Showing papers on "Continuous automaton published in 2007"

A structurally dynamic cellular automaton with memory

Abstract: Major features of conventional cellular automata (CA) include the inalterability of topology and the absence of memory. The effect of simple memory (memory in cells and links) in a particular structurally dynamic CA is explored in this paper.

Model checking propositional projection temporal logic based on SPIN

Abstract: This paper investigates a model checking algorithm for Propositional Projection Temporal Logic (PPTL) with finite models. To this end, a PPTL formula is transformed to a Normal Form Graph (NFG), and then a Nondeterministic Finite Automaton (NFA). The NFA precisely characterizes the finite models satisfying the corresponding formula and can be equivalently represented as a Deterministic Finite Automaton (DFA). When the system to be verified can be modeled as a DFA As, and the property of the system can be specified by a PPTL formula P, then ¬P can be transformed to a DFA Ap. Thus, whether the system satisfies the property or not can be checked by computing the product automaton of As and Ap, and then checking whether or not the product automaton accepts the empty word. Further, this method can be implemented by means of the verification system SPIN.

On a zero speed sensitive cellular automaton

Abstract: Using an unusual yet natural invariant measure we show that there exists a sensitive cellular automaton whose perturbations propagate at an asymptotically null speed for almost all configurations. More specifically, we prove that Lyapunov exponents measuring pointwise or average linear speeds of the faster perturbations are equal to zero. We show that this implies the nullity of the measurable entropy. The measure μ we consider gives the μ-expansiveness property to the automaton. It is constructed with respect to a factor dynamical system based on simple 'counter dynamics'. As a counterpart, we prove that in the case of positively expansive automata, the perturbations move at positive linear speed over all the configurations.

Synchronizing automata preserving a chain of partial orders

Abstract: We present a new class of automata which strictly contains the class of aperiodic automata and shares with the latter certain synchronization properties. In particular, every strongly connected automaton in this new class is synchronizing and has a reset word of length ⌊n(n+1)/6⌋ where n is the number of states of the automaton.

Factor automata of automata and applications

Abstract: An efficient data structure for representing the full index of a set of strings is the factor automaton, the minimal deterministic automaton representing the set of all factors or substrings of these strings. This paper presents a novel analysis of the size of the factor automaton of an automaton, that is the minimal deterministic automaton accepting the set of factors of a finite set of strings, itself represented by a finite automaton. It shows that the factor automaton of a set of strings U has at most 2|Q| - 2 states, where Q is the number of nodes of a prefix-tree representing the strings in U, a bound that significantly improves over 2||U|| - 1, the bound given by Blumer et al. (1987), where ||U|| is the sum of the lengths of all strings in U. It also gives novel and general bounds for the size of the factor automaton of an automaton as a function of the size of the original automaton and the maximal length of a suffix shared by the strings it accepts. Our analysis suggests that the use of factor automata of automata can be practical for large-scale applications, a fact that is further supported by the results of our experiments applying factor automata to a music identification task with more than 15,000 songs.

Normalized expressions and finite automata

Abstract: There exist two well-known quotients of the position automaton of a regular expression. The first one, called the equation automaton, was first introduced by Mirkin from the notion of prebase and has been redefined by Antimirov from the notion of partial derivative. The second one, due to Ilie and Yu and called the follow automaton, can be obtained by eliminating e-transitions in an e-NFA that is always smaller than the classical e-NFAs (Thompson, Sippu and Soisalon–Soininen). Ilie and Yu discussed the difficulty of succeeding in a theoretical comparison between the size of the follow automaton and the size of the equation automaton and concluded that it is very likely necessary to realize experimental studies. In this paper we solve the theoretical question, by first defining a set of regular expressions, called normalized expressions, such that every regular expression can be normalized in linear time, and proving then that the equation automaton of a normalized expression is always smaller than its follow automaton.

Towards a rice theorem on traces of cellular automata

Abstract: The trace subshift of a cellular automaton is the subshift of all possible columns that may appear in a space-time diagram We prove the undecidability of a rather large class of problems over trace subshifts of cellular automata

A Structurally Dynamic Cellular Automaton with Memory in the Triangular Tessellation.

