Showing papers on "Continuous automaton published in 2013"

The finiteness problem for automaton semigroups is undecidable

Abstract: The finiteness problem for automaton groups and semigroups has been widely studied, several partial positive results are known. However we prove that, in the most general case, the problem is undecidable. We study the case of automaton semigroups. Given a NW-deterministic Wang tile set, we construct an Mealy automaton, such that the plane admit a valid Wang tiling if and only if the Mealy automaton generates a finite semigroup. The construction is similar to a construction by Kari for proving that the nilpotency problem for cellular automata is unsolvable. Moreover Kari proves that the tiling of the plane is undecidable for NW-deterministic Wang tile set. It follows that the finiteness problem for automaton semigroup is undecidable.

Periodic Orbits and Dynamical Complexity in Cellular Automata

Abstract: We investigate the relationships between dynamical complexity and the set of periodic configurations of surjective Cellular Automata. We focus on the set of strictly temporally periodic configurations, i.e., the set of those configurations which are temporally but not spatially periodic for a given surjective automaton. The cardinality of this set turns out to be inversely related to the dynamical complexity of the cellular automaton. In particular, we show that for surjective Cellular Automata, the set of strictly temporally periodic configurations has strictly positive measure if and only if the cellular automaton is equicontinuous. Furthermore, we show that the set of strictly temporally periodic configurations is dense for almost equicontinuous surjective cellular automata, while it is empty for the positively expansive ones. In the class of additive cellular automata, the set of strictly temporally periodic points can be either dense or empty. The latter happens if and only if the cellular automaton is topologically transitive. This is not true for general transitive Cellular Automata, where the set of of strictly temporally periodic points can be non-empty and non-dense.

Algorithm for Automaton Specification for Exploring Dynamic Labyrinths

Abstract: A variety of interesting problems arise in the study of finite automata that move about in a two dimensional space. The model proposed by Muller [4] is used here to construct new automaton which can explore any labyrinth and escape through the moving or dynamic obstacles inside over the grid. The earlier results were shown for static obstacles distributed over integer grid and the automaton in this case was constructed to interact on the rectangular grid location endowed with four neighborhood directional states. In this paper we allow obstacles moving in discrete steps and verify that the finite automaton with just five printing symbols can escape or find the exit.

Probabilistic cellular automata, invariant measures, and perfect sampling

Abstract: A probabilistic cellular automaton (PCA) can be viewed as a Markov chain. The cells are updated synchronously and independently, according to a distribution depending on a finite neighborhood. We investigate the ergodicity of this Markov chain. A classical cellular automaton is a particular case of PCA. For a one-dimensional cellular automaton, we prove that ergodicity is equivalent to nilpotency, and is therefore undecidable. We then propose an efficient perfect sampling algorithm for the invariant measure of an ergodic PCA. Our algorithm does not assume any monotonicity property of the local rule. It is based on a bounding process which is shown to also be a PCA. Last, we focus on the PCA majority, whose asymptotic behavior is unknown, and perform numerical experiments using the perfect sampling procedure.

Simultaneous Finite Automata: An Efficient Data-Parallel Model for Regular Expression Matching

Abstract: Automata play important roles in wide area of computing and the growth of multicores calls for their efficient parallel implementation. Though it is known in theory that we can perform the computation of a finite automaton in parallel by simulating transitions, its implementation has a large overhead due to the simulation. In this paper we propose a new automaton called simultaneous finite automaton (SFA) for efficient parallel computation of an automaton. The key idea is to extend an automaton so that it involves the simulation of transitions. Since an SFA itself has a good property of parallelism, we can develop easily a parallel implementation without overheads. We have implemented a regular expression matcher based on SFA, and it has achieved over 10-times speedups on an environment with dual hexa-core CPUs in a typical case.

A continuous–discontinuous cellular automaton method for regular frictional contact problems

Abstract: In the present paper, a continuous–discontinuous cellular automaton method is developed to model the discontinuous problems caused by regular frictional contact, in which level set method, discontinuous enriched shape function, discontinuous cellular automaton method and contact friction theory are combined, by which an analysis from continuity to discontinuity can be achieved, and no assembled stiffness matrix but only cell stiffness is needed in the whole calculation, because of the use of discontinuous cellular automaton method. In the present method, level set method is used to track the discontinuous surface, and discontinuous enriched shape function is employed to describe the discontinuity of displacement and stress. Contact friction theory is applied to construct the Coulomb frictional contact model of discontinuous surfaces; besides, combined with discontinuous cellular automaton method, a new mixed iteration method is proposed to obtain the solution of the problem, and no assembled stiffness matrix is constructed. And the frictional contact iterations are done simultaneously with the updating of cellular automaton, in which the contact states and contact areas can be previously obtained in the cellular automaton updating process, and the efficiency can get much higher. Finally, some numerical examples are given to show that the present method is effective and accurate and can be further extended into some practical engineering.

