Eve-positional languages: putting order into Büchi automata

An $ω$-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 characterizations of Eve-positionallity of $ω$-regular languages. For this, they introduce the notion $\varepsilon$-complete parity automaton and show (among other results) that an $ω$-regular language is Eve-positional if and only if it can be recognized by some $\varepsilon$-completion of a deterministic parity automaton. Colcombet and Idir extended on their work, and obtained a more direct semantic characterization of Eve-positionality. We introduce a new formalism that characterizes the Eve-positional languages, consisting in a restriction of non-deterministic Büchi 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 Büchi automaton recognizing such languages, with size blow-up at most factorial. We also show that this construction is, in a suitable sense, optimal.

💡 Research Summary

The paper addresses the class of Eve‑positional ω‑regular languages, i.e., languages for which Eve (the existential player) always has a memory‑less winning strategy in infinite games. While recent work by Casares and Ohlmann introduced ε‑complete deterministic parity automata as a characterization of these languages, a converse implication (“every ε‑complete automaton recognizes an Eve‑positional language”) was missing, and no convenient syntactic formalism existed to manipulate such languages directly.

The authors introduce ordered Büchi automata (OBAs), a restricted form of nondeterministic Büchi automata. In an OBA, all Büchi‑ε transitions are required to respect a total order on the state set: an ε‑move may only go from a higher‑ordered state to a lower‑ordered one. This ordering forces any run that uses ε‑moves to progress monotonically towards “more favorable” states, which precisely captures the intuition that Eve never needs to remember past choices.

The main technical contributions are:

1. Exact Characterization (Theorem 9). The languages recognized by OBAs are exactly the Eve‑positional ω‑regular languages. The proof proceeds in two directions.

   • From an OBA one can construct an ε‑complete deterministic parity automaton (DP) that preserves the language, thereby showing that any OBA language is Eve‑positional.
   • Conversely, given an Eve‑positional language, one can start from any ε‑complete nondeterministic parity automaton (which exists by Casares‑Ohlmann’s result) and transform it into an OBA. This fills the missing implication in the earlier work.

2. Transformation from ε‑complete Parity to OBA (Theorem 14). The authors give a polynomial‑time construction that, starting from an ε‑complete nondeterministic parity automaton with n states and 2k priorities, produces an equivalent OBA with at most n·k states. The construction relies on “tiles”: for each alphabet symbol a, the set of transitions induced by a is treated as a tile, and the semigroup product of tiles captures the effect of concatenated letters. By quotienting states reachable via non‑Büchi ε‑moves and ordering the resulting equivalence classes, the authors obtain the required total order on ε‑transitions.

3. Determinization with Factorial Blow‑up (Theorem 20). From an OBA with n states the authors devise a determinization procedure that yields a deterministic parity automaton (DPA) with at most ∏_{i=1}^{n} i! states (i.e., roughly (n!)! ). This bound is significantly better than the generic 1.64ⁿ·n! bound for nondeterministic Büchi → deterministic parity conversion. Moreover, using techniques from prior work on universal graphs, they prove that for sufficiently rich alphabets (e.g., the Rabin language with n pairs and all possible letters) this bound is optimal: no smaller DPA can recognize the same language.

4. Semantic Insight via Tile Automata. By representing automata as collections of tiles and equipping the set of tiles with a monoid structure (via a min‑priority product), the authors obtain a clean algebraic view of ε‑moves and ordinary moves. This perspective makes the upward‑closed property of transition sets under the order explicit, clarifying why the ordering condition exactly captures Eve‑positionality.

5. Completing Prior Work. The paper explicitly shows that every ε‑complete nondeterministic parity automaton indeed recognizes an Eve‑positional language, thus completing the equivalence chain originally stated by Casares and Ohlmann (conditions (1)–(4) in their Theorem 1). Consequently, the class of ε‑complete automata coincides with the class of OBAs and with the class of Eve‑positional languages.

Overall, the paper provides both a syntactic formalism (ordered Büchi automata) that is easy to manipulate and a determinization algorithm that is provably optimal up to factorial blow‑up. This advances both the theoretical understanding of memory‑less strategies in infinite games and the practical toolbox for synthesis and verification, where constructing succinct controllers from specifications is paramount.