Abstract: Cellular automata (CAs) are discrete, spatially explicit extended dynamic systems. A CA system is composed of adjacent cells or sites arranged as a regular lattice, which evolves in discrete time steps. Each cell is characterized by an internal state whose value belongs to a finite set. The updating of these states is made simultaneously according to a common local transition rule involving only the neighborhood of each cell. Thus, if Σ(T) i is taken to denote the value of cell i at time step T, the site values evolve by iteration of the mapping: Σ(T 1) i Φ(Σ (T) j i), where Φ is an arbitrary function which specifies the CA rule operating on the neighborhood of the cell i [1]. This paper deals with a particular two-dimensional totalistic CA rule, the parity rule: Σ(T 1) i j i Σ (T) j mod 2, acting on cells with two possible state values (0 and 1). Despite its formal simplicity, the parity rule exhibits complex behavior [2]. We restrict ourselves here to the triangular tessellation, again a simple scenario that allows for complex behavior [3, 4]. Figure 1 shows an example of the parity rule operating on the triangular tessellation starting from an active cell with its three neighbors also active. This is the initial configuration used throughout this paper.

Measure rigidity for algebraic bipermutative cellular automata

Abstract: Let be a bipermutative algebraic cellular automaton. We present conditions that force a probability measure, which is invariant for the -action of F and the shift map σ , to be the Haar measure on Σ, a closed shift-invariant subgroup of the abelian compact group . This generalizes simultaneously results of Host et al (B. Host, A. Maass and S. Martinez. Uniform Bernoulli measure in dynamics of permutative cellular automata with algebraic local rules. Discrete Contin. Dyn. Syst. 9 (6) (2003), 1423–1446) and Pivato (M. Pivato. Invariant measures for bipermutative cellular automata. Discrete Contin. Dyn. Syst. 12 (4) (2005), 723–736). This result is applied to give conditions which also force an ( F , σ )-invariant probability measure to be the uniform Bernoulli measure when F is a particular invertible affine expansive cellular automaton on .

Approximate timed abstractions of hybrid automata

Abstract: Given a hybrid automaton and a desired precision, we aim at constructing an approximate abstraction by means of a timed automaton, whose discrete state trajectories approximate the discrete state trajectories of the original system, with the desired precision on switching times. We show that using the Euclidian metric on reals it is not always possible to construct a timed automaton that is close to a hybrid automaton with finite precision. For this reason, we motivate and introduce relative metrics on reachability time, external language and simulation relation to quantify the precision of our abstraction. Our main result is to propose a novel algorithm to construct a timed automaton that is an approximate timed abstraction of a hybrid automaton with desired precision, and study its convergence properties. For an extended version of this paper refer to [D'Innocenzo et al., 2007].

Sofic Trace Subshift of a Cellular Automaton

Abstract: The trace subshift of a cellular automaton is the subshift of all possible columns that may appear in a space-time diagram. In this paper we study conditions for a sofic subshift to be the trace of a cellular automaton.

Implementation of Hybrid Automata in Scicos

Abstract: Hybrid automaton is a standard model for describing a hybrid system. A hybrid automaton is a state machine augmented with differential equations and is generally represented by a graph composed of vertices and edges where vertices represent continuous activities and edges represent discrete transitions. Modeling a hybrid automaton with large number of vertices may be difficult, time-consuming and error prone using standard modules in modeling and simulation environments such as Scicos. In this paper, we present the new Scicos automaton block used for modeling and simulation of hybrid automata.

Spatial updating, spatial transients, and regularities of a complex automaton with nonperiodic architecture.

Abstract: We study the dynamics of patterns exhibited by rule 52, a totalistic cellular automaton displaying intricate behaviors and wide regions of active/inactive synchronization patches. Systematic computer simulations involving 230 initial configurations reveal that all complexity in this automaton originates from random juxtaposition of a very small number of interfaces delimiting active/inactive patches. Such interfaces are studied with a sidewise spatial updating algorithm. This novel tool allows us to prove that the interfaces found empirically are the only interfaces possible for these periods, independently of the size of the automata. The spatial updating algorithm provides an alternative way to determine the dynamics of automata of arbitrary size, a way of taking into account the complexity of the connections in the lattice.