Characterisation of sets of limit measures after iteration of a cellular automaton on an initial measure

Abstract: The asymptotic behavior of a cellular automaton iterated on a random configuration is well described by its limit probability measure(s). In this paper, we characterize measures and sets of measures that can be reached as limit points after iterating a cellular automaton on a simple initial measure, in the same spirit as SRB measures. In addition to classical topological constraints, we exhibit necessary computational obstructions. With an additional hypothesis of connectivity, we show these computability conditions are sufficient by constructing a cellular automaton realising these sets, using auxiliary states in order to perform computations. Adapting this construction, we obtain a similar characterization for the Cesaro mean convergence, a Rice theorem on the sets of limit points, and we are able to perform computation on the set of measures, i.e. the cellular automaton converges towards a set of limit points that depends on the initial measure. Last, under non-surjective hypotheses, it is possible to remove auxiliary states from the construction.

Epidemic Propagation: An Automaton Model as the Continuous SIR Model

Abstract: The use of the SIR model to predict the time evolution of an epidemic is very frequent and has spatial information about its propagation which may be very useful to contrast its spread. In this paper we take a particular cellular automaton model that well reproduces the time evolution of the disease given by the SIR model; setting the automaton is generally an annoying problem because we need to run a lot of simulations, compare them to the solution of the SIR model and, finally, decide the parameters to use. In order to make this procedure easier, we will show a fast method that, in input, requires the parameters of the SIR continuous model that we want to reproduce, whereas, in output, it yields the parameters to use in the cellular automaton model. The problem of computing the most suitable parameters for the reticular model is reduced to the problem of finding the roots of a polynomial Equation.

Universal pattern generation by cellular automata

Abstract: Cellular automata are mathematical models for massively parallel processing of information by a large number of identical, locally interconnected tiny processors on a regular grid. Extremely simple processors, or cells, are known to be able to generate together complex patterns. In this talk we consider the problem of designing a cellular automaton that can generate all patterns of states from a finite initial seed. We describe a one-dimensional solution that is based on multiplying numbers by a suitable constant. The automaton to multiply by constant 3/2 is shown to be related to some difficult open questions in number theory. We discuss these connections and pose several questions concerning pattern generation in cellular automata.

Arithmetical Analysis of Biomolecular Finite Automaton

Abstract: In the paper we present a theoretical analysis of extension of the finite automaton built on DNA introduced by the Shapiro team to an arbitrary number of states and symbols. In the implementation we use a new idea of several restriction enzymes instead of one. We give arithmetical conditions for the existence of such extensions in terms of ingredients used in the implementation.

Routing alogrithm over semi-regular tessellations

Abstract: Path discovery or routing algorithms are challenging when the nodes are distributed over not on just regular grid like rectangular type but on semiregular grids Investigations in the study of finite state automata that move about in a two dimensional space are suitable to tackle this context The model proposed by H Muller [1] is used here to construct new automaton which can explore the path through obstacles over the grid This model is to be applied for routing phase for data transmission The earlier results were shown for static obstacles distributed over integer grid and the automaton in this case was constructed to interact on the rectangular grid location endowed with four neighborhood directional states In this paper we allow higher degree of neighborhood and mixing the types cells It has been verified that the finite automaton with number of printing (output) symbols determined by the maximum out degree of a cell in the underlying semi-regular grid can find the target

Constructing Cellular Automaton Models from Observation Data

Abstract: We propose a construction method of cellular automaton models based on observation data of phenomena. Because variables and states of cellular automata should be discrete, we discretize the observation data. Under the given number of neighbors and possible states of a site, we estimate the rules of cellular automata by the statistic analysis. In this paper, we apply this method to the discretized diffusion equation and Burgers equation. Both deterministic and stochastic models, which have three neighbors and 2-8 possible states, are obtained. Results of the models are compared with that of the original equations.

