Eve-positional languages: putting order into B\"uchi automata

An $\omega$-regular language is Eve-positional if, in all games with this language as objective, the existential player can play optimally without keeping any information from the previous moves. This notion plays a crucial role in verification, automata theory and synthesis. Casares and Ohlmann recently gave several characterisations of Eve-positionality of $\omega$-regular languages. For this, they introduce the notion of $\varepsilon$-complete parity automaton and show (among other results) that an $\omega$-regular language is Eve-positional if and only if it can be recognised by some $\varepsilon$-completion of a deterministic parity automaton. Colcombet and Idir built on their work, and obtained a more direct algebraic characterisation of Eve-positionality. We introduce a new formalism that characterises the Eve-positional languages, consisting of a restriction of non-deterministic B\"uchi automata. This allows us to complete a missing implication in Casares and Ohlmann's work. We then use this formalism to describe a determinization procedure for non-deterministic B\"uchi automata recognising such languages, with size blow-up at most factorial. We also show that this construction is state-wise optimal for languages over sufficiently complete alphabets.