Reliable Self-Replicating Machines in Asynchronous Cellular Automata

Abstract: We propose a self-replicating machine that is embedded in a two-dimensional asynchronous cellular automaton with von Neumann neighborhood. The machine dynamically encodes its shape into description signals, and despite the randomness of cell updating, it is able to successfully construct copies of itself according to the description signals. Self-replication on asynchronously updated cellular automata may find application in nanocomputers, where reconfigurability is an essential property, since it allows avoidance of defective parts and simplifies programming of such computers.

Chaotic Properties of the Elementary Cellular Automaton Rule 40 in Wolfram's Class I

Abstract: This paper examines the chaotic properties of the elementary cellular automaton rule 40. Rule 40 has been classified into Wolfram’s class I and also into class 1 by G. Braga et al. These classifications mean that the time-space patterns generated by this cellular automaton die out in a finite time and so are not interesting. As such, we may hardly realize that rule 40 has chaotic properties. In this paper we show that the dynamical system defined by rule 40 is Devaney chaos on a class of configurations of some particular type and has every periodic point except prime period one, four, or six. In the process of the proof, it is noticed that the dynamical properties of rule 40 can be related to some interval dynamical systems. These propositions are shown in Theorems 2 and 4.

An exponential gap between Las Vegas and deterministic sweeping finite automata

Abstract: A two-way finite automaton is sweeping if its input head can change direction only on the end-markers. For each n ≥ 2, we exhibit a problem that can be solved by a O(n2)-state sweeping LasVegas automaton, but needs 2Ω(n) states on every sweeping deterministic automaton.

Polynomial cellular neural networks for implementing the game of life

Abstract: One-layer space-invariant Cellular Neural Networks (CNNs) are widely appreciated for their simplicity and versatility; however, such structures are not able to solve non-linearly separable problems. In this paper we show that a polynomial CNN - that has with a direct VLSI implementation - is capable of dealing with the 'Game of Life', a Cellular Automaton with the same computational complexity as a Turing machine. Furthermore, we describe a simple design algorithm that allows to convert the rules of a Cellular Automaton into the weights of a polynomial CNN.

Real time language recognition on 2D cellular automata: dealing with non-convex neighborhoods

Abstract: In this paper we study language recognition by twodimensional cellular automata on different possible neighborhoods. Since it is known that all complete neighborhoods are linearly equivalent we focus on a natural sub-linear complexity class: the real time. We show that any complete neighborhood is sufficient to recognize in real time any language that can be recognized in real-time by a cellular automaton working on the convex hull of V.

Eventually Number-Conserving Cellular Automata

Abstract: 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.

Two Ways of Introducing Alternation into Context-Free Grammars and Pushdown Automata

Abstract: Two ways of introducing alternation for context-free grammars and pushdown automata are compared. One is the usual way which combines “states” with alternation [1], [4], [7], and the other is the way used in [6] to define the alternating context-free grammar, i.e., alternation is governed by the variables of the grammar. In this paper the latter way is taken over to define a new type of alternating pushdown automaton by combining the “pushdown symbols” of the pushdown automaton with alternation. We have derived a characterization of the original alternating context-free grammars in terms of such a new type of alternating pushdown automaton without states. It is also shown that, if (non-alternating) states are introduced as an additional feature for this type of pushdown automaton, then the resulting alternating pushdown automaton has exactly the same expressive power as the original alternating pushdown automaton.

A time hierarchy theorem for nondeterministic cellular automata

Abstract: We present a tight time-hierarchy theorem for nondeterministic cellular automata by using a recursive padding argument. It is shown that, if t2(n) is a time-constructible function and t2(n) grows faster than t1(n+1), then there exists a language which can be accepted by a t2(n)- time nondeterministic cellular automaton but not by any t1(n)-time nondeterministic cellular automaton.

