Half-Positional Objectives Recognized by Deterministic Büchi Automata

A central question in the theory of two-player games over graphs is to understand which objectives are half-positional , that is, which are the objectives for which the protagonist does not need memory to implement winning strategies. Objectives for which both players do not need memory have already been characterized (both in finite and infinite graphs); however, less is known about half-positional objectives. In particular, no characterization of half-positionality is known for the central class of ω -regular objectives . In this paper, we characterize objectives recognizable by deterministic Büchi automata (a class of ω -regular objectives) that are half-positional, in both finite and infinite graphs. Our characterization consists of three natural conditions linked to the language-theoretic notion of right congruence . Furthermore, this characterization yields a polynomial-time algorithm to decide half-positionality of an objective recognized by a given deterministic Büchi automaton.


Introduction
Graph games and reactive synthesis. We study zero-sum turn-based games on graphs confronting two players (a protagonist and its opponent). They interact by moving a pebble in turns through the edges of a graph for an infinite amount of time. Each vertex belongs to a player, and the player controlling the current vertex decides on the next state of the game. Edges of the graph are labeled with colors, and the interaction of the two players therefore produces an infinite sequence of them. The objective of the game is specified by a subset of infinite sequences of colors, and the protagonist wins if the produced sequence belongs to this set. We are interested in finding a winning strategy for the protagonist, that is, a function indicating how the protagonist should move in any situation, guaranteeing the achievement of the objective.
This game-theoretic model is particularly fitted to study the reactive synthesis problem [7]: a system (the protagonist) wants to satisfy a specification (the objective) while interacting continuously with its environment (the opponent). The goal is to build a controller for the system satisfying the specification, whenever possible. This comes down to finding a winning strategy for the protagonist in the derived game.
infinite sequences of elements of A. We denote by ε the empty word.

Games and positionality
Graphs. An (edge-colored) graph G = (V, E) is given by a non-empty set of vertices V (of any cardinality) and a set of edges We assume graphs to be non-blocking: for all v ∈ V , there exists (v ′ , c, v ′′ ) ∈ E such that v = v ′ . We allow graphs with infinite branching. For v ∈ V , an infinite path of G from v is an infinite sequence of edges π = (v 0 , c 1 A finite path of G from v is a finite prefix in E * of an infinite path of G from v. For convenience, we assume that there is a distinct empty path λ v for every v ∈ V . If γ = (v 0 , c 1 , v 1 ) . . . (v n−1 , c n , v n ) is a non-empty finite path of G, we define last(γ) = v n . For an empty path λ v , we define last(λ v ) = v. An infinite (resp. finite) path (v 0 , c 1 A strongly connected component of G is a maximal connected subgraph. Arenas and strategies. We consider two players P 1 and P 2 . An arena is a tuple A = (V, V 1 , V 2 , E) such that (V, E) is a graph and V is the disjoint union of V 1 and V 2 . Intuitively, vertices in V 1 are controlled by P 1 and vertices in V 2 are controlled by P 2 . An arena A = (V, V 1 , V 2 , E) is a one-player arena of P 1 (resp. of P 2 ) if V 2 = ∅ (resp. V 1 = ∅). Finite paths of (V, E) are called histories of A. For i ∈ {1, 2}, we denote by Hists i (A) the set of histories γ of A such that last(γ) ∈ V i .
Let i ∈ {1, 2}. A strategy of P i on A is a function σ i : Hists i (A) → E such that for all γ ∈ Hists i (A), the first component of σ i (γ) coincides with last(γ). Given a strategy σ i of P i , we say that an infinite path π = e 1 e 2 . . . is consistent with σ i if for all finite prefixes γ = e 1 . . . e i of π such that last(γ) ∈ V i , σ i (γ) = e i+1 . A strategy σ i is positional (also called memoryless in the literature) if its outputs only depend on the current vertex and not on the whole history, i.e., if there exists a function f : V i → E such that for γ ∈ Hists i (A), σ i (γ) = f (last(γ)).
