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Sufficiency of non-deterministic ε-complete automata for positionality

Ascertain whether the existence of a non-deterministic ε-complete parity automaton recognising an objective W implies that W is positional over all games.

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Background

The paper introduces ε-complete automata and proves several equivalences for deterministic (or history-deterministic) ε-complete completions in connection with positionality. However, whether a non-deterministic ε-complete witness alone suffices to conclude positionality remains open.

References

We do not know whether the existence of a "non-deterministic" "$\ee$-complete" automaton suffices to prove "positionality".

— Positional $ω$-regular languages (2401.15384 - Casares et al., 27 Jan 2024) in Section 3: Positionality of ω-regular objectives; around Proposition on ε-completable automata