Showing papers on "Continuous automaton published in 2000"

A stochastic cellular automaton modeling gliding and aggregation of myxobacteria

Abstract: The myxobacteria are ubiquitous soil bacteria which aggregate under starvation conditions and build fruiting bodies to survive. Until recently the mechanisms of their social gliding, aggregation, and fruiting body formation have not been well understood. In this paper a stochastic cellular automaton model is presented to describe and provide an understanding of how the bacteria manage to build higher organized structures. In the automaton myxobacteria move on a square grid with periodic boundary conditions. They respond to the four nearest neighbors of their frontal cell poles, mainly through two factors: slime and a diffusing chemoattractant; both are produced by the bacteria themselves. Simulations show the interdependence of the different mechanisms which finally cause aggregation. A simplified version of this model is formally approximated by a system of partial differential equations, a chemotaxis system. A related publication [SIAM J. Appl. Math., 61 (2000), pp. 183--212] deals with the first rigoro...

Optimal amnesic probabilistic automata or how to learn and classify proteins in linear time and space

Abstract: Statistical modeling of sequences is a central paradigm of machine learning that finds multiple uses in computational molecular biology and many other domains. The probabilistic automata typically built in these contexts are subtended by uniform, fixed-memory Markov models. In practice, such automata tend to be unnecessarily bulky and computationally imposing both during their synthesis and use. In [8], much more compact, tree-shaped variants of probabilistic automata are built which assume an underlying Markov process of variable memory length. In [3, 4], these variants, called Probabilistic Suffix Trees (PSTs) were successfully applied to learning and prediction of protein families. The process of learning the automaton from a given training set S of sequences requires Θ (Ln2) worst-case time, where n is the total length of the sequences in S and L is the length of a longest substring of S to be considered for a candidate state in the automaton. Once the automaton is built, predicting the likelihood of a query sequence of m characters may cost time Θ (m2) in the worst case.The main contribution of this paper is to introduce automata equivalent to PSTs but having the following properties: learning the automaton takes O (n) time.prediction of a string of m symbols by the automaton takes O (m) time.Along the way, the paper presents an evolving learning sheme, and addresses notions of empirical probability and related efficient computation,possibly a by-product of more general interest.

New Finite Automaton Constructions Based on Canonical Derivatives

Abstract: Two classical constructions to convert a regular expression into a finite non-deterministic automaton provide complementary advantages: the notion of position of a symbol in an expression, introduced by Glushkov and McNaugthon-Yamada, leads to an efficient computation of the position automaton (there exist quadratic space and time implementations w.r.t. the size of the expression), whereas the notion of derivative of an expression w.r.t. a word, due to Brzozowski, and generalized by Antimirov, yields a small automaton. The number of states of this automaton, called the equation automaton, is less than or equal to the number of states of the position automaton, and in practice it is generally much smaller. So far, algorithms to build the equation automaton, such as Mirkin's or Antimirov's ones, have a high space and time complexity. The aim of this paper is to present new theoretical results allowing to compute the equation automaton in quadratic space and time, improving by a cubic factor Antimirov's construction. These results lay on the computation of a new kind of derivative, called canonical derivative, which makes it possible to connect the notion of continuation in a linear expression due to Berry and Sethi, and the notion of partial derivative of a regular expression due to Antimirov. A main interest of the notion of canonical derivative is that it leads to an efficient computation of the equation automaton via a specific reduction of the position automaton.

Simple deterministic self-organized critical system

Abstract: We introduce a continuous cellular automaton that presents self-organized criticality. It is one-dimensional, totally deterministic, without any embedded randomness, not even in the initial conditions. This system is in the same universality class as the Oslo rice pile, boundary driven interface depinning and the train model for earthquakes. Although the system is chaotic, in the thermodynamic limit chaos occurs only in a microscopic level.

