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Register a new userTwo dimensional outflows for cellular automata with shuffle updates
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In this paper, we explore the two-dimensional behavior of cellular automata with shuffle updates. As a test case, we consider the evacuation of a square room by pedestrians modeled by a cellular automaton model with a static floor field. Shuffle updates are characterized by a variable associated to each particle and called phase, that can be interpreted as the phase in the step cycle in the frame of pedestrian flows. Here we also introduce a dynamics for these phases, in order to modify the properties of the model. We investigate in particular the crossover between low- and high-density regimes that occurs when the density of pedestrians increases, the dependency of the outflow in the strength of the floor field, and the shape of the queue in front of the exit. Eventually we discuss the relevance of these results for pedestrians.
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In studying the predictability of emergent phenomena in complex systems, Israeli & Goldenfeld (Phys. Rev. Lett., 2004; Phys. Rev. E, 2006) showed how to coarse-grain (elementary) cellular automata (CA). Their algorithm for finding coarse-grainings of supercell size $N$ took doubly-exponential $2^{2^N}$-time, and thus only allowed them to explore supercell sizes $N leq 4$. Here we introduce a new, more efficient algorithm for finding coarse-grainings between any two given CA that allows us to systematically explore all elementary CA with supercell sizes up to $N=7$, and to explore individual examples of even larger supercell size. Our algorithm is based on a backtracking search, similar to the DPLL algorithm with unit propagation for the NP-complete problem of Boolean Satisfiability.
Cellular Automaton (CA) and an Integral Value Transformation (IVT) are two well established mathematical models which evolve in discrete time steps. Theoretically, studies on CA suggest that CA is capable of producing a great variety of evolution patterns. However computation of non-linear CA or higher dimensional CA maybe complex, whereas IVTs can be manipulated easily. The main purpose of this paper is to study the link between a transition function of a one-dimensional CA and IVTs. Mathematically, we have also established the algebraic structures of a set of transition functions of a one-dimensional CA as well as that of a set of IVTs using binary operations. Also DNA sequence evolution has been modelled using IVTs.
A transition from asymmetric to symmetric patterns in time-dependent extended systems is described. It is found that one dimensional cellular automata, started from fully random initial conditions, can be forced to evolve into complex symmetrical patterns by stochastically coupling a proportion $p$ of pairs of sites located at equal distance from the center of the lattice. A nontrivial critical value of $p$ must be surpassed in order to obtain symmetrical patterns during the evolution. This strategy is able to classify the cellular automata rules -with complex behavior- between those that support time-dependent symmetric patterns and those which do not support such kind of patterns.
Gauge symmetries play a fundamental role in Physics, as they provide a mathematical justification for the fundamental forces. Usually, one starts from a non-interactive theory which governs `matter, and features a global symmetry. One then extends the theory so as make the global symmetry into a local one (a.k.a gauge-invariance). We formalise a discrete counterpart of this process, known as gauge extension, within the Computer Science framework of Cellular Automata (CA). We prove that the CA which admit a relative gauge extension are exactly the globally symmetric ones (a.k.a the colour-blind). We prove that any CA admits a non-relative gauge extension. Both constructions yield universal gauge-invariant CA, but the latter allows for a first example where the gauge extension mediates interactions within the initial CA.
We investigate number conserving cellular automata with up to five inputs and two states with the goal of comparing their dynamics with diffusion. For this purpose, we introduce the concept of decompression ratio describing expansion of configurations with finite support. We find that a large number of number-conserving rules exhibit abrupt change in the decompression ratio when the density of the initial pattern is increasing, somewhat analogous to the second order phase transition. The existence of this transition is formally proved for rule 184. Small number of rules exhibit infinite decompression ratio, and such rules may be useful for engineering of CA rules which are good models of diffusion, although they will most likely require more than two states.
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