Objectives. An objective is a set W ⊆ C ω (this object is sometimes called a language of infinite words or ω-language in the literature). When an objective W is clear in the context, we say that an infinite word w ∈ C ω is winning if w ∈ W , and losing if w / ∈ W . We write W for the complement C ω \ W of an objective W . An objective W is prefix-independent if for all w ∈ C * and w ′ ∈ C ω , w ′ ∈ W if and only if ww ′ ∈ W . An objective that we will often consider is the Büchi condition: given a subset F ⊆ C, we denote by Büchi(F ) the set of infinite words seeing infinitely many times a color in F . Such an objective is prefix-independent. A game is a tuple (A, W ) of an arena A and an objective W .
Optimality and half-positionality. Let A = (V, V 1 , V 2 , E) be an arena, (A, W ) be a game, and v ∈ V . We say that a strategy σ 1 of P 1 is winning from v if for all infinite paths v 0 A strategy of P 1 is optimal for P 1 in (A, W ) if it is winning from all the vertices from which P 1 has a winning strategy. We often write optimal for P 1 in A if the objective W is clear from the context. We stress that this notion of optimality requires a single strategy to be winning from all the winning vertices (a property sometimes called uniformity).
An objective W is half-positional if for all arenas A, there exists a positional strategy of P 1 on A that is optimal for P 1 in A. We sometimes only consider half-positionality on a restricted set of arenas (typically, finite and/or one-player arenas). An objective W is half-positional over X arenas if for all X arenas A, there exists a positional strategy of P 1 on A that is optimal for P 1 in A.
▶ Remark 1 (ε-edges). Sometimes, arenas are considered to be colored over the alphabet C ∪ {ε}, adding the restriction that no cycle is entirely labeled by ε [27,62,39,35,17]. In that case, an infinite word in (C ∪ {ε}) ω labeling a path is winning if the word obtained by removing the occurrences of ε belongs to W . In this paper, we consider arenas without ε-edges, but all our results apply to this other setting (cf. Remark 40). In general, allowing for arenas with ε-edges has an effect on strategy complexity [17]. ⌟

Büchi automata
Automaton structures and Büchi automata. A non-deterministic automaton structure is a tuple S = (Q, C, Q init , ∆) such that Q is a finite set of states, Q init ⊆ Q is a non-empty set of initial states and ∆ ⊆ Q × C × Q is a set of transitions. We assume that all states of automaton structures are reachable from an initial state in Q init by taking transitions in ∆.
A (transition-based) non-deterministic Büchi automaton (NBA) is an automaton structure S together with a set of transitions α ⊆ ∆. The transitions in α are called Büchi transitions.
Given an NBA B = (Q, C, Q init , ∆, α), a (finite or infinite) run of B on a (finite or infinite) word w = c 1 c 2 . . . ∈ C * ∪ C ω is a sequence (q 0 , c 1 , q 1 )(q 1 , c 2 , q 2 ) . . . ∈ ∆ * ∪ ∆ ω such that q 0 ∈ Q init . An infinite run (q 0 , c 1 , q 1 )(q 1 , c 2 , q 2 ) . . . ∈ ∆ ω of B is accepting if for infinitely many i ≥ 0, (q i , c i+1 , q i+1 ) ∈ α. A word w ∈ C ω is accepted by B if there exists an accepting run of B on w -if not, it is rejected. We denote the set of infinite words accepted by B by L(B), and we then say that L(B) is the objective recognized by B. Here, we take the definition of an ω-regular objective as an objective that can be recognized by an NBA (the classical definition uses ω-regular expressions, but our definition is well-known to be equivalent [47]). Given an automaton structure S = (Q, C, Q init , ∆), we say that an NBA B is built on top of S if there exists α ⊆ ∆ such that B = (Q, C, Q init , ∆, α).