A cellular automaton on a torus

Abstract: In this paper we prove a conjecture of Brian Thwaites concerning the evolution function of a certain cellular automaton on a torus. In the seventies, when more and more people had access to personal computers, John Conway’s game of life became very popular. Since then, the study of this type of game grew up into the theory of cellular automata. In [4] (see also [1, page 311]) Brian Thwaites proposes a conjecture which leads to such a cellular automaton. Thwaites’s conjecture is: Given any finite sequence of rational numbers, take the positive differences of successive members (including differencing the last member with the first); iteration of this operation eventually produces a set of zeros if and only if the size of the set is a power of 2. Our aim in this note is to prove that Thwaites conjecture holds true. Let a0, ..., ad−1 be the given d rational numbers, which we may think as the heights of d poles situated around a circle. These numbers are replaced at the next step by d rational numbers given by the difference in heights of successive poles, and then the process is repeated. Being an iteration of the same operation, it resembles Conway’s life game. For now the field of play is a 1-dimensional torus and since, as we will see only finitely many numbers which depend on the initial configuration are involved, in the long run, we will end up with a cycle. Finding the lengths of these cycles, which depend mostly on d — the size of the torus —, is the interesting problem. In other words, if d is a power of 2, Thwaites conjecture says that the length of any cycle is equal to 1 (the shortest possible). We examine the lengths of the cycles that occur and provide conditions on d which guarantee that these lengths are small or long. A criterion which tests if a given integer is a period for this evolution function is given in section 3. Received : December 20, 1998; Revised : March 22, 1999. AMS Subject Classification: 11B85, 11A07. 312 C.I. COBELI, M. CRÂŞMARU and A. ZAHARESCU 1 – Proof of Thwaites conjecture We begin by making some notations which set the problem in a clearer framework. Let a0, ..., ad−1 be the given rational numbers. For convenience, we unpack this ordered set of numbers by associating to it the infinite sequence (a0, a1, ...), where the components are defined by ak = ak+d for k ≥ 0 . (1) Let us denote by Qd and by Nd the set of all the sequences with rational and natural components respectively, satisfying (1). The evolution function φ :Qd→Qd is defined by φ(a0, a1, ...) = (a ′ 0, a ′ 1, ...), where a′k = |ak − ak+1| for k ≥ 0 . (2) With these notations, Thwaites conjecture says that for any sequence (a0, a1, ...) ∈ Qd, φ(a0, a1, ...) = (0, 0, ...) for all sufficiently large n ∈ N iff d is a power of 2. (Here φ(n) is the repeated composition of n samples of φ.) Let’s note that all the components of φ(a0, a1, ...) are nonnegative if n ≥ 1 and by multiplying all the components of the initial sequence (a0, a1, ...) by the least common multiple of their denominators, we may assume that the domain of our evolution function is Nd. Let M = max {a0, ..., ad−1}. By the definition of φ, it is easy to see that all the components of φ(a0, a1, ...) are integers belonging to [0,M ]. Because there are only finitely many such periodic sequences in Nd, it follows that given any initial configuration (a0, a1, ...), the repeated application of the evolution function will eventually produce a cycle of sequences which keep repeating. The next lemma shows that after sufficiently many steps we always end up with sequences with components having at most 2 distinct values. Lemma 1. Let d be a positive integer, (a0, a1, ...) a sequence of nonnegative integers satisfying (1) and suppose the function φ is defined as above. Then there is a positive integer a such that for sufficiently large n all the components of φ(a0, a1, ...) belong to {0, a}. Proof: The proof is by (inverse) induction. Let us look at a portion of the sequence of numbers we get at some step. We write them on a line as follows: ..., b, 0, ..., 0, } {{ } s zeros m, ...,m, } {{ } u numbers 0, ..., 0, } {{ } t zeros c, ... (3) A CELLULAR AUTOMATON ON A TORUS 313 Here m is the maximum of all our numbers at this step, b and c are nonzero, m > b, m > c, s ≥ 0, t ≥ 0, u ≥ 1 and the part