State-efficient realization of fault-tolerant FSSP algorithms - Umeo, Hiroshi; Kamikawa, Naoki; Maeda, Masashi; Fujita, Gen (Natural computing, 2019)
The firing squad synchronization problem (FSSP, for short) on cellular automata has been studied extensively for more than fifty years, and a rich variety of FSSP algorithms has been proposed. Here we study the classical FSSP on a model of fault-tolerant cellular automata that might have possibly some defective cells and present the first state-efficient implementations of fault-tolerant FSSP algorithms for one-dimensional (1D) and two-dimensional (2D) cellular arrays. It is shown that, under some constraints on the length and distribution of defective cells, any 1D cellular array of length n with p defective cell segments can be synchronized in 2 n - 2 + p steps and the algorithm is realized on a 1D cellular automaton of length n , 2 ≤ n ≤ 50 , having 164 states and 4792 transition rules. In addition, we give by far a smaller-state implementation of a 2D FSSP algorithm that can synchronize any 2D rectangular array of size m × n , possibly including at most O( mn ) isolated defective zones, exactly in 2 ( m + n ) - 4 steps on a cellular automaton with only 6 states and 935 transition rules.