22.4 Reconfigurable broadcast networks (RBN)

Reconfigurable broadcast networks (RBN) are networks consisting of finite-state, anonymous agents that communicate by broadcast. An agent broadcasts a message which is received by its neighbors (one step is a broadcast plus multiple simultaneous receives). The neighbor topology can change (reconfigure) between any step.

Formally, an RBN is \(R = (Q, \Sigma, \delta)\) with \(Q\) a finite set of states, \(\Sigma\) a finite set of messages and \(\delta \subseteq Q \times \{ !a, ?a | a \in \Sigma \} \times Q\) a finite set of transitions. Transitions are broadcasts of the form \((q, !a, q’)\), or receives of the form \((q,?a, q’)\). A configuration \(C\) is a multiset over \(Q\), which intuitively counts the number of agents in each state. Given a letter \(a\in \Sigma\) and two configurations \(C\) and \(C’\) we say that there is a step \(C \xrightarrow{a} C’\) if there exists a multiset \([ t, t_1, \ldots, t_k ]\) of \(\delta\) for some \(k\ge 0\) satisfying \(t=(p, !a, q)\), each \(t_i =(p_i, ?a, q_i)\), and in multiset notation \(C \ge p + \sum_i p_i\), and \(C' = C - p - \sum_i p_i + q + \sum_i q_i\). Intuitively it means that an agent at the state \(p\) broadcasts the message \(a\) and moves to \(q\), and for each \(1 \le i \le k\), there is an agent at the state \(p_i\) which receives this message and moves to \(q_i\). We denote by \(\xrightarrow{*}\) the reflexive and transitive closure of the step relation. \(C’\) is reachable from \(C\) if \(C \xrightarrow{*} C’\).

A cube \(Cube\) over \(Q\) is a set of configurations described by a lower bound \(L \colon Q \rightarrow \mathbb{N}\) and an upper bound \(U \colon Q \rightarrow \mathbb{N} \cup \{ \infty \}\) such that \(Cube = \{ C : L \le C \le U \}\). A counting set is a finite union of cubes. The reachability set \(post^*(S)\) of a counting set \(S\) is all the configurations reachable from a configuration of \(S\). The open question is whether \(post^*(S)\), for \(S\) a counting set, is still a counting set, i.e. are counting sets closed under reachability in RBN? And if so, then what can we say about the size of its bounds with respect to the size of the bounds of \(S\).

In a paper by A. R. Balasubramanian and I in Fossacs22, we answered yes, and that the size is exponential in the size of \(S\) and \(Q\). But there was an error in the proof (Theorem 2), kindly pointed out by Nicolas Waldburger. If we solve this problem for RBN then we get some nice results like closing a complexity gap from an ICALP paper by Bouyer et al. on an equivalent-for-these-questions model called asynchronous shared-memory systems or register protocols. This relates to a result for immediate observation (IO) Petri nets, which can be seen as a subclass of RBN. For IO nets, the answer is yes and the size is polynomial in the size of \(S\) and \(Q\).

Link to FoSSaCS article: https://arxiv.org/abs/2201.10432