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Henryk Fuks

Publications and source records attributed to Henryk Fuks.

At least 19 recordsLinked to original sources
arXiv2007

Catalan numbers and power laws in cellular automaton rule 14

We discuss example of an elementary cellular automaton for which the density of ones decays toward its limiting value as a power of the number of iterations $n$. Using the fact that this rule conserves the number of blocks 10 and that preimages of some other blocks exhibit patterns closely related to patterns observed in rule 184, we derive expressions for the number of $n$-step preimages of all blocks of length 3. These expressions involve Catalan numbers, and together with basic properties of iterated probability measures they allow us to to compute the density of ones after $n$ iterations, as well as probabilities of occurrence of arbitrary block of length smaller or equal to 3.

nlin.CG↗
arXiv2007

Enumeration of number-conserving cellular automata rules with two inputs

We show that there exists a one-to-one correspondence between the set of number-conserving cellular automata (CA) with $q$ inputs and the set of balanced sequences with $q$ terms. This allows to enumerate number-conserving CA. We also show that number-conserving rules are becoming increasingly rare as the number of states increases.

nlin.CG↗
arXiv2006

Motion representation of one-dimensional cellular automaton rules

Generalizing the motion representation we introduced for number-conserving rules, we give a systematic way to construct a generalized motion representation valid for non-conservative rules using the expression of the current, which appears in the discrete version of the continuity equation, completed by the discrete analogue of the source term. This new representation is general, but not unique, and can be used to represent, in a more visual way, any one-dimensional cellular automaton rule. A few illustrative examples are presented.

nlin.CG↗
arXiv2006

Dynamics of the Cellular Automaton Rule 142

We investigate dynamics of the cellular automaton rule 142. This rule possesses additive invariant of the second order, namely it conserves the number of blocks 10. Rule 142 can be alternatively described as an operation on a binary string in which we simultaneously flip all symbols which have dissenting right neighbours. We show that the probability of having a dissenting neighbour can be computed exactly using the fact that the surjective rule 60 transforms rule 142 into rule 226. We also demonstrate that the conservation of the number of 10 blocks implies that these blocks move with speed -1 or stay in the same place, depending on the state of the preceding site. At the density of blocks 10 equal to 0.25, the rule 142 exhibits a phenomenon similar to the jamming transitions occurring in discrete models of traffic flow.

nlin.CG↗
arXiv2006

Critical behavior of number-conserving cellular automata with nonlinear fundamental diagrams

We investigate critical properties of a class of number-conserving cellular automata (CA) which can be interpreted as deterministic models of traffic flow with anticipatory driving. These rules are among the only known CA rules for which the shape of the fundamental diagram has been rigorously derived. In addition, their fundamental diagrams contain nonlinear segments, as opposed to majority of number-conserving CA which exhibit piecewise-linear diagrams. We found that the nature of singularities in the fundamental diagram of these rules is the same as for rules with piecewise-linear diagrams. The current converges toward its equilibrium value like $t^{-1/2}$, and the critical exponent $\beta$ is equal to 1. This supports the conjecture of universal behavior at singularities in number-conserving rules. We discuss properties of phase transitions occurring at singularities as well as properties of the intermediate phase.

nlin.CG↗
arXiv2003

Non-deterministic density classification with diffusive probabilistic cellular automata

We present a probabilistic cellular automaton (CA) with two absorbing states which performs classification of binary strings in a non-deterministic sense. In a system evolving under this CA rule, empty sites become occupied with a probability proportional to the number of occupied sites in the neighborhood, while occupied sites become empty with a probability proportional to the number of empty sites in the neighborhood. The probability that all sites become eventually occupied is equal to the density of occupied sites in the initial string.

nlin.CG↗
arXiv2002

A class of cellular automata equivalent to deterministic particle systems

We demonstrate that a local mapping f in a space of bisequences over {0,1} which conserves the number of nonzero sites can be viewed as a deterministic particle system evolving according to a local mapping in a space of increasing bisequences over Z. We present an algorithm for determination of the local mapping in the space of particle coordinates corresponding to the local mapping f.

nlin.CG↗
arXiv2002

Packet delay in models of data networks

We investigate individual packet delay in a model of data networks with table-free, partial table and full table routing. We present analytical estimation for the average packet delay in a network with small partial routing table. Dependence of the delay on the size of the network and on the size of the partial routing table is examined numerically. Consequences for network scalability are discussed.

