123 research outputs found
Non-Uniform Cellular Automata: classes, dynamics, and decidability
The dynamical behavior of non-uniform cellular automata is compared with the
one of classical cellular automata. Several differences and similarities are
pointed out by a series of examples. Decidability of basic properties like
surjectivity and injectivity is also established. The final part studies a
strong form of equicontinuity property specially suited for non-uniform
cellular automata.Comment: Paper submitted to an international journal on June 9, 2011. This is
an extended and improved version of the conference paper: G. Cattaneo, A.
Dennunzio, E. Formenti, and J. Provillard. "Non-uniform cellular automata".
In Proceedings of LATA 2009, volume 5457 of Lecture Notes in Computer
Science, pages 302-313. Springe
Nilpotency and periodic points in non-uniform cellular automata
Nilpotent cellular automata have the simplest possible dynamics: all initial configurations lead in bounded time into the unique fixed point of the system. We investigate nilpotency in the setup of one-dimensional non-uniform cellular automata (NUCA) where different cells may use different local rules. There are infinitely many cells in NUCA but only a finite number of different local rules. Changing the distribution of the local rules in the system may drastically change the dynamics. We prove that if the available local rules are such that every periodic distribution of the rules leads to nilpotent behavior then so do also all eventually periodic distributions. However, in some cases there may be non-periodic distributions that are not nilpotent even if all periodic distributions are nilpotent. We demonstrate such a possibility using aperiodic Wang tile sets. We also investigate temporally periodic points in NUCA. In contrast to classical uniform cellular automata, there are NUCA-even reversible equicontinuous ones-that do not have any temporally periodic points. We prove the undecidability of this property: there is no algorithm to determine if a NUCA with a given finite distribution of local rules has a periodic point
A Survey of Cellular Automata: Types, Dynamics, Non-uniformity and Applications
Cellular automata (CAs) are dynamical systems which exhibit complex global
behavior from simple local interaction and computation. Since the inception of
cellular automaton (CA) by von Neumann in 1950s, it has attracted the attention
of several researchers over various backgrounds and fields for modelling
different physical, natural as well as real-life phenomena. Classically, CAs
are uniform. However, non-uniformity has also been introduced in update
pattern, lattice structure, neighborhood dependency and local rule. In this
survey, we tour to the various types of CAs introduced till date, the different
characterization tools, the global behaviors of CAs, like universality,
reversibility, dynamics etc. Special attention is given to non-uniformity in
CAs and especially to non-uniform elementary CAs, which have been very useful
in solving several real-life problems.Comment: 43 pages; Under review in Natural Computin
Causal Dynamics of Discrete Surfaces
We formalize the intuitive idea of a labelled discrete surface which evolves
in time, subject to two natural constraints: the evolution does not propagate
information too fast; and it acts everywhere the same.Comment: In Proceedings DCM 2013, arXiv:1403.768
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