Modeling linear dynamical systems by continuous-valued cellular automata

Abstract: This paper exposes a procedure for modeling and solving linear systems using continuous-valued cellular automata. The original part of this work consists on showing how the cells in the automaton may contain both real values and operators for carrying out numerical calculations and solve a desired problem. In this sense the automaton acts as a program, where data and operators are mixed in the evolution space for obtaining the correct calculations. As an example, Euler's integration method is implemented in the configuration space in order to achieve an approximated solution for a dynamical system. Three examples showing linear behaviors are presented.

Self-Reproduction Model for Communication-Restricted Cellular Automata

Abstract: Many researchers have constructed a self-reproduction model on cellular automata so far. C. G. Langton constructed a simple model on 2-D cellular automata, and it is a dynamic-loop . We consider an embedding of Langton's model on 1-bit communication cellular automata. The 1-bit communication cellular automaton is a 2-D cellular automaton where communication-capacity is restricted to 1-bit at one-step. The Langton's model has 8 internal states and generally it is impossible to transfer 8-bit by 1-bit communication without any less of time overhead. In this paper, we present two models for the self-reproduction. The first model reproduces a self-reproduction model that behaves likes Langton-loop on the 2-D communication-restricted cellular automata. However the embedding model has a large number of internal states. The second model, by removing some properties from Langton's model, we reduced internal states.

Effect of Successive Observation on Quantum Cellular Automaton

Abstract: A quantum cellular automaton, which is extended from a classical cellular automaton with Wolfram's rule 150, is studied. In contrast to the classical cellular automaton, all possible configurations exist in a quantum superposition before measurement. We show that measuring the state of only one qubit significantly aeffects the time evolution in the quantum cellular automaton. In particular, we demonstrate that, occasionally, repeating the measurement enhances the appearance of the configurations found in the classical cellular automaton. The occurrence of this enhancement is primarily determined by the results of the measurement in the early time steps, and it is sustained by a feedback mechanism.

Catalan numbers and power laws in cellular automaton rule 14

Abstract: 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.

Limit Language Complexity of Elementary Cellular Automation of Rule 164 Examined from Particles

Abstract: The limit set of elementary cellular automaton is found through defining some kinds of particles and its regularity is proved.The result shows that the finite automaton can accept this limit set.

Generic method for solving partial differential equations through the design of problem-specific Cellular Neural Networks

Abstract: This paper presents an original and generic numerical method for solving partial differential equations. A new mathematical and systematic method stemming from the local very nature of any differential problem is proposed: a custom tailored continuous automaton is purposely derived from any given differential problem so that its steady state yields the solution in a quantitatively correct way. The combined use of formal computing and continuous automata thus offers the unique possibility to completely automate the process from formal problem specification to its numerical solution.

Two-dimensional Four Color Cellular Automaton: Surface Explorations

Abstract: An implementation of a two bit Gray-coded temporal logic based cellular automaton created several emergent properties. A program implementing the two-dimensional four color cellular automaton using Ant Farm is demonstrated. New and previously described emergent integer sequences are also identified. Possible applications in spatial modeling, knowledge domain modeling, and mobile robot movement are suggested. This paper studies the movement of a simple cellular automaton called the lean quaternary temporal logic (LQTL) ant. In particular, I study the movement of an ant that is searching a “space” using a Gray-code logic to direct it. In this paper a minimal interaction context is modeled and some outcomes reported about the interaction between two entities. In the first section the fundamental premises on which the ant’s movement is based are described. In the second section, the two entities used to model a minimal interaction are described and their behavior is detailed. In the third section, observations are reported on the results of running an algorithm that models the behavior of the two entities. In the fourth section, the behavior of the interacting entities is described and further work identified. The task of identifying the simplest emergent algorithmic spatial search behavior that also included a return-to-base attribute led to my study of cellular automata. No other spatial search strategies are reported here from the literature. Research, over some 40 or more years into cellular automata, is reviewed in Sarkar [1] and Ganguly et al .[ 2]. Cellular automata take many forms but underlying them all is a set of simple rules and the behavioral interactions that result from playing the rules out in a modeling facility, usually a computer. A two-dimensional (2D) cellular automaton can be viewed as a “virtual ant” interacting with a grid of cells. Little communication occurs between adjacent cells.