cs.NI↗
arXiv2002

Individual-based lattice model for spatial spread of epidemics

We present a lattice gas cellular automaton (LGCA) to study spatial and temporal dynamics of an epidemic of SIR (susceptible-infected-removed) type. The automaton is fully discrete, i.e. space, time and number of individuals are discrete variables. The automaton can be applied to study spread of epidemics in both human and animal populations. We investigate effects of spatial inhomogeneities in initial distribution of infected and vaccinated populations on the dynamics of epidemic of SIR type. We discuss vaccination strategies which differ only in spatial distribution of vaccinated individuals. Also, we derive an approximate, mean-field type description of the automaton, and discuss differences between the mean-field dynamics and the results of LGCA simulation.

nlin.CG↗
arXiv2001

Convergence to equilibrium in a class of interacting particle systems evolving in discrete time

We conjecture that for a wide class of interacting particle systems evolving in discrete time, namely conservative cellular automata with piecewise linear flow diagram, relaxation to the limit set follows the same power law at critical points. We further describe the structure of the limit sets of such systems as unions of shifts of finite type. Relaxation to the equilibrium resembles ballistic annihilation, with ``defects'' propagating in opposite direction annihilating upon collision.

nlin.CG↗
arXiv2000

Number-conserving cellular automaton rules

A necessary and sufficient condition for a one-dimensional q-state n-input cellular automaton rule to be number-conserving is established. Two different forms of simpler and more visual representations of these rules are given, and their flow diagrams are determined. Various examples are presented and applications to car traffic are indicated. Two nontrivial three-state three-input self-conjugate rules have been found. They can be used to model the dynamics of random walkers.

adap-org↗
arXiv1999

Exact results for deterministic cellular automata traffic models

We present a rigorous derivation of the flow at arbitrary time in a deterministic cellular automaton model of traffic flow. The derivation employs regularities in preimages of blocks of zeros, reducing the problem of preimage enumeration to a well known lattice path counting problem. Assuming infinite lattice size and random initial configuration, the flow can be expressed in terms of generalized hypergeometric function. We show that the steady state limit agrees with previously published results.

comp-gas↗
arXiv1999

Performance of data networks with random links

We investigate simplified models of computer data networks and examine how the introduction of additional random links influences the performance of these net works. In general, the impact of additional random links on the performance of the network strongly depends on the routing algorithm used in the network. Significant performance gains can be achieved if the routing is based on "geometrical distance" or shortest path reduced table routing. With shortest path full table routing degradation of performance is observed.

adap-org↗
arXiv1999

Exact results for deterministic cellular automata traffic models

We present a rigorous derivation of the flow at arbitrary time in a deterministic cellular automaton model of traffic flow. The derivation employs regularities in preimages of blocks of zeros, reducing the problem of preimage enumeration to a well known lattice path counting problem. Assuming infinite lattice size and random initial configuration, the flow can be expressed in terms of generalized hypergeometric function. We show that the steady state limit agrees with previously published results.

math.DS↗
arXiv1999

Exact results for deterministic cellular automata traffic models

We present a rigorous derivation of the flow at arbitrary time in a deterministic cellular automaton model of traffic flow. The derivation employs regularities in preimages of blocks of zeros, reducing the problem of preimage enumeration to a well known lattice path counting problem. Assuming infinite lattice size and random initial configuration, the flow can be expressed in terms of generalized hypergeometric function. We show that the steady state limit agrees with previously published results.

math.DS↗
arXiv1998

Cellular automaton rules conserving the number of active sites

This paper shows how to determine all the unidimensional two-state cellular automaton rules of a given number of inputs which conserve the number of active sites. These rules have to satisfy a necessary and sufficient condition. If the active sites are viewed as cells occupied by identical particles, these cellular automaton rules represent evolution operators of systems of identical interacting particles whose total number is conserved. Some of these rules, which allow motion in both directions, mimic ensembles of one-dimensional pseudo-random walkers. Numerical evidence indicates that the corresponding stochastic processes might be non-Gaussian.

adap-org↗
arXiv1997

Modeling diffusion of innovations with probabilistic cellular automata

We present a family of one-dimensional cellular automata modeling the diffusion of an innovation in a population. Starting from simple deterministic rules, we construct models parameterized by the interaction range and exhibiting a second-order phase transition. We show that the number of individuals who eventually keep adopting the innovation strongly depends on connectivity between individuals.

adap-org↗
arXiv1997

Generalized Deterministic Traffic Rules

We study a family of deterministic models for highway traffic flow which generalize cellular automaton rule 184. This family is parametrized by the speed limit $m$ and another parameter $k$ that represents a ``degree of aggressiveness'' in driving, strictly related to the distance between two consecutive cars. We compare two driving strategies with identical maximum throughput: ``conservative'' driving with high speed limit and ``aggressive'' driving with low speed limit. Those two strategies are evaluated in terms of accident probability. We also discuss fundamental diagrams of generalized traffic rules and examine limitations of maximum achievable throughput. Possible modifications of the model are considered.

adap-org↗