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In this paper, we look at two ways to implement determinisitic one dimensional cellular automata into hyperbolic cellular automata in three contexts: the pentagrid, the heptagrid and the dodecagrid, these tilings being classically denoted by $\{5,4\}$, $\{7,3\}$ and $\{5,3,4\}$ respectively.
Causality among events is widely recognized as a most fundamental structure of spacetime, and causal sets have been proposed as discrete models of the latter in the context of quantum gravity theories, notably in the Causal Set Programme. In the rather different context of what might be called the 'Computational Universe Programme' -- one which associates the complexity of physical phenomena to the emergent features of models such as cellular automata -- a choice problem arises with respect to the variety of formal systems that, in virtue of their computational universality (Turing-completeness), qualify as equally good candidates for a computational, unified theory of physics. This paper proposes Causal Sets as the only objects of physical significance and relevance to be considered under the 'computational universe' perspective, and as the appropriate abstraction for shielding the unessential details of the many different computationally universal candidate models. At the same time, we propose a fully deterministic, radical alternative to the probabilistic techniques currently considered in the Causal Set Programme for growing discrete spacetimes. We investigate a number of computation models by grouping them into two broad classes, based on the support on which they operate; in one case this is linear, like a tape or a string of symbols; in the other, it is a two-dimensional grid or a planar graph. For each model we identify the causality relation among computation events, implement it, and conduct a possibly exhaustive exploration of the associated causal set space, while examining quantitative and qualitative features such as dimensionality, curvature, planarity, emergence of pseudo-randomness, causal set substructures and particles.
A class of additive cellular automata (ACA) on a finite group is defined by an index-group $\m g$ and a finite field $\m F_p$ for a prime modulus $p$ \cite{Bul_arch_1}. This paper deals mainly with ACA on infinite commutative groups and direct products of them with some non commutative $p$-groups. It appears that for all abelian groups, the rules and initial states with finite supports define behaviors which being restricted to some infinite regular series of time moments become significantly simplified. In particular, for free abelian groups with $n$ generators states $V^{[t]}$ of ACA with a rule $R$ at time moments $t=p^k,k>k_0,$ can be viewed as $||R||$ copies of initial state $V^{[0]}$ moving through an $n$-dimensional Euclidean space. That is the behavior is similar to gliders from J.Conway's automaton {\sl Life}. For some other special infinite series of time moments the automata states approximate self-similar structures and the approximation becomes better with time. An infinite class $\mathrm{DHC}(\mbf S,θ)$ of non-commutative $p$-groups is described which in particular includes quaternion and dihedral $p$-groups. It is shown that the simplification of behaviors takes place as well for direct products of non-commutative groups from the class $\mathrm{DHC}(\mbf S,θ)$ with commutative groups. Finally, an automaton on a non-commutative group is constructed such that its behavior at time moments $2^k,k\ge2,$ is similar to a glider gun. It is concluded that ACA on non-commutative groups demonstrate more diverse variety of behaviors comparing to ACA on commutative groups.
Random Boolean networks (RBNs) are models of genetic regulatory networks. It is useful to describe RBNs as self-organizing systems to study how changes in the nodes and connections affect the global network dynamics. This article reviews eight different methods for guiding the self-organization of RBNs. In particular, the article is focussed on guiding RBNs towards the critical dynamical regime, which is near the phase transition between the ordered and dynamical phases. The properties and advantages of the critical regime for life, computation, adaptability, evolvability, and robustness are reviewed. The guidance methods of RBNs can be used for engineering systems with the features of the critical regime, as well as for studying how natural selection evolved living systems, which are also critical.
