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Synchronous Counting and Computational Algorithm Design
"... Abstract. Consider a complete communication network on n nodes, each of which is a state machine with s states. In synchronous 2counting, 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 selfstabilising ( ..."
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Abstract. Consider a complete communication network on n nodes, each of which is a state machine with s states. In synchronous 2counting, 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 selfstabilising (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 nontrivial 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 2state solution for all values n ≥ 6. 1