Stably computable relations and predicates
James Aspnes
TL;DR
This work analyzes the relationship between stably computable relations $R(x,y)$ and predicates $R(\langle x,y\rangle)$ in population protocols on complete graphs. It shows that, for $n \ge 3$, a stably computable, total predicate $R(\langle x,y\rangle)$ guarantees that the corresponding relation $R(x,y)$ is stably computable, via a nondeterministic exploration of possible outputs and an embedded semilinear tester; however, the converse is not true in general, as reachability provides a counterexample. A positive result holds for single-valued relations: if $R(x,y)$ is single-valued and stably computable, then $R(\langle x,y\rangle)$ is stably computable. The paper also presents a wrapper construction to extend results to small population sizes and discusses the potential generalization to other interaction graphs, highlighting nondeterministic freezing as a useful technique. Overall, it clarifies when stably computable predicates transfer to relations and identifies precise conditions under which the reverse direction holds, informing the design and analysis of distributed protocols.
Abstract
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.