System and method for implementing reservoir computing using cellular automata

Abstract: An implementation of a reservoir computing based recurrent neural network is disclosed. Cellular automaton is used as the reservoir of dynamical systems. Input is projected onto the initial conditions of automaton cells and nonlinear computation is performed on the input via application of a rule in the automaton for a period of time. The evolution of the automaton creates a space-time volume of the automaton state space, and it is used as the output of the reservoir. The output is further processed according to reservoir computing principles to achieve the assigned task. The reservoir is trained for the specific task and dataset via optimizing the rule of the automaton.

About Strongly Universal Cellular Automata

Abstract: In this paper, we construct a strongly universal cellular automaton on the line with 11 states and the standard neighbourhood. We embed this construction into several tilings of the hyperbolic plane and of the hyperbolic 3D space giving rise to strongly universal cellular automata with 10 states.

2-D Reversible Cellular Automata with Nearest and Prolonged Next Nearest Neighborhoods under Periodic Boundary †

Abstract: In this paper, we study a 2-dimensional cellular automaton generated by a new local rule with the nearest neighborhoods and prolonged next nearest neighborhoods under periodic boundary condition over the ternary field (Z 3). We obtain the rule matrix of this cellular automaton and characterize this family by exploring some of their important characteristics. We get some recurrence equations which simplifies the computation of the rank of the rule matrix related to the 2-dimensional cellular automaton drastically. Next, we propose an algorithm to determine the rank of the rule matrix. Finally, we conclude by presenting an application to error correcting codes.

Complex system modeling using TSK fuzzy cellular automata and differential evolution

Abstract: This paper presents a proposal to model the dynamics of complex systems that can be represented as homogeneous continuous cellular automata. The method includes a homogeneous fuzzy cellular automaton whose representative cell is a Takagi-Sugeno-Kang fuzzy system. This cell is tuned by means of the differential evolution algorithm, which tries to optimize a similarity measure between the dynamics of the target complex system and the automaton. The experiments show that our approach is able to obtain a valid model for a complex system that exhibits a two-dimensional nonlinear wave scheme.

Flexible time and ether in one-dimensional cellular automata

Abstract: A one-dimensional cellular automaton is an infinite row of identical machines---the cells---which depend for their behaviour only on the states of their direct neighbours. This thesis introduces a new way to think about one-dimensional cellular automata. The formalism of Flexible Time allows one to unify the states of of a finite number of cells into a single object, even if they occur at different times. This gives greater flexibility to handle the structures that occur in the development of a cellular automaton. Flexible Time makes it possible to calculate in an algebraic way the fate of a finite number of cells. In the first part of this thesis the formalism is developed in detail. Then it is applied to a specific problem of one-dimensional cellular automata, namely ether formation. The so-called ether is a periodic pattern of cells that occurs in some cellular automata: It arises from almost all randomly chosen initial configurations, and why this happens is not clear. For one of these cellular automata, the elementary cellular automaton with rule code 54, ether formation is expressed in the formalism of Flexible Time. Then a partial result about ether formation is proved: There is a certain fragment of the ether that arises with probability 1 from every random initial configuration, and it is then propagated with probability 1 to any later time. The persistence of the ether fragment is a strong argument that the ether under Rule 54 indeed arises from almost all input configurations. The result only requires that the states of the cells are chosen independently and with equal probability distributions, and that all cell states can occur. This is not yet a full proof of ether formation, but it is derived by formal means, not just by computer simulations.

Some more on the basis finite automaton

Abstract: We consider in this paper the basis nite automaton and its some properties We shall also consider some properties of special binary relation dened on the sets of states of canonical automata for the given language and for its mirror image We shall also consider an algorithm of constructing the basis automaton dening the language which has a priory given variant of this relation

A Uniquely Ergodic Cellular Automaton

Abstract: We construct a one-dimensional uniquely ergodic cellular automaton which is not nilpotent. This automaton can perform asymptotically infinitely sparse computation, which nevertheless never disappears completely. The construction builds on the self-simulating automaton of Gacs. We also prove related results of dynamical and computational nature, including the undecidability of unique ergodicity, and the undecidability of nilpotency in uniquely ergodic cellular automata.

Computing on a Simple Asynchronous Cellular Automaton

Abstract: An asynchronous cellular automaton (ACA) is a cellular automaton which allows cells to undergo state transitions independently at random timings. This paper gives an efficient scheme to compute on a special ACA model that takes a substantially simple transition function. This ACA has much less complexity as compared to other models with respect to the number of transition rules.

