Weakly synchronous systems with three machines are Turing powerful

TL;DR

This paper demonstrates that weakly synchronous systems with three processes can generate complex behaviors, making the configuration reachability problem undecidable, thus extending understanding of the computational power of such systems.

Contribution

It proves that three-process weakly synchronous systems can produce MSCs of arbitrarily large treewidth, leading to undecidability of reachability.

Findings

Configuration reachability is undecidable for three-process systems.

Three-process systems can generate MSCs with arbitrarily large treewidth.

Weak synchronization increases system complexity significantly.

Abstract

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.