4,855 research outputs found
Balanced simplices
An additive cellular automaton is a linear map on the set of infinite
multidimensional arrays of elements in a finite cyclic group
. In this paper, we consider simplices appearing in the
orbits generated from arithmetic arrays by additive cellular automata. We prove
that they are a source of balanced simplices, that are simplices containing all
the elements of with the same multiplicity. For any
additive cellular automaton of dimension or higher, the existence of
infinitely many balanced simplices of appearing in
such orbits is shown, and this, for an infinite number of values . The
special case of the Pascal cellular automata, the cellular automata generating
the Pascal simplices, that are a generalization of the Pascal triangle into
arbitrary dimension, is studied in detail.Comment: 33 pages ; 11 figures ; 1 tabl
Impartial games emulating one-dimensional cellular automata and undecidability
We study two-player \emph{take-away} games whose outcomes emulate two-state
one-dimensional cellular automata, such as Wolfram's rules 60 and 110. Given an
initial string consisting of a central data pattern and periodic left and right
patterns, the rule 110 cellular automaton was recently proved Turing-complete
by Matthew Cook. Hence, many questions regarding its behavior are
algorithmically undecidable. We show that similar questions are undecidable for
our \emph{rule 110} game.Comment: 22 pages, 11 figure
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
Gliders and Ether in Rule 54
This is a study of the one-dimensional elementary cellular automaton rule 54
in the new formalism of "flexible time". We derive algebraic expressions for
groups of several cells and their evolution in time. With them we can describe
the behaviour of simple periodic patterns like the ether and gliders in an
efficient way. We use that to look into their behaviour in detail and find
general formulas that characterise them.Comment: 10 pages, 6 figures, 3 tables. Some errors of the printed version are
correcte
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