Consider a complete communication network on n nodes, each of which is a state machine with s states. In synchronous 2-counting, the nodes receive a common clock pulse and they have to agree on which pulses are “odd” and which are “even”. We require that the solution is self-stabilising (reaching the correct operation from any initial state) and it tolerates f Byzantine failures (nodes that send arbitrary misinformation). Prior algorithms are expensive to implement in hardware: they require a source of random bits or a large number of states s. We use computational techniques to construct very compact deterministic algorithms for the first non-trivial case of f = 1. While no algorithm exists for n < 4, we show that as few as 3 states are sufficient for all values n ≥ 4. We prove that the problem cannot be solved with only 2 states for n = 4, but there is a 2-state solution for all values n ≥ 6.
Teruo Higashino, Yoshiaki Katayama, Toshimitsu Masuzawa, Maria Potop-Butucaru, and Masafumi Yamashita (Eds.): Stabilization, Safety, and Security of Distributed Systems, 15th International Symposium, SSS 2013, Osaka, Japan, November 13–16, 2013, Proceedings, volume 8255 of Lecture Notes in Computer Science, pages 237–250, Springer, Berlin, 2013