.2020.12 dblp:conf/mfcs/BeaurK20 fatcat:gymyhpgusrfnheallmxxwjc42e
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