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arXiv2019

Universal One-Dimensional Cellular Automata Derived for Turing Machines and its Dynamical Behaviour

Universality in cellular automata theory is a central problem studied and developed from their origins by John von Neumann. In this paper, we present an algorithm where any Turing machine can be converted to one-dimensional cellular automaton with a 2-linear time and display its spatial dynamics. Three particular Turing machines are converted in three universal one-dimensional cellular automata, they are: binary sum, rule 110 and a universal reversible Turing machine.

nlin.CG↗

arXiv2019

Efficient methods to determine the reversibility of general 1D linear cellular automata in polynomial complexity

In this paper, we study reversibility of one-dimensional(1D) linear cellular automata(LCA) under null boundary condition, whose core problems have been divided into two main parts: calculating the period of reversibility and verifying the reversibility in a period. With existing methods, the time and space complexity of these two parts are still too expensive to be employed. So the process soon becomes totally incalculable with a slightly big size, which greatly limits its application. In this paper, we set out to solve these two problems using two efficient algorithms, which make it possible to solve reversible LCA of very large size. Furthermore, we provide an interesting perspective to conversely generate 1D LCA from a given period of reversibility. Due to our methods' efficiency, we can calculate the reversible LCA with large size, which has much potential to enhance security in cryptography system.

cs.CC↗

arXiv2022

Freezing, Bounded-Change and Convergent Cellular Automata

This paper studies three classes of cellular automata from a computational point of view: freezing cellular automata where the state of a cell can only decrease according to some order on states, cellular automata where each cell only makes a bounded number of state changes in any orbit, and finally cellular automata where each orbit converges to some fixed point. Many examples studied in the literature fit into these definitions, in particular the works on cristal growth started by S. Ulam in the 60s. The central question addressed here is how the computational power and computational hardness of basic properties is affected by the constraints of convergence, bounded number of change, or local decreasing of states in each cell. By studying various benchmark problems (short-term prediction, long term reachability, limits) and considering various complexity measures and scales (LOGSPACE vs. PTIME, communication complexity, Turing computability and arithmetical hierarchy) we give a rich and nuanced answer: the overall computational complexity of such cellular automata depends on the class considered (among the three above), the dimension, and the precise problem studied. In particular, we show that all settings can achieve universality in the sense of Blondel-Delvenne-Kůrka, although short term predictability varies from NLOGSPACE to P-complete. Besides, the computability of limit configurations starting from computable initial configurations separates bounded-change from convergent cellular automata in dimension~1, but also dimension~1 versus higher dimensions for freezing cellular automata. Another surprising dimension-sensitive result obtained is that nilpotency becomes decidable in dimension~ 1 for all the three classes, while it stays undecidable even for freezing cellular automata in higher dimension.

cs.DM↗

arXiv2019

Max-Plus Generalization of Conway's Game of Life

We propose a max-plus equation which includes Conway's Game of Life (GoL) as a special case. There are some special solutions to the equation which include and unify those to GoL. Moreover, the multi-value extension of GoL is derived from the equation and the behavior of solutions is discussed.

nlin.CG↗

arXiv2019

Periodic solutions of one-dimensional cellular automata with random rules

We study cellular automata with randomly selected rules. Our setting are two-neighbor rules with a large number $n$ of states. The main quantity we analyze is the asymptotic probability, as $n \to \infty$, that the random rule has a periodic solution with given spatial and temporal periods. We prove that this limiting probability is non-trivial when the spatial and temporal periods are confined to a finite range. The main tool we use is the Chen-Stein method for Poisson approximation. The limiting probability distribution of the smallest temporal period for a given spatial period is deduced as a corollary and relevant empirical simulations are presented.

math.PR↗

arXiv2019

The Variant-Rule, Another Logically Universal Rule

The Variant-rule derives from the Precursor-rule by interchanging two classes of its 28 isotropic mappings. Although this small mutation conserves most glider types and stable blocks, glider-gun engines are changed, as are most large scale pattern behaviors, illustrating both the robustness and fragility of evolution. We demonstrate these newly discovered structures and dynamics, and utilising two different glider types, build the logical gates required for universality in the logical sence.

nlin.CG↗

arXiv2019

Hurwitz Orbits with a Cellular Automaton

Hurwitz orbits are the orbits of the braid group action on the powers of a rack. Hurwitz orbits for the action of the braid group on three strands are used in \cite{21} and \cite{22} for the classification of Nichols algebras. This classification is based on a combinatorial invariant called plague on the Hurwitz orbits. The method to calculate plagues on the Hurwitz orbits is formulated in \cite{22} by using a cellular automaton on the Hurwitz orbits and their quotients. By using this cellular automaton-based method we estimate the minimal plagues on the Hurwitz orbits.

math.GR↗

arXiv2019

A hierarchical structure in the motion representation of 2-state number-conserving cellular automata