of the sequence that begins and ends with m contains only 0’s or m’s. Then, after at most s + u + t steps, the maximum of the numbers that are produced out by this portion of the sequence will be ≤ max {m− b,m− c} < m. Of course at a given step the sequence of numbers we obtain might contain several subsequences of the form (3), but what happens is that after at most d steps the maximum of the numbers at that step will be strictly less than m. The lemma then follows by induction. By multiplying all the components of the initial configuration by a−1, where a is given by Lemma 1, we may assume that after sufficiently many steps all the components of the sequences we obtain are 0 or 1. Then our operation (taking the positive differences of successive members of the sequence) is nothing else than addition in the group (Z/2Z,+). Now there is a transparent way to generalize the game by replacing Z/2Z by a more general finite monoid and also by playing on a multidimensional field. The operation in this case is to take the sum (or product if the multiplicative notation is used) of the closest neighbors. We only mention here that if we keep the same group Z/2Z, but play on a multidimensional torus, then we eventually obtain a sequence of zeros if and only if the size of one of the dimensions is a power of 2. This can be showed by following the same lines of proof. Returning to our problem, let us observe that by starting with an arbitrary sequence of 0’s and 1’s, by applying repeatedly the evolution function, we obtain the following table which is filled with the beginning of the sequences obtained in the first few iterations. Step 1 2 3 · · · 0. a0 a1 a2 · · · 1. a0 + a1 a1 + a2 a2 + a3 · · · 2. a0 + a2 a1 + a3 a2 + a4 · · · 3. a0 + a1 + a2 + a3 a1 + a2 + a3 + a4 a2 + a3 + a4 + a5 · · · 4. a0 + a4 a1 + a5 a2 + a6 · · · 5. a0 + a1 + a4 + a5 a1 + a2 + a5 + a6 a2 + a3 + a6 + a7 · · · 6. a0 + a2 + a4 + a6 a1 + a3 + a5 + a7 a2 + a4 + a6 + a8 · · · 7. a0 + a1 + · · · + a7 a1 + a2 + · · · + a8 a2 + a3 + · · · + a9 · · · 8. a0 + a8 a1 + a9 a2 + a10 · · · 9. a0 + a1 + a8 + a9 a1 + a2 + a9 + a10 a2 + a3 + a10 + a11 · · · 10. a0 + a2 + a8 + a10 a1 + a3 + a9 + a11 a2 + a4 + a10 + a12 · · · · · · · · · · · · · · · · · · 314 C.I. COBELI, M. CRÂŞMARU and A. ZAHARESCU Now it is easy to see by induction that in the above table if d=2m then the d-th row (and consequently all that follow after it) contains only 0’s. Also, there are sequences of 0’s and 1’s, namely those containing an odd number of 1’s, for which on the (d−1)-th row all the numbers are equal to 1. Thus, if d is a power of 2 and we start with an arbitrary set of 0’s and 1’s, then the process will produce 0’s in d steps and only for particular ak’s in less then d steps. The outcome in the case d 6= 2m can be also deduced easily by induction. Thus, if we start for example with the periodic sequence given by a0 = 1 and ak = 0 for 1 ≤ k ≤ d−1, then 1 will always be the first number on the rows representing the steps of order a power of 2. Therefore, if d is not a power of 2 then there are sequences which will never produce a set of 0’s. We summarize our result in the following theorem which proves the Thwaites conjecture. Theorem 1. Let d be a positive integer and suppose the evolution function φ is defined as above. Then there is a rational number r>0 such that the repeated application of φ to any initial sequence of rational numbers (a0, a1, ...) satisfying (1) will eventually produce a cycle of sequences with the property (1) with all their components in {0, r}. Moreover, the cycle will contain only the sequence (0, 0, ...) independently on the initial sequence if and only if d is a power of 2. 2 – The length of cycles We assume in this section that the evolution function φ is defined on Ud, where Ud, is the set of all the sequences with components in {0, 1}, satisfying (1). This is not restrictive as we saw above and also has the advantage that it makes φ to be additive. Theorem 1 shows that if d is a power of 2 then the length of any cycle is equal to 1. Suppose from now on that d is not a power of 2. Write d = 2k r with k ≥ 0 and r odd, r > 1. Let s be the order of 2 modulo r. Thus 2s−1 ≡ 0 (mod r). Let F = {0, 1} be the field with 2 elements and let I be the ideal of F [X] generated by Xd−1. Map the sequence a = (a0, ..., ad−1) to the coset

