Kévin Perrot

Feedback Arc Set Problem and NP-Hardness of Minimum Recurrent Configuration Problem of Chip-Firing Game on Directed Graphs

In this paper we present further studies of recurrent configurations of chip-firing games on Eulerian directed graphs (simple digraphs), a class on the way from undirected graphs to general directed graphs. A computational problem that arises naturally from this model is to find the minimum number of chips of a recurrent configuration, which we call the minimum recurrent configuration (MINREC) problem. We point out a close relationship between MINREC and the minimum feedback arc set (MINFAS) problem on Eulerian directed graphs, and prove that both problems are NP-hard.

Transduction on Kadanoff sand pile model avalanches, application to wave pattern emergence

Sand pile models are dynamical systems describing the evolution from N stacked grains to a stable configuration. It uses local rules to depict grain moves and iterate it until reaching a fixed configuration from which no rule can be applied. The main interest of sand piles relies in their Self Organized Criticality (SOC), the property that a small perturbation -- adding some sand grains -- on a fixed configuration has uncontrolled consequences on the system, involving an arbitrary number of grain fall. Physicists L. Kadanoff et al inspire KSPM, a model presenting a sharp SOC behavior, extending the well known Sand Pile Model. In KSPM(D), we start from a pile of N stacked grains and apply the rule: D-1 grains can fall from column i onto the D-1 adjacent columns to the right if the difference of height between columns i and i+1 is greater or equal to D. This paper develops a formal background for the study of KSPM fixed points. This background, resumed in a finite state word transducer, is used to provide a plain formula for fixed points of KSPM(3).

Avalanche structure in the kadanoff sand pile model

Sand pile models are dynamical systems emphasizing the phenomenon of Self Organized Criticality (SOC). From N stacked grains, iterating evolution rules leads to some critical configuration where a small disturbance has deep consequences on the system, involving numerous steps of grain fall. Physicists L. Kadanoff et al inspire KSPM, a model presenting a sharp SOC behavior, extending the well known Sand Pile Model. In KSPM with parameter D we start from a pile of N stacked grains and apply the rule: D-1 grains can fall from column i onto the D-1 adjacent columns to the right if the difference of height between columns i and i+1 is greater or equal to D. We propose an iterative study of KSPM evolution where one single grain addition is repeated on a heap of sand. The sequence of grain falls following a single grain addition is called an avalanche. From a certain column precisely studied for D = 3, we provide a plain process describing avalanches. We hope that this process is a first stone toward the study of KSPM fixed points structure.

Emergence of Wave Patterns on Kadanoff Sandpiles

Emergence is a concept that is easy to exhibit, but very hard to formally handle. This paper is about cubic sand grains moving around on nicely packed columns in one dimension (the physical sandpile is two dimensional, but the support of sand columns is one dimensional). The Kadanoff Sandpile Model is a discrete dynamical system describing the evolution of a finite number of stacked grains —as they would fall from an hourglass— to a stable configuration (fixed point). Grains move according to the repeated application of a simple local rule until reaching a fixed point. The main interest of the model relies in the difficulty of understanding its behavior, despite the simplicity of the rule. In this paper we prove the emergence of wave patterns periodically repeated on fixed points. Remarkably, those regular patterns do not cover the entire fixed point, but eventually emerge from a seemingly highly disordered segment. The proof technique we set up associated arguments of linear algebra and combinatorics, which interestingly allow to formally state the emergence of regular patterns without requiring a precise understanding of the chaotic initial segment’s dynamic.

Kadanoff sand pile model. Avalanche structure and wave shape

Sand pile models are dynamical systems describing the evolution from NN stacked grains to a stable configuration. It uses local rules to depict grain moves and iterate it until reaching a fixed configuration from which no rule can be applied. Physicists L. Kadanoff et al. inspire KSPM, extending the well known Sand Pile Model (SPM). In KSPM(DD), we start from a pile of NN stacked grains and apply the rule: D−1D−1 grains can fall from column ii onto columns i+1,i+2,...,i+D−1i+1,i+2,...,i+D−1 if the difference of height between columns ii and i+1i+1 is greater or equal to DD. Toward the study of fixed points (stable configurations on which no grain can move) obtained from NN stacked grains, we propose an iterative study of KSPM evolution consisting in the repeated addition of one grain on a heap of sand, triggering an avalanche at each iteration. We develop a formal background for the study of avalanches, resumed in a finite state word transducer, and explain how this transducer may be used to predict the form of fixed points. Further precise developments provide a plain formula for fixed points of KSPM(3), showing the emergence of a wavy shape.

On the complexity of two-dimensional signed majority cellular automata

We study the complexity of signed majority cellular automata on the planar grid. We show that, depending on their symmetry and uniformity, they can simulate different types of logical circuitry under different modes. We use this to establish new bounds on their overall complexity, concretely: the uniform asymmetric and the non-uniform symmetric rules are Turing universal and have a P-complete prediction problem; the non-uniform asymmetric rule is in-trinsically universal; no symmetric rule can be intrinsically universal. We also show that the uniform asymmetric rules exhibit cycles of super-polynomial length, whereas symmetric ones are known to have bounded cycle length.

On the Flora of Asynchronous Locally Non-monotonic Boolean Automata Networks

Boolean automata networks (BANs) are a well established model for biological regulation systems such as neural networks or genetic networks. Studies on the dynamics of BANs, whether it is synchronous or asynchronous, have mainly focused on monotonic networks, where fundamental questions on the links relating their static and dynamical properties have been raised and addressed. This paper explores analogous questions on asynchronous non-monotonic networks, xor-BANs, that are BANs where all the local transition functions are xor-functions. Using algorithmic tools, we give a general characterisation of the asynchronous transition graphs for most of the cactus xor-BANs and strongly connected xor-BANs. As an illustration of the results, we provide a complete description of the asynchronous dynamics of two particular classes of xor-BAN, namely xor-Flowers and xor-Cycle Chains. This work also leads to new bisimulation equivalences specific to xor-BANs.

Linearity is Strictly More Powerful than Contiguity for Encoding Graphs

Linearity and contiguity are two parameters devoted to graph encoding. Linearity is a generalisation of contiguity in the sense that every encoding achieving contiguity k induces an encoding achieving linearity k, both encoding having size \(\Theta (k.n)\), where n is the number of vertices of G. In this paper, we prove that linearity is a strictly more powerful encoding than contiguity, i.e. there exists some graph family such that the linearity is asymptotically negligible in front of the contiguity. We prove this by answering an open question asking for the worst case linearity of a cograph on n vertices: we provide an \(O(\log n/\log \log n)\) upper bound which matches the previously known lower bound.

Computational Complexity of the Avalanche Problem on One Dimensional Kadanoff Sandpiles

In this paper we prove that the general avalanche problem AP is in NC for the Kadanoff sandpile model in one dimension, answering an open problem of [2]. Thus adding one more item to the (slowly) growing list of dimension sensitive problems since in higher dimensions the problem is P-complete (for monotone sandpiles).

A framework for (de)composing with Boolean automata networks

Boolean automata networks (BANs) are a generalisation of Boolean cellular automata. In such, any theorem describing the way BANs compute information is a strong tool that can be applied to a wide range of models of computation. In this paper we explore a way of working with BANs which involves adding external inputs to the base model (via modules), and more importantly, a way to link networks together using the above mentioned inputs (via wirings). Our aim is to develop a powerful formalism for BAN (de)composition. We formulate two results: the first one shows that our modules/wirings definition is complete; the second one uses modules/wirings to prove simulation results amongst BANs.