A one-dimensional two-state number-conserving cellular automaton (NCCA) is a cellular automaton whose states are 0 or 1 and where cells take states 0 and 1 and updated their states by the rule which keeps overall sum of states constant. It can be regarded as a kind of particle based modeling of physical systems and has another intuitive representation, motion representation, based on the movement of each particle. We introduced a kind of hierarchical interpretation of motion representations to understand the necessary pattern size to each motion. We show any NCCA of its neighborhood size n can be hierarchically represented by NCCAs of their neighborhood size from n-1 to 1.

nlin.CG↗

arXiv2019

On the Complexity of Asynchronous Freezing Cellular Automata

In this paper we study the family of freezing cellular automata (FCA) in the context of asynchronous updating schemes. A cellular automaton is called freezing if there exists an order of its states, and the transitions are only allowed to go from a lower to a higher state. A cellular automaton is asynchronous if at each time-step only one cell is updated. Given configuration, we say that a cell is unstable if there exists a sequential updating scheme that changes its state. In this context, we define the problem AsyncUnstability, which consists in deciding if a cell is unstable or not. In general AsyncUnstability is in NP, and we study in which cases we can solve the problem by a more efficient algorithm. We begin showing that AsyncUnstability is in NL for any one-dimensional FCA. Then we focus on the family of life-like freezing CA (LFCA), which is a family of two-dimensional two-state FCA that generalize the freezing version of the game of life, known as life without death. We study the complexity of AsyncUnstability for all LFCA in the triangular and square grids, showing that almost all of them can be solved in NC, except for one rule for which the problem is NP-complete.

cs.CC↗

arXiv2019

Reversible Computation in Wireless Communications

This chapter presents the pioneering work in applying reversible computation paradigms to wireless communications. These applications range from developing reversible hardware architectures for underwater acoustic communications to novel distributed optimisation procedures in large radio-frequency antenna arrays based on reversing Petri nets. Throughout the chapter, we discuss the rationale for introducing reversible computation in the domain of wireless communications, exploring the inherently reversible properties of communication channels and systems formed by devices in a wireless network.

cs.ET↗

arXiv2019

On the Complexity of the Stability Problem of Binary Freezing Totalistic Cellular Automata

In this paper we study the family of two-state Totalistic Freezing Cellular Automata (TFCA) defined over the triangular and square grids with von Neumann neighborhoods. We say that a Cellular Automaton is Freezing and Totalistic if the active cells remain unchanged, and the new value of an inactive cell depends only on the sum of its active neighbors. We classify all the Cellular Automata in the class of TFCA, grouping them in five different classes: the Trivial rules, Turing Universal rules,Algebraic rules, Topological rules and Fractal Growing rules. At the same time, we study in this family the Stability problem, consisting in deciding whether an inactive cell becomes active, given an initial configuration.We exploit the properties of the automata in each group to show that: - For Algebraic and Topological Rules the Stability problem is in $\text{NC}$. - For Turing Universal rules the Stability problem is $\text{P}$-Complete.

cs.DS↗

arXiv2020

Bunch width versus macrostep height: A quantitative study of the effects of step-step repulsion

Bunching of steps at the surface of growing crystals can be induced by both directions of the driving force: step up and step down. The processes happen in different adatom concentrations and differ in character. In this study we show how the overall picture of the bunching process depends on the strength of short range step-step repulsion. The repulsive interaction between steps, controlled by an additional parameter, is introduced into the recently studied atomistic scale model of vicinal crystal growth, based on cellular automata. It is shown that the repulsion modifies bunching process in a different way, depending on the direction of the destabilizing force. In particular, bunch profiles, stability diagrams and time-scaling dependences of various bunch properties are affected when the step-step repulsion increases. The repulsion between steps creates a competition between two characteristic sizes - bunch width and macrostep height, playing the role of the second length scale that describes the step bunching phenomenon. A new characteristic time scale dependent on the step-step repulsion parameter emerges as an effect of interplay between (01) faceted macrosteps and (11) faceted bunches. The bunch height being the major characteristic size of the bunches is not influenced dramatically by the repulsion.

cond-mat.mtrl-sci↗

arXiv2020

Elementary Cellular Automata along with delay sensitivity can model communal riot dynamics

This work explores the potential of elementary cellular automata to model the dynamics of riot. Here, to model such dynamics, we introduce probabilistic loss of information and delay perturbation in the updating scheme of automata to capture sociological parameters - presence of anti-riot population and organizational presence of communal forces in the rioting society respectively. Moreover, delay has also been incorporated in the model to capture the non-local interaction of neighbours. Finally, the model is verified by a recent event of riot that occurred in Baduria of West Bengal, India.

