Stability of Multivalued Continuous Consensus

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Multivalued consensus functions defined from a vector of inputs over the set V of possible input values (and possibly from the previous input and output values) to a single output are investigated. The consensus functions are designed to tolerate t faulty inputs. Two classes of multivalued consensus functions are defined, the exact value and the range value, which require the output to be one of the nonfaulty inputs or in the range of the nonfaulty inputs, respectively. The instability of consensus functions is examined, counting the maximal number of output changes along a geodesic path of input changes, a path in which each input is changed at most once. Lower and upper bounds for the instability of multivalued consensus functions as a function of n, the number of sensors, t, and $|V|$ are presented. A new technique for obtaining such lower bounds, using edgewise simplex subdivision, is presented.