Revisiting the cutting of the firing squad synchronization

Abstract

Various synchronization algorithms have been introduced in literature during the last decades to deal with the firing squad synchronization problem on cellular automata (CA). Among others defective CA algorithms, where the CA cell is able to transmit information without previous processing, have been also presented. In our case, originating from the classical Mazoyer’s paper, where a minimum-time solution is presented with 6 states, a one-dimensional CA where one cell may permanently fail is presented. In the proposed algorithm, the defective cell can neither process nor transmit information any longer, while it is considered that such dynamic defects may become apparent in any time step of computation. A thorough analysis of the synchronization, in terms of location and time at which cell fails, for the cells found in both sides of defective cell is delivered to decipher the corresponding maximal possible number of synchronized cells in each part of the cut, due to defect, CA array. The proposed algorithm is properly extended to consider more than one defective cells that may occur in the under study one-dimensional CA. Based on the aforementioned analysis, we provide the generalization of synchronization with multiple totally defective cells, while application examples of the generalized CA algorithm in case of two defective cells are also presented. Finally, another intriguing aspect refers to handling of states that could be tentatively characterized as unknown, in a confrontation similar to the previous defective state but also different, since now this(these) cell(s) are not stated as faulty but unknown. As a result, a new one-dimensional CA with less states, compared to the previous CA defective algorithms, able to synchronize the maximal possible number of cells in each part occurs.