In this paper, we investigate the non-signalized intersection issue considering traffic flow and energy dissipation in terms of game theory based on the Nagel-Schreckenberg (NaSch) model. There are two types of driver agents at the intersection when vehicles on the two streets are approaching to it simultaneously: C agents (cooperative strategy) pulling up to avoid collision and D agents (defective strategy) crossing the intersection audaciously. Phase diagram of the system, which describes free-flow phase, segregation phase, jammed phase and maximum current curve representing the social maximum payoff, is presented. Dilemma game is observed at the phase-segregated state except for the case of $v_{\max }=1.$
The Fermi Paradox is the apparent contradiction between the high probability extraterrestrial civilizations' existence and the lack of contact with such civilizations. In general, solutions to Fermi's paradox come down to either estimation of Drake equation parameters i.e. our guesses about the potential number of extraterrestrial civilizations or simulation of civilizations development in the universe. We consider a new type of cellular automata, that allows to analyze Fermi paradox. We introduce bonus stimulation model (BS-model) of development in cellular space (Universe) of objects (Civilizations). When civilizations get in touch they stimulate development each other, increasing their life time. We discovered nonlinear threshold behaviour of total volume of civilizations in universe and on the basis of our model we built analogue of Drake equation.
This is a study of the one-dimensional elementary cellular automaton rule 54 in the new formalism of "flexible time". We derive algebraic expressions for groups of several cells and their evolution in time. With them we can describe the behaviour of simple periodic patterns like the ether and gliders in an efficient way. We use that to look into their behaviour in detail and find general formulas that characterise them.
A large number of models for pedestrian dynamics have been developed over the years. However, so far not much attention has been paid to their quantitative validation. Usually the focus is on the reproduction of empirically observed collective phenomena, as lane formation in counterflow. This can give an indication for the realism of the model, but practical applications, e.g. in safety analysis, require quantitative predictions. We discuss the current experimental situation, especially for the fundamental diagram which is the most important quantity needed for calibration. In addition we consider the implications for the modelling based on cellular automata. As specific example the floor field model is introduced. Apart from the properties of its fundamental diagram we discuss the implications of an egress experiment for the relevance of conflicts and friction effects.
In the present article we emphasize the importance of modeling time in the context of agent-based models. To this end, we present a (selective) survey of the Cellular Automata-literature on updating and draw parallels to the issue of agent activation in agent-based models. By means of two simple models, Schelling's segregation model and Epstein's demographic prisoner's dilemma we investigate the influence of choosing different regimes of agent activation. Our experiments indicate that timing is not a critical issue for very simple models but bears huge influence on model behavior and results as soon as the degree of complexity increases only so slightly. After a brief review of the way commonly used ABM simulation environments handle the issue of timing, we draw some tentative conclusions about the importance of timing and the need for more research towards that direction, similar to the concerted effort on updating in cellular automata.
In this paper the theory of 2-Variable Boolean Operation (2-VBO) has been discussed on a pair of n-bit strings. 2-VBO serves to bring out the relation between numbers which when plot on a 2-D surface form interesting patterns; patterns that may be fixed, periodic, chaotic or complex. Some of these patterns represent natural fractals. This paper also provides mathematical analysis corresponding to each of the obtained patterns, which would aid to understanding their formation. 2-VBO is an attempt towards the production and classification of patterns which represent various mathematical models and naturally occurring phenomena.
We present both simulated and partial empirical evidence for the computational utility of many connected vesicle analogs of an encapsulated non-linear chemical processing medium. By connecting small vesicles containing a solution of sub-excitable Belousov-Zhabotinsky (BZ) reaction, sustained and propagating wave fragments are modulated by both spatial geometry, network connectivity and their interaction with other waves. The processing ability is demonstrated through the creation of simple Boolean logic gates and then by the combination of those gates to create more complex circuits.
We propose a method for deriving networks from one-dimensional binary cellular automata. The derived networks are usually directed and have structural properties corresponding to the dynamical behaviors of their cellular automata. Network parameters, particularly the efficiency and the degree distribution, show that the dependence of efficiency on the grid size is characteristic and can be used to classify cellular automata and that derived networks exhibit various degree distributions. In particular, a class IV rule of Wolfram's classification produces a network having a scale-free distribution.
