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.

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)