SSS 2013 · 15th International Symposium on Stabilization, Safety, and Security of Distributed Systems, Osaka, Japan, November 2013 · doi:10.1007/978-3-319-03089-0_17

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

ISBN 978-3-319-03088-3

- Danny Dolev, Keijo Heljanko, Matti Järvisalo, Janne H. Korhonen, Christoph Lenzen, Joel Rybicki, Jukka Suomela, and Siert Wieringa: Synchronous counting and computational algorithm design ·
*Journal of Computer and System Sciences*82:310–332, 2016