Topic
Continuous automaton
About: Continuous automaton is a research topic. Over the lifetime, 947 publications have been published within this topic receiving 17417 citations.
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TL;DR: It is explained why replication is observed so frequently and a method for collecting the smallest periodic seeds is presented, assisted by algorithmic searches, for the Exactly 1 cellular automaton.
Abstract: In the Exactly 1 cellular automaton (also known as Rule 22), every site of the one-dimensional lattice is either in state 0 or in state 1, and a synchronous update rule dictates that a site is in state 1 next time if and only if it sees a single 1 in its three-site neighborhood at the current time. We analyze this rule started from finite seeds, i.e., those initial configurations that have only finitely many 1’s. Three qualitatively different types of evolution are observed: replication, periodicity, and chaos. We focus on rigorous results, assisted by algorithmic searches, for the first two behaviors. In particular, we explain why replication is observed so frequently and present a method for collecting the smallest periodic seeds. Some empirical observations about chaotic seeds are also presented.
18 citations
15 Nov 1997
TL;DR: A class of binary linear cellular automata whose corresponding minimal automata exhibit full exponential blow-up and corresponding minimal Fischer automata as well as the minimal DFAs have maximal complexity are constructed.
Abstract: We study the sizes of minimal finite state machines associated with linear cellular automata. In particular, we construct a class of binary linear cellular automata whose corresponding minimal automata exhibit full exponential blow-up. These cellular automata have Hamming distance 1 to a permutation automaton. Moreover, the corresponding minimal Fischer automata as well as the minimal DFAs have maximal complexity. By contrast, the complexity of higher iterates of a cellular automaton always stays below the theoretical upper bound.
18 citations
27 Feb 1997
TL;DR: It is proven that, for general cellular automata, ergodicity is equivalent to topological chaos (transitivity and sensitivity to initial conditions) and it is proved that linear CA over Zp with p prime have dense periodic orbits.
Abstract: We study the ergodic behavior of linear cellular automata over Zm. The main contribution of this paper is an easy-to-check necessary and sufficient condition for a linear cellular automaton over Zm to be ergodic. We prove that, for general cellular automata, ergodicity is equivalent to topological chaos (transitivity and sensitivity to initial conditions). Finally we prove that linear CA over Zp with p prime have dense periodic orbits.
18 citations
TL;DR: The goal of this paper is to design a reversible d -dimensional cellular automaton which is capable of simulating the behavior of any given d-dimensional cellular Automaton over any given configuration with respect to a well suited notion of simulation.
18 citations
Journal Article•
TL;DR: It is shown that the first order theory of a one-dimensional cellular automaton, construed as a structure with the global map and equality, is decidable.
Abstract: We show that the first order theory of a one-dimensional cellular automaton, construed as a structure with the global map and equality, is decidable. The argument employs bi-infinite versions of Büchi automata that can also be used to demonstrate that the spectra of cellular automata on finite grids are regular. For existential properties our method can be used to produce witnesses.
18 citations