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.

翻译：若群体协议的输出总能稳定，且当且仅当y是输入x的可能输出时R(x,y)成立，则该协议稳定地计算关系R(x,y)。或者，若给定输入<x,y>时，群体协议输出稳定于谓词的真值，则该协议计算关于对<x,y>的谓词R(<x,y>)。我们探讨稳定计算R(x,y)与R(<x,y>)之间的关联。我们证明，对于在节点数n≥2的完全交互图上运行的群体协议，若R(<x,y>)是稳定可计算的谓词，且对每个x至少存在一个y使R(x,y)成立，则R(x,y)是稳定可计算的关系。反之则未必成立，除非对每个x恰好存在一个y使R(x,y)成立。