Weakly synchronous systems with three machines are Turing powerful
This resolves an open problem in distributed systems theory, establishing the boundary between decidability and undecidability for such systems, which is incremental but precise.
The paper tackled the decidability of the configuration reachability problem in weakly synchronous systems with three processes using p2p FIFO communication, showing it is undecidable by constructing a system that generates Message Sequence Charts with arbitrarily large treewidth.
Communicating finite-state machines (CFMs) are a Turing powerful model of asynchronous message-passing distributed systems. In weakly synchronous systems, processes communicate through phases in which messages are first sent and then received, for each process. Such systems enjoy a limited form of synchronization, and for some communication models, this restriction is enough to make the reachability problem decidable. In particular, we explore the intriguing case of p2p (FIFO) communication, for which the reachability problem is known to be undecidable for four processes, but decidable for two. We show that the configuration reachability problem for weakly synchronous systems of three processes is undecidable. This result is heavily inspired by our study on the treewidth of the Message Sequence Charts (MSCs) that might be generated by such systems. In this sense, the main contribution of this work is a weakly synchronous system with three processes that generates MSCs of arbitrarily large treewidth.