23.6 Target for pushdown RBN

Reconfigurable Broadcast Networks (RBN) are a model for large groups of identical agents communicating via unreliable broadcast. A pushdown RBN (PRBN) is simply one where each agent is modeled by a pushdown transition system. A PRBN on a message alphabet \(\mathcal{M}\) is described by a pushdown automaton \(\mathcal{A}\) over the alphabet \( \{br(m), rec(m) | m \in \mathcal{M} \}\). Configurations of \(\mathcal{A}\), i.e., pairs \((q, \sigma)\) of state and stack content, are called local configurations.\(\\ \)

A configuration of the PRBN is a function from a finite set of agents \(\mathbb{A}\) to local configurations. A run starts with an arbitrarily large set of agents, all in the initial state with an empty stack . A step consists of one agent taking a transition with a broadcast \(br(m)\), and each other agent either not moving or taking a transition \(rec(m)\) (in other words, one agent broadcasts and other agents non-deterministically receive the broadcast or not).\(\\ \)

Classic problems on such models ask whether a given set of configurations is reachable. A particular case of interest is whether we can reach a configuration where all agents are in a given state \(q_f\). This problem is called Target. The open problem is to find tight complexity bounds on the Target problem for PRBN. The problem on finite-state RBN is known to be solvable in polynomial time. An easy reduction from Horn satisfiability shows PTIME- completeness.\(\\ \)

On the other hand, one can show that the problem for PRBN is in NP: intuitively, to witness Target, it suffices to one can guess the set of messages \(\mathcal{M}' \subseteq \mathcal{M}\) of messages used, and two orders on this set, an order of appearance \(\leq_a\) and one of disappearance \(\leq_d\). Those orders are the one in which messages are first broadcast in the run, and the one in which messages are last broadcast. Then one checks, for each \(m \in \mathcal{M}'\), that there are local runs that broadcast \(m\) and only receive messages lower for \(<_a\) beforehand (resp. greater for \(<_d\) afterwards). < p>

If they exist, one can construct a run witnessing Target by making many agents follow each of those runs. If a run witnessing Target exists, they can be obtained by taking the local runs of agents broadcasting first and last each message.\(\\ \)

Is the Target problem for PRBN NP-hard? in PTIME?