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Computational Complexity of Avalanches in the Kadanoff Sandpile Model

Formenti E., Goles E., Martin B.

Fundamenta Informaticae

2012

Vol. 115, nr 1

107-124

This paper investigates the avalanche problem AP for the Kadanoff sandpile model (KSPM). We prove that (a slight restriction of) AP is in NC1 in dimension one, leaving the general case open. Moreover, we prove that AP is P-complete in dimension two. The proof of this latter result is based on a reduction from the monotone circuit value problem by building logic gates and wires which work with an initial sand distribution in KSPM. These results are also related to the known prediction problem for sandpiles which is in NC1 for one-dimensional sandpiles and P-complete for dimension 3 or higher. The computational complexity of the prediction problem remains open for the Bak’s model of two-dimensional sandpiles.

A Geometrical Hierarchy on Graphs via Cellular Automata

Martin B.

Fundamenta Informaticae

2002

Vol. 52, nr 1-3

157-181

Historically, cellular automata were defined on the lattices \mathbbZn, but the definition can be extended to bounded degree graphs. Given a notion of simulation between cellular automata defined on different structures (namely graphs of automata), we can deduce an order on graphs. In this paper, we link this order to graph properties and explicit the order for most of the common graphs.