▶ Remark 2. Notice that for generality, we allow the set of colors C to be infinite. This is uncommon for ω-regular objectives and automata, but has no real impact: when an objective is specified by an NBA, there will be at most finitely many equivalence classes of colors for the equivalence relation "inducing exactly the same transitions in the NBA". In Section 4.3.2, it will be helpful to consider infinite sets of colors. ⌟ Deterministic automata. An automaton structure S = (Q, C, Q init , ∆) is deterministic if |Q init | = 1 and, for each q ∈ Q and c ∈ C, there is exactly one q ′ ∈ Q such that (q, c, q ′ ) ∈ ∆ (we remark that, without loss of generality, we define deterministic automaton structures so that for each state and each color there is one outgoing transition -such automata are sometimes called complete or total). A deterministic Büchi automaton (DBA) is an NBA whose underlying automaton structure is deterministic. For a DBA B = (Q, C, {q init }, ∆, α), we denote by q init the unique initial state (and we will drop the braces around q init in the tuple), and by δ : Q × C → Q the update function that associates to (q, c) ∈ Q × C the only q ′ ∈ Q such that (q, c, q ′ ) ∈ ∆. We denote by δ * the natural extension of δ to finite words -by induction, the function δ * : Q × C * → Q is such that δ * (q, ε) = q, and for w ∈ C * , c ∈ C, δ * (q, wc) = δ(δ * (q, w), c). As transitions are uniquely determined by their first two components, we also assume for brevity that α ⊆ Q × C.
An objective W is DBA-recognizable if there exists a DBA B such that W = L(B).
▶ Remark 3. The fact that a single state suffices to recognize Büchi(F ) relies on the assumption that our DBA are transition-based and not state-based (α is a set of transitions, not of states). Indeed, apart from the trivial cases F = ∅ and F = C, a state-based DBA recognizing Büchi(F ) requires two states. The third condition of our upcoming characterization (Theorem 19) would therefore not apply to this simple example if we only considered state-based DBA. ⌟ In this paper, all automata will be deterministic, and the term "automaton" will stand for "deterministic automaton" by default.
▶ Remark 4. Deterministic Büchi automata recognize a proper subset of the ω-regular objectives. That is, not every non-deterministic Büchi automaton can be converted into a deterministic one recognizing the same objective [61]. ⌟ We restate a well-known lemma about ω-regular objectives: if two ω-regular objectives are not equal, then they are distinguished by an ultimately periodic word. Ultimately periodic words can easily be finitely represented, and this lemma will be used in Section 4 to force some behaviors to appear in finite arenas.
When ∼ has finitely many equivalence classes, we can associate a natural deterministic [58,46]. The transition function δ ∼ is welldefined since it follows from the definition of ∼ that if w 1 ∼ w 2 , then for all c ∈ C, w 1 c ∼ w 2 c. Hence, the choice of representatives for the equivalence classes does not have an impact on this definition. We call the automaton structure S ∼ the prefix-classifier of W . ▶ Remark 6. Equivalence relation ∼ W has only one equivalence class if and only if W is prefix-independent. In particular, an objective has a prefix-classifier with a single state if and only if it is prefix-independent. ⌟ An important property of ∼ is the following.
Proof. If w 1 and w 2 have the same winning continuations, they have in particular the same winning continuations starting with w. ◀ We define the prefix preorder ⪯ W of W : for w 1 , w 2 ∈ C * , we write w 1 ⪯ W w 2 if w −1 1 W ⊆ w −1 2 W (meaning that any continuation that is winning after w 1 is also winning after w 2 ). Intuitively, w 1 ⪯ W w 2 means that a game starting with w 2 is always preferable to a game starting with w 1 for P 1 , as there are more ways to win after w 2 . When W is clear from the context, we write ⪯ for ⪯ W . Relation ⪯ ⊆ C * × C * is a preorder. Notice that ∼ is equal to ⪯ ∩ ⪰. We also define the strict preorder ≺ = ⪯ \ ∼.
