Number-conserving Cellular Automata Research Articles

A decomposition theorem for number-conserving multi-state cellular automata on triangular grids

This paper concerns two-dimensional cellular automata on a triangular grid that preserve the sum of the states of all the cells. To study such cellular automata, we adapt the idea of the split-and-perturb decomposition of a number-conserving local rule, developed first for square grids, to the setting of triangular grids. As a result, we obtain a new mathematical tool that allows, for example, to enumerate all so-called k-ary (i.e., binary, ternary, quaternary, quinary, etc.) number-conserving cellular automata on a triangular grid, regardless of the value of k.

Seven-state rotation-symmetric number-conserving cellular automaton that is not isomorphic to any septenary one.

We consider two-dimensional cellular automata with the von Neumann neighborhood that satisfy two properties of interest from a modeling viewpoint: rotation symmetry (i.e., the local rule is invariant under rotation of the neighborhood by 90^{∘}) and number conservation (i.e., the sum of all the cell states is conserved upon every update). It is known that if the number of states k is smaller than or equal to six, then each rotation-symmetric number-conserving cellular automaton is isomorphic to some k-ary one, i.e., one with state set {0,1,...,k-1}. In this paper, we exhibit an example of a seven-state rotation-symmetric number-conserving cellular automaton that is not isomorphic to any septenary one. This example strongly supports our plea that research into multistate cellular automata should not only focus on those that have {0,1,...,k-1} as a state set.

Non-uniform number-conserving elementary cellular automata

In this paper, we investigate non-uniform elementary cellular automata (i.e., one-dimensional cellular automata whose cells can use different Wolfram rules to update their states) in the context of number conservation. As a result, we obtain an exhaustive characterization of such number-conserving cellular automata on all finite grids both with periodic and null boundary conditions. The characterization obtained allows, inter alia, to enumerate all number-conserving non-uniform elementary cellular automata, in particular those that are reversible. Surprisingly, the numbers obtained are closely related to the Fibonacci sequence.

Number Conservation via Particle Flow in One-dimensional Cellular Automata

A number-conserving cellular automaton is a simplified model for a system of interacting particles. This paper contains two related constructions by which one can find all one-dimensional number-conserving cellular automata with one kind of particle. The output of both methods is a “flow function”, which describes the movement of the particles. In the first method, one puts increasingly stronger restrictions on the particle flow until a single flow function is specified. There are no dead ends, every choice of restriction steps ends with a flow. The second method uses the fact that the flow functions can be ordered and then form a lattice. This method consists of a recipe for the slowest flow that enforces a given minimal particle speed in one given neighbourhood. All other flow functions are then maxima of sets of these flows. Other questions, like that about the nature of non-deterministic number-conserving rules, are treated briefly at the end.

Construction of One-Dimensional Nonuniform Number Conserving Elementary Cellular Automata Rules

An effort to study one-dimensional nonuniform elementary number conserving cellular automata (NCCA) rules from an exponential order rule space of cellular automata is an excellent computational task. To perform this task effectively, a mathematical heritage under the number of conserving functions over binary strings of length [Formula: see text] has been highlighted along with their number conserving cellular automata rules (either uniform or nonuniform). A basic approach for the construction of some feasible nonuniform NCCA rules of any finite configuration with the assistance of nine uniform elementary CA rules has been investigated. From our construction procedure, recurrence equations are formulated as suitably solved to ascertain the actual range of NCCA rules. The state transition diagrams (STDs) of NCCA rules are analyzed. While classifying the binary strings through STDs, we found a fascinating optical insight that equal weight strings from a class whose cardinality is the same as the binomial coefficient [Formula: see text] where [Formula: see text] is the length and [Formula: see text] is the weight of the binary string.

