360 research outputs found
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
A Simple Optimum-Time FSSP Algorithm for Multi-Dimensional Cellular Automata
The firing squad synchronization problem (FSSP) on cellular automata has been
studied extensively for more than forty years, and a rich variety of
synchronization algorithms have been proposed for not only one-dimensional
arrays but two-dimensional arrays. In the present paper, we propose a simple
recursive-halving based optimum-time synchronization algorithm that can
synchronize any rectangle arrays of size m*n with a general at one corner in
m+n+max(m, n)-3 steps. The algorithm is a natural expansion of the well-known
FSSP algorithm proposed by Balzer [1967], Gerken [1987], and Waksman [1966] and
it can be easily expanded to three-dimensional arrays, even to
multi-dimensional arrays with a general at any position of the array.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249
A Genetically Evolved Solution to the Firing Squad Problem
In 1957, J. Myhill presented the firing squad problem. A special case of k-color cellular automata (CA) synchronization, the firing squad problem offers more stringent rules allowing for a provable minimal running time. To date, CA solutions have been found that run in minimal time using as many as sixteen states and as few as six [5]. There have also been arguments against the existence of solutions using only 4 states [11]. Due to the extremely large search space involved with such problems, the existing solutions have all been analytic in nature. We attempt to apply genetic algorithms and genetic programming to create transition tables that solve the firing squad problem. Ideally, the solutions would run in minimal time. No generalized solutions were found, but progress was made towards determining the best strategies for an evolved solution
Dynamic neighbourhood cellular automata.
We propose a definition of cellular automaton in which each cell can change its neighbourhood during a computation. This is done locally by looking not farther than neighbours of neighbours and the number of links remains bounded by a constant throughout. We suggest that dynamic neighbourhood cellular automata can serve as a theoretical model in studying algorithmic and computational complexity issues of ubiquitous computations. We illustrate our approach by giving an optimal, logarithmic time solution of the Firing Squad Synchronization problem in this setting
Dynamic Neighbourhood Cellular Automata
We propose a definition of cellular automaton in which each cell can change its neighbourhood during a computation. This is done locally by looking not farther than neighbours of neighbours and the number of links remains bounded by a constant throughout. We suggest that dynamic neighbourhood cellular automata can serve as a theoretical model in studying algorithmic and computational complexity issues of ubiquitous computations. We illustrate our approach by giving an optimal, logarithmic time solution of the Firing Squad Synchronization problem in this setting
Exploring Millions of 6-State FSSP Solutions: the Formal Notion of Local CA Simulation
In this paper, we come back on the notion of local simulation allowing to
transform a cellular automaton into a closely related one with different local
encoding of information. This notion is used to explore solutions of the Firing
Squad Synchronization Problem that are minimal both in time (2n -- 2 for n
cells) and, up to current knowledge, also in states (6 states). While only one
such solution was proposed by Mazoyer since 1987, 718 new solutions have been
generated by Clergue, Verel and Formenti in 2018 with a cluster of machines. We
show here that, starting from existing solutions, it is possible to generate
millions of such solutions using local simulations using a single common
personal computer
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
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