Damage Spreading and µ-Sensitivity on Cellular Automata

Abstract: We show relations between new notions on cellular automata based on topological and measure-theoretical concepts: almost everywhere sensitivity to initial conditions for Besicovitch pseudo-distance, damage spreading (which measures the information (or damage) propagation) and the destruction of the initial configuration information. Through natural examples, we illustrate the links between these formal definitions and Wolfram's empirical classification.

A family of multi-value cellular automaton model for traffic flow

Abstract: A family of multi-value cellular automaton (CA) associated with traffic flow is presented in this paper. The family is obtained by extending the rule-184 CA, which is an ultradiscrete analogue to the Burgers equation. CA models in the family show both metastable states and stop-and-go waves, which are often observed in real traffic flow. Metastable states in the models exist not only on a prominent part of a free phase but also in a congested phase.

Directly Constructing Minimal DFAs: Combining Two Algorithms by Brzozowski

Abstract: In this paper, we combine (and refine) two of Brzozowski's algorithms -- yielding a single algorithm which constructs a minimal deterministic finite automaton (DFA) from a regular expression.

Inductive Synthesis of an Automaton from Its Specification in the Logical Language mathfrak {L}

Abstract: An approach to the synthesis of an automaton specified by a formula of the logical language mathfrak {L} is proposed. This approach makes it possible to construct the automaton inductively, i.e., according to the structure of the formula, starting from automata corresponding to subformulas represented in the form of conjunction or disjunction of literals.

The impact of the rule structure on the algorithm of 2D cellular automata implementation

Abstract: In case of the 2D cellular automata (CA) the whole rule f can be considered the set of sub-junctions grouped due to the number of "ones" in the neighbourhood. Such decomposition enables indication of sub-functions, which are more important for the global dynamics of the automaton then others. The cellular automata can be implemented on a cellular neural network (CNN). The simplest way of such implementation on the CNN universal machine was proposed by Cronuse and Chua (1995). We present the modification of this method due to which the information about sub-functions can be saved; and from this we are able to simplify the architecture of a CNN-universal machine implementing CA. Furthermore, the shortage of the time of a new state evaluation is possible. The advantage of the modified CA implementation is the possibility to receive the new automaton by manipulation of this part of the structure, which is connected with the sub-function.

On the existence of a variational principle for deterministic cellular automaton models of highway traffic flow

Abstract: 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 non-decreasing 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.

The completeness criterion for systems which contain all one-place finite-automaton functions

Abstract: We consider the completeness problem for the functional system P whose elements are finite-automaton functions (f.-a. functions) and the only operations are the operations of superposition. It is known that P does not contain finite complete systems. However D. N. Babin constructed an example of a finite set of f.-a. functions which together with the set P(l) of all one-place f.-a. functions forms a complete system in P. In this paper, the completeness criterion of systems of f.-a. functions which contain P(l) is given. It allows us to construct nontrivial examples of complete systems. The research was supported by the Russian Foundation for Basic Research, grant 00-01-00374. 1. BASIC NOTIONS AND RESULTS Let P be the set of f.-a. functions whose input and output variables take values from the set E% of all infinite sequences consisting of zeros and ones. We assume that on the set P the operation of superposition is defined. Let M C P. The closure of a set M with respect to this operation is denoted by [M]. A set M C P is called complete if [M] = P. Let f(xl , . . . ,*Λ) = y be the f.-a. function of Ρ determined by a system of canonical equations (see [1,3])

Computer simulations in the frame of mean-field theory and its application to a surface-reaction-like cellular automaton model

Abstract: Computer simulations are introduced in the frame of site-approximation mean-field rate equations and applied to an A–B2 surface-reaction-like cellular automaton model. There appears the second-order (continuous) transition between B-saturated and steady reactive phase, which fails to be predicted by pure site-approximation mean-field approach. The conclusion is made that such a kinetic behavior may be explained by fluctuations in number space.

Systems of generators of automaton permutations groups

Abstract: Systems of induced generating actions of automaton permutations groups on words of length r are investigated. A family of irreducible systems of generators is constructed, and the cardinality of such systems is found to be related with r. The infinite generativity (even in the topological sense) of groups of all automaton permutations, finite automaton permutations, and finitary automaton permutations is established; the well-known proof of this fact contained an error.

Symmetric Fractals Generated by Cellular Automata

Abstract: : We demonstrate how to construct fractals which are generated by a combination of a cellular automaton and a substitution. Moreover, if the substitution and the cellular automaton exhibit certain symmetry features, the fractal will inherit these symmetries.

Virtual scanning: A method for testing one class of self-organizing cellular automata

Abstract: The problem of formation of diagnosis about a cellular automaton (an array of processor elements) with a large number of processors and multiple failures is solved on a self-organization basis. We suggest a virtual scan method in which the flip-flop link configuration is determined from the actual layout of good cells and intercellular links. Starting from the initial cell at the periphery of the array, the scan chain is constructed as the result of the group behavior of identically functioning cells. Along this chain, diagnosis about all cells and cell interfaces that have a good route to the initial cell is formed. This method is an alternative to the conventional scan method based on an end-to-end shift register, where deterministic control and fixed flip-flop links in the shift chains are used.

Cellular Automata with Activation and Inhibition

Abstract: We investigate simple cellular automata models with activation and inhibition which show interesting patterns at certain parameter values. The dynamical behaviour is classified using monotony properties of the automaton rule. Also discrete equivalents of Lyapunov functions are applied to show convergence to stationary states in some cases. Finally the emerging patterns are characterised.