generalized to CFMs with bounded channels: Over *universally* bounded MSCs (where all possible linear extensions meet a given channel bound), deterministic CFMs are expressively equivalent to MSO logic [@HenriksenJournal; @Kuske01]. Over *existentially* bounded MSCs (some linear extension meets the channel bound), CFMs are still expressively equivalent to MSO logic [@GKM06], but inherently nondeterministic [@GKM07]. The proofs of these characterizations reduce message-passing systems to finite-state shared-memory systems so that deep results from Mazurkiewicz trace theory [@DiekertRozenberg95] can be applied.

This generic approach is no longer applicable when the restriction on the channel capacity is dropped. Actually, in general, CFMs do not capture MSO logic [@BolligJournal]. On the other hand, they are expressively equivalent to existential MSO logic when we discard the happened-before relation [@BolligJournal] or when restricting to two first-order variables [@BFG-stacs18]. Both results rely on normal forms of FO logic, due to Hanf [@Hanf1965] and Scott [@GradelO99], respectively. However, MSCs with the happened-before relation are structures of *unbounded* degree (while Hanf’s normal form requires structures of bounded degree), and we consider FO logic with *arbitrarily* many variables (while Scott’s normal form only applies to two-variable logic). That is, neither approach is applicable in our case.

Finally, there exists a translation 