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Minimisation and succinctness of parity automata for positional languages

Prove that deterministic and history-deterministic parity automata recognising positional languages can be minimised in polynomial time, and show that history-deterministic parity automata are not more succinct than deterministic ones for this class of languages.

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Background

Building on structural results, the authors conjecture efficient minimisation for positional languages and that history-determinism brings no size advantage over determinism in this setting, hinting at canonical/deterministic representatives.

References

Conjecture "Deterministic" and "history-deterministic" "parity automata" "recognising" "positional" languages can be minimised in polynomial time. Moreover, "history-deterministic" "parity automata" for this class of languages are not more succinct than "deterministic" ones.

— Positional $ω$-regular languages (2401.15384 - Casares et al., 27 Jan 2024) in Conclusions, Minimisation and canonisation of parity automata (Conjecture)