TY  - JOUR
A1  - Guetschow, Johannes
A1  - Nesme, Vincent
A1  - Werner, Reinhard F.
T1  - Self-similarity of cellular automata on abelian groups
JF  - Journal of cellular automata
N2  - It is well known that the spacetime diagrams of some cellular automata have a self-similar fractal structure: for instance Wolfram's rule 90 generates a Sierpinski triangle. Explaining the self-similarity of the spacetime diagrams of cellular automata is a well-explored topic, but virtually all of the results revolve around a special class of automata, whose typical features include irreversibility, an alphabet with a ring structure, a global evolution that is a ring homomorphism, and a property known as (weakly) p-Fermat. The class of automata that we study in this article has none of these properties. Their cell structure is weaker, as it does not come with a multiplication, and they are far from being p-Fermat, even weakly. However, they do produce self-similar spacetime diagrams, and we explain why and how.
KW  - fractal
KW  - abelian group
KW  - linear cellular automaton
KW  - substitution system
KW  - self-similarity
Y1  - 2012
SN  - 1557-5969
VL  - 7
IS  - 2
SP  - 83
EP  - 113
PB  - Old City Publishing Science
CY  - Philadelphia
ER  - 
TY  - JOUR
A1  - Arrighi, Pablo
A1  - Nesme, Vincent
A1  - Werner, Reinhard F.
T1  - One-Dimensional quantum cellular automata
JF  - International journal of unconventional computing : non-classical computation and cellular automata
N2  - We define and study quantum cellular automata (QCA). We show that they are reversible and that the neighborhood of the inverse is the opposite of the neighborhood. We also show that QCA always admit, modulo shifts, a two-layered block representation. Note that the same two-layered block representation result applies also over infinite configurations, as was previously shown for one-dimensional systems in the more elaborate formalism of operators algebras [18]. Here the proof is simpler and self-contained, moreover we discuss a counterexample QCA in higher dimensions.
KW  - cellular automata
KW  - quantum
KW  - neighborhood
KW  - block representation
Y1  - 2011
SN  - 1548-7199
VL  - 7
IS  - 4
SP  - 223
EP  - 244
PB  - Old City Publishing Science
CY  - Philadelphia
ER  -