Given a DBA B = (Q, C, q init , ∆, α) recognizing the objective W , observe that for w, w ′ ∈ C * such that δ * (q init , w) = δ * (q init , w ′ ), we have w ∼ w ′ . In this case, equivalence relation ∼ has at most |Q| equivalence classes. For q ∈ Q, we write q −1 W for the objective recognized by the DBA (Q, C, q, ∆, α). Objective q −1 W equals w −1 W for any word w ∈ C * such that δ * (q init , w) = q. We extend the equivalence relation ∼ and preorder ⪯ to elements of Q (we sometimes write ∼ B and ⪯ B to avoid any ambiguity).
Safe words. Let B = (Q, C, q init , ∆, α) be a DBA. We say that a run ϱ ∈ ∆ * of B is α-safe, or simply safe for brevity, if it does not contain any transition from α. For q ∈ Q, we define We call the words in the first set the safe words from q, and the words in the second set the safe cycles from q. We state an important property of safe words.

Saturation of DBA with Büchi transitions.
In what follows, we will make extensive use of a "normal form" of Büchi automata verifying that any safe path can be extended to a safe cycle. Such a normal form can be produced by saturating a given DBA B with Büchi transitions [42,1,2]. To do so, we add to α all transitions that do not appear in a safe cycle of B. Safe cycles can be easily identified by decomposing in strongly connected components the structure obtained by removing the Büchi transitions from B. We say that B = (Q, C, q init , ∆, α) is saturated if for every α ′ ⊋ α, the automaton obtained by replacing α with α ′ does not recognize L(B).
A safe component of B is a strongly connected component of the graph obtained by removing the Büchi transitions from B. That is, a strongly connected component of (Q, ∆ safe ), with ∆ safe = ∆ \ α. Proof. We first prove the existence of such an α sat . Let (Q 1 , ∆ 1 ), . . . , (Q k , ∆ k ) be the safe components of B. We consider the automaton B sat whose Büchi transitions are those that do not appear in any safe component (Q i , ∆ i ), that is, we let We show that L(B sat ) = L(B). Since α ⊆ α sat , it is verified that L(B) ⊆ L(B sat ). For the other inclusion, let w / ∈ L(B). There are w 0 ∈ C * , w ′ ∈ C ω such that w = w 0 w ′ and the run B(q 0 , w ′ ) produced by reading w ′ from q 0 = δ * (q init , w 0 ) does not visit any Büchi transition. In particular, B(q 0 , w ′ ) is an infinite path in the finite graph (Q, ∆ safe ). This implies that eventually, run B(q 0 , w ′ ) reaches and stays in the same strongly connected component of graph (Q, ∆ safe ). Formally, there are w 1 ∈ C * , w 2 ∈ C ω such that w ′ = w 1 w 2 and the run B(q 1 , w 2 ) produced by reading w 2 from q 1 = δ * (q 0 , w 1 ) lies entirely in some safe component (Q i , ∆ i ). Let ∆ q1,w2 be the transitions appearing in B(q 1 , w 2 ). We have ∆ q1,w2 ⊆ ∆ i . Therefore, ∆ q1,w2 ∩ α sat = ∅, so B sat (q init , w) is also a rejecting run of B sat .
We prove that B sat is saturated and the uniqueness of α sat at the same time. Let α ′ be another acceptance set such that α ′ ⊈ α sat and let B ′ be the automaton obtained by replacing We can therefore consider a word w ∈ SafeCycles Bsat (q) labeling a safe cycle including the transition (q, c, q ′ ). Let w 0 ∈ C * such that δ * (q init , w 0 ) = q. Then, w 0 w ω / ∈ L(B sat ), whereas w 0 w ω ∈ L(B ′ ), so B ′ does not recognize the same objective as B sat .
