The complexity of verifying population protocols

Abstract

Population protocols (Angluin et al. in PODC, 2004) are a model of distributed computation in which indistinguishable, finite-state agents interact in pairs to decide if their initial configuration, i.e., the initial number of agents in each state, satisfies a given property. In a seminal paper Angluin et al. classified population protocols according to their communication mechanism, and conducted an exhaustive study of the expressive power of each class, that is, of the properties they can decide (Angluin et al. in Distrib Comput 20(4):279–304, 2007). In this paper we study the correctness problem for population protocols, i.e., whether a given protocol decides a given property. A previous paper (Esparza et al. in Acta Inform 54(2):191–215, 2017) has shown that the problem is decidable for the main population protocol model, but at least as hard as the reachability problem for Petri nets, which has recently been proved to have non-elementary complexity. Motivated by this result, we study the computational complexity of the correctness problem for all other classes introduced by Angluin et al., some of which are less powerful than the main model. Our main results show that for the class of observation models the complexity of the problem is much lower, ranging from $$\varPi _2^p$$ Π 2 p to .

Parameterized Analysis of Reconfigurable Broadcast Networks (Long Version)

“…This is also true for RBN. Indeed Proposition 2.12 of [13] states the above result for "well-behaved generalized protocols" (Definition 2.1 of [13]). Fix an arbitrary RBN protocol P. By definition 2.1 of [13], it is a generalized protocol by setting Conf to be the set of configurations of P, Σ to be its set of states, and Step to be the step relation of the underlying RBN.…”

“…Indeed Proposition 2.12 of [13] states the above result for "well-behaved generalized protocols" (Definition 2.1 of [13]). Fix an arbitrary RBN protocol P. By definition 2.1 of [13], it is a generalized protocol by setting Conf to be the set of configurations of P, Σ to be its set of states, and Step to be the step relation of the underlying RBN. By definition 2.8 of [13], it is well-behaved, i.e., every fair execution eventually ends up in a bottom strongly connected component of the reachability graph.…”

“…Proposition 2.12 of [13] entails that an IO protocol computes a predicate ϕ if and only if post * (I b ) ⊆ pre * (ST b ) for every b ∈ {0, 1}. This is also true for RBN.…”

“…We show that there exist PSPACE algorithms to evaluate boolean combinations over counting sets and reachability set of counting sets. This result and its proof are adapted from a similar result for population protocols in [13].…”

“…We introduce a model of computation using RBN called RBN protocols. We take inspiration from the extensively-studied model of population protocols [1,2,13]. The reader can consult the above references for more details on population protocols.…”