Diverging junctions are important network bottlenecks, and a better understanding of diverging traffic dynamics has both theoretical and practical implications. In this paper, we first introduce a continuous multi-commodity kinematic wave model of diverging traffic and then present a new framework for constructing kinematic wave solutions to its Riemann problem with jump initial conditions. In supply-demand space, the solutions on a link consist of an interior state and a stationary state, subject to admissible conditions such that there are no positive and negative kinematic waves on the upstream and downstream links respectively. In addition, the solutions have to satisfy entropy conditions consistent with various discrete diverge models. In the proposed analytical framework, kinematic waves on each link can be uniquely determined by the stationary and initial conditions, and we prove that the stationary states and boundary fluxes exist and are unique for the Riemann problem of diverge models when all or partial of vehicles have predefined routes. We show that the two diverge models by Lebacque and Daganzo are asymptotically equivalent. We also prove that the supply-proportional and priority-based diverge models are locally optimal evacuation strategies. With numerical examples, we demonstrate the validity of the analytical solutions of interior states, stationary states, and corresponding kinematic waves. This study presents a unified framework for analyzing traffic dynamics arising in diverging traffic and could be helpful for developing emergency evacuation strategies.
A highly efficient three-dimensional (3D) Lattice Boltzmann (LB) model for high speed compressible flows is proposed. This model is developed from the original one by Kataoka and Tsutahara[Phys. Rev. E 69, 056702 (2004)]. The convection term is discretized by the Non-oscillatory, containing No free parameters and Dissipative (NND) scheme, which effectively damps oscillations at discontinuities. To be more consistent with the kinetic theory of viscosity and to further improve the numerical stability, an additional dissipation term is introduced. Model parameters are chosen in such a way that the von Neumann stability criterion is satisfied. The new model is validated by well-known benchmarks, (i) Riemann problems, including the problem with Lax shock tube and a newly designed shock tube problem with high Mach number; (ii) reaction of shock wave on droplet or bubble. Good agreements are obtained between LB results and exact ones or previously reported solutions. The model is capable of simulating flows from subsonic to supersonic and capturing jumps resulted from shock waves.
In this report, we present a formal approach that addresses the problem of emergence of phase transitions in stochastic and attractive nonlinear threshold Boolean automata networks. Nonlinear networks considered are informally defined on the basis of classical stochastic threshold Boolean automata networks in which specific interaction potentials of neighbourhood coalition are taken into account. More precisely, specific nonlinear terms compose local transition functions that define locally the dynamics of such networks. Basing our study on nonlinear networks, we exhibit new results, from which we derive conditions of phase transitions.
Cellular automata have been mainly studied on very regular graphs carrying the vertices (like lines or grids) and under synchronous dynamics (all vertices update simultaneously). In this paper, we study how the asynchronism and the graph act upon the dynamics of the classical Minority rule. Minority has been well-studied for synchronous updates and is thus a reasonable choice to begin with. Yet, beyond its apparent simplicity, this rule yields complex behaviors when asynchronism is introduced. We investigate the transitory part as well as the asymptotic behavior of the dynamics under full asynchronism (also called sequential: only one random vertex updates at each time step) for several types of graphs. Such a comparative study is a first step in understanding how the asynchronous dynamics is linked to the topology (the graph). Previous analyses on the grid [1,2] have observed that Minority seems to induce fast stabilization. We investigate here this property on arbitrary graphs using tools such as energy, particles and random walks. We show that the worst case convergence time is, in fact, strongly dependent on the topology. In particular, we observe that the case of trees is non trivial.
Existence of large-amplitude time-periodic breathers localized near a single site is proved for the discrete Klein--Gordon equation, in the case when the derivative of the on-site potential has a compact support. Breathers are obtained at small coupling between oscillators and under nonresonance conditions. Our method is different from the classical anti-continuum limit developed by MacKay and Aubry, and yields in general branches of breather solutions that cannot be captured with this approach. When the coupling constant goes to zero, the amplitude and period of oscillations at the excited site go to infinity. Our method is based on near-identity transformations, analysis of singular limits in nonlinear oscillator equations, and fixed-point arguments.
Based on a Boltzmann-like traffic equation for aggressive drivers we construct in this paper a second-order continuum traffic model which is similar to the Navier-Stokes equations for viscous fluids by applying two well-known methods of gas-kinetic theory, namely: the Chapman-Enskog method and the method of moments of Grad. The viscosity coefficient appearing in our macroscopic traffic model is not introduced in an ad hoc way - as in other second-order traffic flow models - but comes into play through the derivation of a first-order constitutive relation for the traffic pressure. Numerical simulations show that our Navier-Stokes-like traffic model satisfies the anisotropy condition and produces numerical results which are consistent with our daily experiences in real traffic.