An 8-state minimal time solution to the firing squad synchronization problem

This paper proposes an optimization approach for solving a classical problem in cellular automata theory: the 6-states Firing Squad Synchronization Problem (FSSP). To this purpose, we introduce an original optimization function which quantifies the quality of solutions according only to the main goal of the problem without taking into account any side information about cellular automata computations. This function is used for a dedicated Iterated Local Search algorithm which finds several hundreds of new solutions of the FSSP. Note, that up to present only one human-designed solution was known which is optimal in time. Most of the new solutions found by our algorithm have lower complexity (in terms of number of transitions rules used). An analysis of the fitness landscape for FSSP explains why, counter-intuitively, local search strategy can achieve good results for FSSP.

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“…Few years after Goto's solution, Waksman [16] and Balzer [17] proposed optimal time solutions with 16 and 8 states, respectively.…”

Section: Known Solutionsmentioning

confidence: 99%

“…Balzer [17], and later Sanders [20], proved that there is no optimal time solution for 4 states able to synchronize all the lines for any n. Remark that there are solutions which synchronize lines of length 2 n but not lines of any length [21].…”

Section: Known Solutionsmentioning

confidence: 99%

An Iterated Local Search to find many solutions of the 6-states Firing Squad Synchronization Problem

Clergue

Vérel

Formenti

2018

Applied Soft Computing

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“…We make the assumption that the border of this patch is synchronized, as done by a firing-squad-like algorithm (See [2] or [12]). This assumption is satisfied at the beginning of the computation, because this beginning in itself 315 trivially is a synchronizing event.…”

Section: Ongoing Behaviourmentioning

confidence: 99%

Leader election on two-dimensional periodic cellular automata

Bacquey

2017

Theoretical Computer Science

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This article explores the computational power of bi-dimensional cellular automata acting on periodical configurations. It extends in some sense the results of a similar paper dedicated to the one-dimensional case. More precisely, we present an algorithm that computes a "minimal pattern network", i.e. a minimal pattern and the two translation vectors it can use to tile the entire configuration. This problem is equivalent to the computation of a leader, which is one equivalence class of the cells of the periodical configuration.

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“…13 C X C X C X C X ] t = 30 1 2 3 4 5 6 7 8 9 31 1 2 3 4 5 6 7 8 9 1 C C C C C C C C C 2 C X X X X X X X X 3 C X C C C C C C C 4 C X C X X X X X X 5 C X C X C C C C C 6 C X C X C X X X X 7 C X C X C X C C C 8 C X C X C X C X X 9 C X C X C X C X C 10 C X C X C X C X C 11 C X C X C X C X C 12 C X C X C X C X C 13 C X C X C X C X C t = 32 1 2 3 4 5 6 7 8 9 1 X X X X X X X X X 2 X X X X X X X X X 3 X X X X X X X X X 4 X X X X X X X X X 5 X X X X X X X X X 6 X X X X X X X X X 7 X X X X X X X X X 8 X X X X X X X X X 9 X X X X X X X X X 10 X X X X X X X X X 11 X X X X X X X X X 12 X X X X X X X X X 13 X X X X X X X X X t = 33 1 2 3 4 5 6 7 8 9 Fig. 4.…”

Section: Zebra-like Mapping On Rectangle Arraysmentioning

confidence: 99%

A State-Efficient Zebra-Like Implementation of Synchronization Algorithms for 2D Rectangular Cellular Arrays

Umeo

Nomura²

2012

BIOMATH

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BIOMATH h t t p : / / w w w. b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h /Abstract-The firing squad synchronization problem 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 (1D) but two-dimensional (2D) arrays. In the present paper, we propose a simple and state-efficient mapping scheme: zebra-like mapping for implementing 2D synchronization algorithms for rectangular arrays. The zebra-like mapping we propose embeds two types of configurations alternately onto a 2D array like a zebra-like pattern, one configuration is a synchronization configuration of 1D arrays and the other is a stationary configuration which keeps its state unchanged until the final synchronization. It is shown that the mapping gives us a smallest, known at present, implementation of 2D FSSP algorithms for rectangular arrays. The implementation itself has a nice property that the correctness of the constructed transition rule set is clear and transparent. It is shown that there exists a ninestate 2D cellular automaton that can synchronize any (m x n) rectangle in (m+n+max(m,n)-3) steps.

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An 8-state minimal time solution to the firing squad synchronization problem

Cited by 166 publications

References 2 publications

An Iterated Local Search to find many solutions of the 6-states Firing Squad Synchronization Problem

An Iterated Local Search to find many solutions of the 6-states Firing Squad Synchronization Problem

Leader election on two-dimensional periodic cellular automata

A State-Efficient Zebra-Like Implementation of Synchronization Algorithms for 2D Rectangular Cellular Arrays

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