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Decidability of complementability of a global type

Determine whether, given a global type over arrows (a deterministic finite automaton specifying an existential MSC language), it is decidable to check the existence of a complement global type whose existential MSC language equals the set-theoretic complement within the synchronous MSC universe.

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Background

The paper proves that not all global types are complementable and presents complementation procedures for certain subclasses (e.g., commutation-closed types via duality and sender-driven types via renunciation).

However, whether the property of being complementable is decidable for arbitrary global types remains unresolved. Establishing decidability (or undecidability) would significantly advance the automata-theoretic foundations of MPST.

References

some old works on Mazurkiewicz traces could possibly shade some light on a few questions that arise from our work, and which, to the best of our knowledge, remain open, like the decidability of the complementability of a global type, or concise complementation procedures for non commutation-closed global types beyond sender-driven choices.

— Realisability and Complementability of Multiparty Session Types (2507.17354 - Giusto et al., 23 Jul 2025) in Section 6: Related works