Stably computable relations and predicates

TL;DR

This paper explores the relationship between stably computable relations and predicates in population protocols, establishing conditions under which one form implies the other on complete interaction graphs.

Contribution

It characterizes when stably computable predicates imply stably computable relations in population protocols, highlighting differences based on uniqueness conditions.

Findings

If R(<x,y>) is a stably computable predicate with at least one y for each x, then R(x,y) is a stably computable relation.

The converse holds only if R(x,y) holds for exactly one y per x.

The results depend on the structure of the interaction graph and the uniqueness of outputs.

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.