Approximating dynamics of a number-conserving cellular automaton by a finite-dimensional dynamical system

The local structure theory for cellular automata (CA) can be viewed as an finite-dimensional approximation of infinitely dimensional system. While it is well known that this approximation works surprisingly well for some CA, it is still not clear why it is the case, and which CA rules have this property. In order to shed some light on this problem, we present an example of a four input CA for which probabilities of occurrence of short blocks of symbols can be computed exactly. This rule is number conserving and possesses a blocking word. Its local structure approximation correctly predicts steady-state probabilities of small length blocks, and we present a rigorous proof of this fact, without resorting to numerical simulations. We conjecture that the number-conserving property together with the existence of the blocking word are responsible for the observed perfect agreement between the finite-dimensional approximation and the actual infinite-dimensional dynamical system.

Efficient enumeration of three-state two-dimensional number-conserving cellular automata

We present a simple approach for finding all number-conserving two-dimensional cellular automata with the von Neumann neighborhood. We demonstrate the efficiency of this approach by enumerating and describing four special cases: totalistic, outer-totalistic, binary and three-state cellular automata. The last result was not published before. We then proceed to find all reversible two-dimensional number-conserving cellular automata with three states with the von Neumann neighborhood and show there are only trivial ones.

Ternary reversible number-conserving cellular automata are trivial

We introduce a novel method to study the reversibility of d-dimensional number-conserving multi-state cellular automata with the von Neumann neighborhood. We apply this method to ternary such cellular automata, for which, up to now, nothing was known about their reversibility. It turns out that they are all trivial: the only reversible such cellular automata are shifts that are intrinsically 1-dimensional.

All binary number-conserving cellular automata based on adjacent cells are intrinsically one-dimensional.

A binary number-conserving cellular automaton is a discrete dynamical system that models the movement of particles in a d-dimensional grid. Each cell of the grid is either empty or contains a particle. In subsequent time steps the particles move between the cells, but in one cell there can be at most one particle at a time. In this paper, the von Neumann neighborhood is considered, which means that in each time step a particle can move to an adjacent cell only. It is proven that regardless of the dimension d, all of these cellular automata are trivial, as they are intrinsically one-dimensional. Thus, for given d, there are only 4d+1 binary number-conserving cellular automata with the von Neumann neighborhood: the identity rule and the shift and traffic rules in each of the 2d possible directions.

A two-layer representation of four-state reversible number-conserving 2D cellular automata

We present a novel representation of 1D reversible and number-conserving cellular automata with four states. Carrying this view over to two dimensions, we are able to construct 65 four-state reversible and number-conserving 2D cellular automata with the von Neumann neighborhood. A clever use of the split-and-perturb decomposition of number-conserving CAs allows to prove by elimination that this list is complete.

Number-conserving cellular automata with a von Neumann neighborhood of range one

We present necessary and sufficient conditions for a cellular automaton with a von Neumann neighborhood of range one to be number-conserving. The conditions are formulated for any dimension and for any set of states containing zero. The use of the geometric structure of the von Neumann neighborhood allows for computationally tractable conditions even in higher dimensions.

Universality of One-Dimensional Reversible and Number-Conserving Cellular Automata

We study one-dimensional reversible and number-conserving cellular automata (RNCCA) that have both properties of reversibility and number-conservation. In the case of 2-neighbor RNCCA, Garc\'ia-Ramos proved that every RNCCA shows trivial behavior in the sense that all the signals in the RNCCA do not interact each other. However, if we increase the neighborhood size, we can find many complex RNCCAs. Here, we show that for any one-dimensional 2-neighbor reversible partitioned CA (RPCA) with s states, we can construct a 4-neighbor RNCCA with 4s states that simulates the former. Since it is known that there is a computationally universal 24-state 2-neighbor RPCA, we obtain a universal 96-state 4-neighbor RNCCA.