New aspects of symmetry of elementary cellular automata

Abstract: We present a new classification of elementary cellular automata. It is based on the structure of the network of states, connected with the transitions between them; the latter are determined by the automaton rule. Recently an algorithm has been proposed to compress the network of states (M. J. Krawczyk, Physica A 390 (2011) 2181). In this algorithm, states are grouped into classes, according to the local symmetry of the network. In the new classification, an automaton is described by the number of classes #(N) as dependent on the system size N. In most cases, the results reflect the known classification into 88 groups. However, the function #(N) also appears to be the same for some rules which have not been grouped together yet. In this way, the automaton 23 is equivalent to 232, 77 to 178, 105 to 150, the pair (43, 113) to the pair (142, 212) and the group (12, 68, 207, 221) to the group (34, 48, 187, 243). Furthermore, automata 51, 204, the pair (15, 85) and the pair (170,240) are all mutually equivalent. Results are also presented on the structure of networks of states.

On Cardinality of the Group of Weak Fuzzy Automaton Isomorphisms

Abstract: Recent studies on fuzzy automata are influenced by algebraic techniques to tackle issues like reduction, minimization and their languages. Fuzzy automaton homomorphism is one such majorally discussed technique. This paper is concerned with the group of (weak) fuzzy automaton automorphisms and constructions of all (weak) fuzzy automaton automorphisms over arbitrary fuzzy automaton. It is shown that (1) every arbitrary fuzzy automaton is decomposed into distinct primaries, (2) primaries are maximal singly generated fuzzy automata and (3) every weak fuzzy automaton homomorphism on an arbitrary fuzzy automaton is uniquely determined into weak fuzzy automaton homomorphisms on all its primaries. Therefore, the discussion begun over strongly connected fuzzy automaton and continue constructions as well as characterizations of (weak) fuzzy automaton homomorphisms, isomorphisms, endomorphisms and automorphisms sequentially over perfect fuzzy automaton, singly generated fuzzy automaton and primaries of fuzzy automaton. Finally, it is obtained that the group of weak fuzzy automaton automorphisms and its cardinality over arbitrary fuzzy automaton.

Hard Asymptotic Sets for One-Dimensional Cellular Automata.

Abstract: We prove that the (language of the) asymptotic set (and the nonwandering set) of a one-dimensional cellular automaton can be $\SIGMA^1_1$-hard. We do not go into much detail, since the constructions are relatively standard.

Transfer matrix analysis of one-dimensional majority cellular automata with thermal noise

Abstract: Thermal noise in a cellular automaton refers to a random perturbation to its function which eventually leads this automaton to an equilibrium state controlled by a temperature parameter. We study the 1-dimensional majority-3 cellular automaton under this model of noise. Without noise, each cell in this automaton decides its next state by majority voting among itself and its left and right neighbour cells. Transfer matrix analysis shows that the automaton always reaches a state in which every cell is in one of its two states with probability 1/2 and thus cannot remember even one bit of information. Numerical experiments, however, support the possibility of reliable computation for a long but finite time.

Determinization of ordinal automata

Abstract: It is proved that for each nondeterministic ordinal automaton there exists a deterministic ordinal automaton which is equivalent to the original one for all countable ordinals. An upper bound for the number of states of the deterministic automaton is double exponential in the number of states of the nondeterministic automaton.

Modeling "quantum" field theory through "classical" cellular automaton

Abstract: The purpose of this paper is to illustrate how cellular automaton can be used in order to arrive at discrete, but "classical", model leading to the emergent "quantum" field theory.

Isomorphisms on strongly connected group automaton

Abstract: Direct product of a permutation strongly connected automaton and a synchronizing strongly connected Aleshin Type automaton is also a strongly connected automaton An automaton is called quasi-ideal automaton if and only if all the following conditions are satisfied; (i) It is strongly connected (ii) the minimal ideal of its transition semi group is a right group (iii) the ranges of the idempotent elements of the minimal ideal of its transition group form a merging of a partition on its set of states An automaton is isomorphic to the direct product of a permutation strongly connected automaton and synchronizing strongly connected Aleshin type automaton if and only if it is a quasi ideal automaton

Computable Throughput and Metastability in a Cellular Automaton Model for Traffic Flow

Abstract: A cellular automaton model for traffic flow is analyzed. For this model, it is shown that under ergodic initial configurations, the distribution of cars will converge in time to a mixture of free flow and solid blocks. Furthermore, the nature of the free flow and solid block distributions is fully described, thus allowing for a specific computation of throughput in terms of the parameters. The model is also shown to exhibit a hysteresis phenomenon, which is similar to what has been observed on actual highways.