613 research outputs found
A Survey of Cellular Automata: Types, Dynamics, Non-uniformity and Applications
Cellular automata (CAs) are dynamical systems which exhibit complex global
behavior from simple local interaction and computation. Since the inception of
cellular automaton (CA) by von Neumann in 1950s, it has attracted the attention
of several researchers over various backgrounds and fields for modelling
different physical, natural as well as real-life phenomena. Classically, CAs
are uniform. However, non-uniformity has also been introduced in update
pattern, lattice structure, neighborhood dependency and local rule. In this
survey, we tour to the various types of CAs introduced till date, the different
characterization tools, the global behaviors of CAs, like universality,
reversibility, dynamics etc. Special attention is given to non-uniformity in
CAs and especially to non-uniform elementary CAs, which have been very useful
in solving several real-life problems.Comment: 43 pages; Under review in Natural Computin
Revisiting the Rice Theorem of Cellular Automata
A cellular automaton is a parallel synchronous computing model, which
consists in a juxtaposition of finite automata whose state evolves according to
that of their neighbors. It induces a dynamical system on the set of
configurations, i.e. the infinite sequences of cell states. The limit set of
the cellular automaton is the set of configurations which can be reached
arbitrarily late in the evolution.
In this paper, we prove that all properties of limit sets of cellular
automata with binary-state cells are undecidable, except surjectivity. This is
a refinement of the classical "Rice Theorem" that Kari proved on cellular
automata with arbitrary state sets.Comment: 12 pages conference STACS'1
A Minimal Time Solution to the Firing Squad Synchronization Problem with Von Neumann Neighborhood of Extent 2
Cellular automata provide a simple environment in which to study global behaviors. One example of a problem that utilizes cellular automata is the Firing Squad Synchronization Problem, first proposed in 1957. This paper provides an overview of the standard Firing Squad Synchronization Problem and a commonly used technique in solving it. This paper also provides a statement of a new extension of the Standard Firing Squad Synchronization Problem to a different neighborhood definition - a Von Neumann neighborhood of extent 2. An 8 state 651 rule minimal time solution to the extended problem is described, presented and proven, along with Python code used in running simulations of the solution
self-stabilizing
Consider a fully-connected synchronous distributed system consisting of n nodes, where up to f nodes may be faulty and every node starts in an arbitrary initial state. In the synchronous C-counting problem, all nodes need to eventually agree on a counter that is increased by one modulo C in each round for given C>1. In the self-stabilising firing squad problem, the task is to eventually guarantee that all non-faulty nodes have simultaneous responses to external inputs: if a subset of the correct nodes receive an external “go” signal as input, then all correct nodes should agree on a round (in the not-too-distant future) in which to jointly output a “fire” signal. Moreover, no node should generate a “fire” signal without some correct node having previously received a “go” signal as input. We present a framework reducing both tasks to binary consensus at very small cost. For example, we obtain a deterministic algorithm for self-stabilising Byzantine firing squads with optimal resilience f<n/3, asymptotically optimal stabilisation and response time O(f), and message size O(log f). As our framework does not restrict the type of consensus routines used, we also obtain efficient randomised solutions
Near-optimal self-stabilising counting and firing squads
Consider a fully-connected synchronous distributed system consisting of n nodes, where up to f nodes may be faulty and every node starts in an arbitrary initial state. In the synchronous C-counting problem, all nodes need to eventually agree on a counter that is increased by one modulo C in each round for given C>1. In the self-stabilising firing squad problem, the task is to eventually guarantee that all non-faulty nodes have simultaneous responses to external inputs: if a subset of the correct nodes receive an external “go” signal as input, then all correct nodes should agree on a round (in the not-too-distant future) in which to jointly output a “fire” signal. Moreover, no node should generate a “fire” signal without some correct node having previously received a “go” signal as input. We present a framework reducing both tasks to binary consensus at very small cost. For example, we obtain a deterministic algorithm for self-stabilising Byzantine firing squads with optimal resilience f<n/3, asymptotically optimal stabilisation and response time O(f), and message size O(log f). As our framework does not restrict the type of consensus routines used, we also obtain efficient randomised solutions
Distributed algorithms for hard real-time systems
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