nlin.CG↗

arXiv2020

On computing the Lyapunov exponents of reversible cellular automata

We consider the problem of computing the Lyapunov exponents of reversible cellular automata (CA). We show that the class of reversible CA with right Lyapunov exponent $2$ cannot be separated algorithmically from the class of reversible CA whose right Lyapunov exponents are at most $2-δ$ for some absolute constant $δ>0$. Therefore there is no algorithm that, given as an input a description of an arbitrary reversible CA $F$ and a positive rational number $ε>0$, outputs the Lyapunov exponents of $F$ with accuracy $ε$. We also compute the average Lyapunov exponents (with respect to the uniform measure) of the CA that perform multiplication by $p$ in base $pq$ for coprime $p,q>1$.

math.DS↗

arXiv2020

On a class of new nonlocal traffic flow models with look-ahead rules

This paper presents a new class of one-dimensional (1D) traffic models with look-ahead rules that take into account of two effects: nonlocal slow-down effect and right-skewed non-concave asymmetry in the fundamental diagram. The proposed 1D cellular automata (CA) models with the Arrhenius type look-ahead interactions implement stochastic rules for cars' movement following the configuration of the traffic ahead of each car. In particular, we take two different look-ahead rules: one is based on the distance from the car under consideration to the car in front of it; the other one depends on the car density ahead. Both rules feature a novel idea of multiple moves, which plays a key role in recovering the non-concave flux in the macroscopic dynamics. Through a semi-discrete mesoscopic stochastic process, we derive the coarse-grained macroscopic dynamics of the CA model. We also design a numerical scheme to simulate the proposed CA models with an efficient list-based kinetic Monte Carlo (KMC) algorithm. Our results show that the fluxes of the KMC simulations agree with the coarse-grained macroscopic averaged fluxes for the different look-ahead rules under various parameter settings.

nlin.CG↗

arXiv2020

LUNAR: Cellular Automata for Drifting Data Streams

With the advent of huges volumes of data produced in the form of fast streams, real-time machine learning has become a challenge of relevance emerging in a plethora of real-world applications. Processing such fast streams often demands high memory and processing resources. In addition, they can be affected by non-stationary phenomena (concept drift), by which learning methods have to detect changes in the distribution of streaming data, and adapt to these evolving conditions. A lack of efficient and scalable solutions is particularly noted in real-time scenarios where computing resources are severely constrained, as it occurs in networks of small, numerous, interconnected processing units (such as the so-called Smart Dust, Utility Fog, or Swarm Robotics paradigms). In this work we propose LUNAR, a streamified version of cellular automata devised to successfully meet the aforementioned requirements. It is able to act as a real incremental learner while adapting to drifting conditions. Extensive simulations with synthetic and real data will provide evidence of its competitive behavior in terms of classification performance when compared to long-established and successful online learning methods.

cs.LG↗

arXiv2020

$q$-VFCA: $q$-state Vector-valued Fuzzy Cellular Automata

Elementary fuzzy Cellular Automata (CA) are known as continuous counterpart of elementary CA, which are 2-state CA, via the polynomial representation of local rules. In this paper, we first develop a new fuzzification methodology for $q$-state CA. It is based on the vector representation of $q$-state CA, that is, the $q$-states are assigned to the standard basis vectors of the $q$-dimensional real space and the local rule can be expressed by a tuple of $q$ polynomials. Then, the $q$-state vector-valued fuzzy CA are defined by expanding the set of the states to the convex hull of the standard basis vectors in the $q$-dimensional real space. The vector representation of states enables us to enumerate the number-conserving rules of 3-state vector-valued fuzzy CA in a systematic way.

nlin.CG↗

arXiv2020

The program-size complexity of self-assembled paths

We prove a Pumping Lemma for the noncooperative abstract Tile Assembly Model, a model central to the theory of algorithmic self-assembly since the beginning of the field. This theory suggests, and our result proves, that small differences in the nature of adhesive bindings between abstract square molecules gives rise to vastly different expressive capabilities. In the cooperative abstract Tile Assembly Model, square tiles attach to each other using multi-sided cooperation of one, two or more sides. This precise control of tile binding is directly exploited for algorithmic tasks including growth of specified shapes using very few tile types, as well as simulation of Turing machines and even self-simulation of self-assembly systems. But are cooperative bindings required for these computational tasks? The definitionally simpler noncooperative (or Temperature 1) model has poor control over local binding events: tiles stick if they bind on at least one side. This has led to the conjecture that it is impossible for it to exhibit precisely controlled growth of computationally-defined shapes. Here, we prove such an impossibility result. We show that any planar noncooperative system that attempts to grow large algorithmically-controlled tile-efficient assemblies must also grow infinite non-algorithmic (pumped) structures with a simple closed-form description, or else suffer blocking of intended algorithmic structures. Our result holds for both directed and nondirected systems, and gives an explicit upper bound of $(8|T|)^{4|T|+1}(5|σ| + 6)$, where $|T|$ is the size of the tileset and $|σ|$ is the size of the seed assembly, beyond which any path of tiles is pumpable or blockable.

cs.CC↗