Fluctuation-driven computing on number-conserving cellular automata

A number-conserving cellular automaton (NCCA) is a cellular automaton in which the states of cells are denoted by integers, and the sum of all of the numbers in a configuration is conserved throughout its evolution. NCCAs have been widely used to model physical systems that are ruled by conservation laws of mass or energy. Imai et al. [13] showed that the local transition function of NCCA can be effectively translated into the sum of a binary flow function over pairs of neighboring cells. In this paper, we explore the computability of NCCAs in which the pairwise number flows are performed at fully asynchronous timings. Despite the randomness that is associated with asynchronous transitions, useful computation still can be accomplished efficiently in the cellular automata through the active exploitation of fluctuations [18]. Specifically, certain numbers may flow randomly fluctuating between forward and backward directions in the cellular space, as if they were subject to Brownian motion. Because random fluctuations promise a powerful resource for searching through a computational state space, the Brownian-like flow of the numbers allows for efficient embedding of logic circuits into our novel asynchronous NCCA.

Product decomposition for surjective 2-block NCCA

In this paper we define products of one-dimensional Number Conserving Cellular Automata (NCCA) and show that surjective NCCA with 2 blocks (i.e radius 1/2) can always be represented as products of shifts and identites. In particular, this shows that surjective 2-block NCCA are injective.

On the Non-existance of Rotation-Symmetric von Neumann Neighbor Number-Conserving Cellular Automata of Which the State Number is Less than Four

A number-conserving cellular automaton (NCCA) is a cellular automaton such that all states of cells are represented by integers and the total number of its configuration is conserved throughout its computing process. In constrast to normal cellular automata, there are infinitely many assignments of states for NCCAs with a constant state number. As for von Neumann neighbor(radius one) NCCAs with rotation-symmetry, a local function can be represented by summation of four binary functions. In this paper, we show that the minimum size of state set of rotation-symmetric von Neumann neighbor NCCA is 5 by using this representation.

EVENTUALLY NUMBER-CONSERVING CELLULAR AUTOMATA

Although it is undecidable whether a one-dimensional cellular automaton obeys a given conservation law over its limit set, it is however possible to obtain sufficient conditions to be satisfied by a one-dimensional cellular automaton to be eventually number-conserving. We present a preliminary study of two-input one-dimensional cellular automaton rules called eventually number-conserving cellular automaton rules whose limit sets, reached after a number of time steps of the order of the cellular automaton size, consist of states having a constant number of active sites. In particular, we show how to find rules having given limit sets satisfying a conservation rule. Viewed as models of systems of interacting particles, these rules obey a kind of Darwinian principle by either annihilating unnecessary particles or creating necessary ones.

Universality and decidability of number-conserving cellular automata

Number-conserving cellular automata (NCCA) are particularly interesting, both because of their natural appearance as models of real systems, and because of the strong restrictions that number-conservation implies. Here we extend the definition of the property to include cellular automata with any set of states in Z , and show that they can be always extended to “usual” NCCA with contiguous states. We show a way to simulate any one dimensional CA through a one-dimensional NCCA, proving the existence of intrinsically universal NCCA. Finally, we give an algorithm to decide, given a CA, if its states can be labeled with integers to produce a NCCA, and to find this relabeling if the answer is positive.

ON THE EXISTENCE OF A VARIATIONAL PRINCIPLE FOR DETERMINISTIC CELLULAR AUTOMATON MODELS OF HIGHWAY TRAFFIC FLOW

It is shown that a variety of deterministic cellular automaton models of highway traffic flow obey a variational principle which states that, for a given car density, the average car flow is a nondecreasing function of time. This result is established for systems whose configurations exhibits local jams of a given structure. If local jams have a different structure, it is shown that either the variational principle may still apply to systems evolving according to some particular rules, or it could apply under a weaker form to systems whose asymptotic average car flow is a well-defined function of car density. To establish these results, it has been necessary to characterize among all number-conserving cellular automaton rules which ones may reasonably be considered as acceptable traffic rules. Various notions such as free-moving phase, perfect and defective tiles, and local jam play an important role in the discussion. Many illustrative examples are given.