When C is finite, the set of transitions B sat can be computed in time O(|C| · |Q|), as it consists of decomposing a graph with at most |C| · |Q| transitions into strongly connected components [59]. In Figure 1, we show an example of the saturation process presented in the proof of Lemma 9.
The following simple lemma follows, which holds true in saturated DBA: every word that is safe from a state can be completed into a safe cycle from the same state. This is a key technical lemma used many times in the upcoming proofs.
Proof. Let q ′ = δ * (q, w). By Lemma 9, α contains all transitions that do not belong to a safe component. Therefore, q ′ must belong to the same safe component as q, so there exists a safe run from q ′ to q. Taking the word w ′ labeling this run, we obtain the desired result. ◀

Half-positionality characterization for DBA-recognizable objectives
In this section, we present our main contribution in Theorem 19, by giving three conditions that exactly characterize half-positional DBA-recognizable objectives. These conditions are presented in Section 3.1. Theorem 19 and several consequences of it are stated in Section 3.2 (the proof of Theorem 19 is postponed to Sections 4 and 5). In Section 3.3, we use this characterization to show that we can decide the half-positionality of a DBA in polynomial time.

Three conditions for half-positionality
We define the three conditions on objectives at the core of our characterization.
▶ Condition 1 (Total prefix preorder). We say that an objective W ⊆ C ω has a total prefix preorder if for all w 1 , w 2 ∈ C * , w 1 ⪯ W w 2 or w 2 ⪯ W w 1 .
▶ Example 11 (Not total prefix preorder). Let C = {a, b}. We consider the objective W recognized by the DBA B depicted in Figure 2 (left). It consists of the infinite words starting with aa or bb. This objective does not have a total prefix preorder: words a and b are incomparable for ⪯ W . Indeed, a ω is winning after a but not after b, and b ω is winning after b but not after a. In terms of automaton states, we have that q a and q b are incomparable for ⪯ B . This objective is not half-positional, as witnessed by the arena on the right of Figure 2.
In this arena, P 1 is able to win when the game starts in v 1 by playing a in v 3 , and when the game starts in v 2 by playing b. However, no positional strategy wins from both v 1 and v 2 . ⌟ ▶ Remark 12. The prefix preorder of an objective W is total if and only if the prefix preorder of its complement W is total. ⌟ ▶ Remark 13. Having a total prefix preorder is equivalent to the strong monotony notion [6] in general, and equivalent to monotony [33] for ω-regular objectives. We discuss in more depth the relation between the conditions appearing in the characterization and other properties from the literature studying half-positionality in Appendix A. ⌟ A straightforward result for an objective W recognized by a DBA B is that it has a total prefix preorder if and only if the (reachable) states of B are totally ordered for ⪯ B . Moreover, transitions of B following a given word are non-decreasing w.r.t. states.
Intuitively, this means that whenever a word w 2 can be used to make progress after seeing a word w 1 (in the sense of getting to a position in which more continuations are winning), then repeating this word has to be winning.
▶ Example 15 (Not progress-consistent). Let C = {a, b}. We consider the objective W = C * aaC ω recognized by the DBA with three states in Figure 3 (left). This objective contains the words seeing, at some point, twice the color a in a row. Notice that the prefix preorder of this objective is total (q init ≺ q a ≺ q aa ). This objective is not progress-consistent: we have ε ≺ ba, but (ba) ω / ∈ W . This objective is not half-positional: if P 1 plays in an arena with a choice among two cycles ba and ab depicted in Figure 3 (right), it is possible to win by playing ba and then ab, but a positional strategy can only achieve words (ba) ω or (ab) ω , which are both losing.
• v ab ba ▶ Example 16 (Progress-consistent objective). We consider a slight modification of the previous example by adding two Büchi transitions: see the DBA in Figure 4. The objective recognized by this DBA is W = Büchi({a}) ∪ C * aaC ω : W contains the words seeing a infinitely often, or that see a twice in a row at some point. The equivalence classes for This objective is progress-consistent: any word reaching q aa is straightforwardly accepted when repeated infinitely often, and any word w such that δ * (q init , w) = q a necessarily contains at least one a, and thus is accepted when repeated infinitely often. Objective W is half-positional, which will be readily shown with our upcoming characterization (Theorem 19).
