Continuous automaton

About: Continuous automaton is a research topic. Over the lifetime, 947 publications have been published within this topic receiving 17417 citations.

The inactive-active phase transition in the noisy additive (exclusive-or) probabilistic cellular automaton

Abstract: We investigate the inactive-active phase transition in an array of additive (exclusive-or) cellular automata under noise. The model is closely related with the Domany-Kinzel probabilistic cellular automaton, for which there are rigorous as well as numerical estimates on the transition probabilities. Here we characterize the critical behavior of the noisy additive cellular automaton by mean field analysis and finite-size scaling and show that its phase transition belongs to the directed percolation universality class of critical behavior. As a by-product of our analysis, we argue that the critical behavior of the noisy elementary CA 90 and 102 (in Wolfram's enumeration scheme) must be the same. We also perform an empirical investigation of the mean field equations to assess their quality and find that away from the critical point (but not necessarily very far away) the mean field approximations provide a reasonably good description of the dynamics of the PCA.

Gliders and Ether in Rule 54

Abstract: This is a study of the one-dimensional elementary cellular automaton rule 54 in the new formalism of "flexible time". We derive algebraic expressions for groups of several cells and their evolution in time. With them we can describe the behaviour of simple periodic patterns like the ether and gliders in an efficient way. We use that to look into their behaviour in detail and find general formulas that characterise them.

Simulations between Programs as Cellular Automata

Abstract: We present cellular automata on appropriate digraphs and show that any covered normal logic program is a cellular automaton. Seeing programs as cellular automata shifts attention from classes of Herbrand models to orbits of Herbrand interpretations. Orbits capture both the declarative, model-theoretic meaning of programs as well as their inferential behavior. Logically and intentionally different programs can produce orbits that simulate each other. Simple examples of such behavior are compellingly exhibited with space-time diagrams of the programs as cellular automata. Construing a program as a cellular automaton leads to a general method for simulating any covered program with a Horn clause program. This means that orbits of Horn programs are completely representative of orbits of covered normal programs.

Physical approach to the ergodic behavior of stochastic cellular automata with generalization to random processes with infinite memory

Abstract: The large time behavior of stochastic cellular automata is investigated by means of an analogy with Van Kampen's approach to the ergodic behavior of Markov processes in continuous time and with a discrete state space. A stochastic cellular automaton with a finite number of cells may display an extremely large, but however finite number M of lattice configurations. Since the different configurations are evaluated according to a stochastic local rule connecting the variables corresponding to two successive time steps, the dynamics of the process can be described in terms of an inhomogeneous Markovian random walk among the M configurations of the system. An infinite Lippman-Schwinger expansion for the generating function of the total times q 1 , …, q M spent by the automaton in the different M configurations is used for the statistical characterization of the system. Exact equations for the moments of all times q 1 , …, q M are derived in terms of the propagator of the random walk. It is shown that for large values of the total time q = Σ u q u the average individual times 〈 q u 〉 attached to the different configurations u = 1, …, M are proportional to the corresponding stationary state probabilities P u st : 〈 q u 〉 ∼ q P u st , u = 1, …, M . These asymptotic laws show that in the long run the cellular automaton is ergodic, that is, for large times the ensemble average of a property depending on the configurations of the lattice is equal to the corresponding temporal average evaluated over a very large time interval. For large total times q the correlation functions of the individual sojourn times q 1 , …, q M increase linearly with the total number of time steps q : 〈 Δq u Δq u ′ 〉 ∼ q as q → ∞ which corresponds to non-intermittent fluctuations. An alternative approach for investigating the ergodic behavior of Markov processes in discrete space and time is suggested on the basis of a multiple averaging of a Kronecker symbol; this alternative approach can be extended to non-Markovian random processes with infinite memory. The implications of the results for the numerical analysis of the large time behavior of stochastic cellular automata are also investigated.

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.