Basic properties for sand automata

Presented at MFCS 2005 (Gdansk, POLAND). Long version with complete proofs published in Theoretical Computer Science, 2006, under the title "From Sandpiles to Sand Automata".International audienceWe prove several results about the relations between injectivity and surjectivity for sand automata. Moreover, we begin the exploration of the dynamical behavior of sand automata proving that the property of nilpotency is undecidable. We believe that the proof technique used for this last result might reveal useful for many other results in this context

Energy constrained sandpile models

We study two driven dynamical systems with conserved energy. The two automata contain the basic dynamical rules of the Bak, Tang and Wiesenfeld sandpile model. In addition a global constraint on the energy contained in the lattice is imposed. In the limit of an infinitely slow driving of the system, the conserved energy $E$ becomes the only parameter governing the dynamical behavior of the system. Both models show scale free behavior at a critical value $E_c$ of the fixed energy. The scaling with respect to the relevant scaling field points out that the developing of critical correlations is in a different universality class than self-organized critical sandpiles. Despite this difference, the activity (avalanche) probability distributions appear to coincide with the one of the standard self-organized critical sandpile.Comment: 4 pages including 3 figure

Short period attractors and non-ergodic behavior in the deterministic fixed energy sandpile model

We study the asymptotic behaviour of the Bak, Tang, Wiesenfeld sandpile automata as a closed system with fixed energy. We explore the full range of energies characterizing the active phase. The model exhibits strong non-ergodic features by settling into limit-cycles whose period depends on the energy and initial conditions. The asymptotic activity $\rho_a$ (topplings density) shows, as a function of energy density $\zeta$, a devil's staircase behaviour defining a symmetric energy interval-set over which also the period lengths remain constant. The properties of $\zeta$-$\rho_a$ phase diagram can be traced back to the basic symmetries underlying the model's dynamics.Comment: EPL-style, 7 pages, 3 eps figures, revised versio

Large-Scale Synchrony in Weakly Interacting Automata

We study the behavior of two spatially distributed (sandpile) models which are weakly linked with one another. Using a Monte-Carlo implementation of the renormalization group and algebraic methods, we describe how large-scale correlations emerge between the two systems, leading to synchronized behavior.Comment: 6 pages, 3 figures; to appear PR

Cluster Statistics of BTW automata

The cluster statistics of BTW automata in the SOC states are obtained by extensive computer simulation. Various moments of the clusters are calculated and few results are compared with earlier available numerical estimates and exact results. Reasonably good agreement is observed. An extended statistical analysis has been made.Comment: 8 pages Latex, To appear in Acta Physica Polonica B (2011

Minimizing finite automata is computationally hard

It is known that deterministic finite automata (DFAs) can be algorithmically minimized, i.e., a DFA M can be converted to an equivalent DFA M' which has a minimal number of states. The minimization can be done efficiently [6]. On the other hand, it is known that unambiguous finite automata (UFAs) and nondeterministic finite automata (NFAs) can be algorithmically minimized too, but their minimization problems turn out to be NP-complete and PSPACE-complete [8]. In this paper, the time complexity of the minimization problem for two restricted types of finite automata is investigated. These automata are nearly deterministic, since they only allow a small amount of non determinism to be used. On the one hand, NFAs with a fixed finite branching are studied, i.e., the number of nondeterministic moves within every accepting computation is bounded by a fixed finite number. On the other hand, finite automata are investigated which are essentially deterministic except that there is a fixed number of different initial states which can be chosen nondeterministically. The main result is that the minimization problems for these models are computationally hard, namely NP-complete. Hence, even the slightest extension of the deterministic model towards a nondeterministic one, e.g., allowing at most one nondeterministic move in every accepting computation or allowing two initial states instead of one, results in computationally intractable minimization problems

Cellular Automaton for Realistic Modelling of Landslides

A numerical model is developed for the simulation of debris flow in landslides over a complex three dimensional topography. The model is based on a lattice, in which debris can be transferred among nearest neighbors according to established empirical relationships for granular flows. The model is then validated by comparing a simulation with reported field data. Our model is in fact a realistic elaboration of simpler ``sandpile automata'', which have in recent years been studied as supposedly paradigmatic of ``self-organized criticality''. Statistics and scaling properties of the simulation are examined, and show that the model has an intermittent behavior.Comment: Revised version (gramatical and writing style cleanup mainly). Accepted for publication by Nonlinear Processes in Geophysics. 16 pages, 98Kb uuencoded compressed dvi file (that's the way life is easiest). Big (6Mb) postscript figures available upon request from [email protected] / [email protected]