Here, notice that the complement W of W is not progress-consistent. Indeed, a ≺ W a(bab), but a(bab) ω / ∈ W . Unlike having a total prefix preorder, progress-consistency can hold for an objective but not its complement.
Note that half-positionality of W cannot be shown using existing half-positionality criteria [38,6] (it is neither prefix-independent nor concave) nor bipositionality criteria, as it is simply not bipositional. ⌟ ▶ Condition 3 (Recognizability by the prefix-classifier). Being recognized by a Büchi automaton built on top of the prefix-classifier is our third condition. In other words, for a DBA-recognizable objective W ⊆ C ω and its prefix-classifier S ∼ = (Q ∼ , C, q init , ∆ ∼ ), this condition requires that there exists • Figure 4 A DBA recognizing the set of words seeing a infinitely many times, or aa at some point.
We show an example of a DBA-recognizable objective that satisfies the first two conditions (having a total prefix preorder and progress-consistency), but not this third condition, and which is not half-positional.
▶ Example 17 (Not recognizable by the prefix-classifier). Let C = {a, b}. We consider the objective W = Büchi({a}) ∩ Büchi({b}) recognized by the DBA in Figure 5. This objective is prefix-independent: as such (Remark 6), there is only one equivalence class for ∼. This implies that the prefix preorder is total, and that W is progress-consistent (the premise of the progress-consistency property can never be true). This objective is not half-positional, as witnessed by the arena in Figure 5 (right): P 1 has a winning strategy from v, but it needs to take infinitely often both a and b. Any DBA recognizing this objective has at least two states, but all their (reachable) states are equivalent for ∼ -no matter the state we choose as an initial state, the recognized objective is the same (by prefix-independence). As it is prefix-independent, its prefix-classifier S ∼ has only one state. As will be shown formally, being recognized by a DBA built on top of the prefix-classifier is necessary for half-positionality of DBA-recognizable objectives over finite one-player arenas. Unlike the two other conditions, it is in general not necessary for half-positionality of general objectives, including objectives recognized by other standard classes of ω-automata.
▶ Example 18. We consider the complement W of the objective W = Büchi({a})∩Büchi({b}) of Example 17, which consists of the words ending with a ω or b ω . Objective W is not DBArecognizable (a close proof can be found in [5,Theorem 4.50]). Still, it is recognizable by a deterministic coBüchi automaton similar to the automaton in Figure 5, but which accepts infinite words that visit transitions labeled by • only finitely often. This objective is half-positional, which can be shown using [27,Theorem 6]. However, its prefix-classifier has just one state, and there is no way to recognize W by building a coBüchi (or even parity) automaton on top of it. ⌟

Characterization and corollaries
We have now defined the three conditions required for our characterization.

over all arenas) if and only if its prefix preorder ⪯ is total, it is progress-consistent, and it can be recognized by a Büchi automaton built on top of its prefix-classifier S ∼ .
Proof. The proof of the necessity of the three conditions can be found in Section 4, respectively in Propositions 26, 27, and 28. The proof of the sufficiency of the conjunction of the three conditions can be found in Section 5, Proposition 41. ◀ This characterization is valuable to prove (and disprove) half-positionality of DBArecognizable objectives. Examples 11,15, and 17 are all not half-positional, and each of them falsifies exactly one of the three conditions from the statement. On the other hand, Example 16 is half-positional. We have already discussed its progress-consistency, but it is also straightforward to verify that its prefix preorder is total and that it is recognizable by its prefix-classifier: the right congruence has three totally ordered equivalence classes corresponding to the states of the automaton of Figure 4.
