Stably computable relations and predicates

A population protocol stably computes a relation R(x,y) if its output always stabilizes and R(x,y) holds if and only if y is a possible output for input x. Alternatively, a population protocol computes a predicate R(<x,y>) on pairs <x,y> if its output stabilizes on the truth value of the predicate when given <x,y> as input. We consider how stably computing R(x,y) and R(<x,y>) relate to each other. We show that for population protocols running on a complete interaction graph with n>=2, if R(<x,y>) is a stably computable predicate such that R(x,y) holds for at least one y for each x, then R(x,y) is a stably computable relation. In contrast, the converse is not necessarily true unless R(x,y) holds for exactly one y for each x.