We state two notable consequences of Theorem 19 and of its proof technique. The first one is the specialization of Theorem 19 to prefix-independent objectives. It states that all prefix-independent, DBA-recognizable objectives that are half-positional are of the kind Büchi(F ) for some F ⊆ C. Prefix-independence of objectives is a frequent assumption in the literature [38,26,30,24] -we show that under this assumption, half-positionality of DBA-recognizable objectives is very easy to understand and characterize.
Proof. The right-to-left implication follows from the known half-positionality of objectives of the kind Büchi(F ) (this is a special case of a parity game [29]). For the left-to-right implication, let W be a prefix-independent, DBA-recognizable, half-positional objective. By Theorem 19, it is recognized by a DBA B built on top of S ∼ . As W is prefix-independent (Remark 6), its prefix-classifier has just one state, and there is a single transition from and to this single state for each color. Hence, W = Büchi(F ), where F is the set of colors whose only transition is a Büchi transition in B. ◀ ▶ Remark 21. A corollary of this result is that when W is prefix-independent, DBArecognizable and half-positional, we also have that W is half-positional. Indeed, the complement of objective W = Büchi(F ) is a so-called coBüchi objective, which is also known to be half-positional [29]. This statement does not hold in general when W is not prefixindependent, as was shown in Example 16. Moreover, the reciprocal of the statement also does not hold, as was shown in Example 18. ⌟ ▶ Remark 22. A second corollary is that prefix-independent DBA-recognizable half-positional objectives are closed under finite union (since a finite union of Büchi conditions is a Büchi condition). This settles Kopczyński's conjecture for DBA-recognizable objectives. ⌟ A second consequence of Theorem 19 and its proof technique shows that half-positionality of DBA-recognizable objectives can be reduced to half-positionality over the restricted class of finite, one-player arenas. Results reducing strategy complexity in two-player arenas to the easier question of strategy complexity in one-player arenas are sometimes called one-to-two-player lifts and appear in multiple places in the literature [33,9,41,12].
▶ Proposition 23 (One-to-two-player and finite-to-infinite lift). Let W ⊆ C ω be a DBArecognizable objective. If objective W is half-positional over finite one-player arenas, then it is half-positional over all arenas (of any cardinality).

Proof.
When showing the necessity of the three conditions for half-positionality of DBArecognizable objectives in Section 4 (Propositions 26, 27, and 28), we actually show their necessity for half-positionality over finite one-player arenas. Hence, assuming half-positionality over finite one-player arenas, we have the three conditions from the characterization of Theorem 19, so we have half-positionality over all arenas. ◀ One-to-two-player lifts from the literature all require an assumption on the strategy complexity of both players, and are either stated solely over finite arenas, or solely over infinite arenas. Proposition 23, albeit set in the more restricted context of DBA-recognizable objectives, displays stronger properties than the known one-to-two-player lifts.
It is asymmetric in the sense that we simply need a hypothesis on one player: halfpositionality of DBA-recognizable objectives over one-player arenas implies their halfpositionality over two-player arenas. It shows that half-positionality of DBA-recognizable objectives over finite arenas implies half-positionality over infinite arenas. Both these properties do not hold for general objectives.
Some objectives are half-positional over one-player but not over two-player arenas [31, Section 7] -we have that this is not possible for DBA-recognizable objectives. Some objectives are half-positional over finite but not over infinite arenas (see, e.g., the mean-payoff objective [28,54]) -we have that this is not possible for DBA-recognizable objectives.

Deciding half-positionality in polynomial time
In this section, we assume that C is finite. We show that the problem of deciding, given a DBA B = (Q, C, q init , ∆, α) as an input, whether L(B) is half-positional can be solved in polynomial time, and more precisely in time O(|C| 2 · |Q| 4 ). We investigate how to verify each property used in the characterization of Theorem 19. Let B = (Q, C, q init , ∆, α) be a DBA (we assume w.l.o.g. that all states in Q are reachable from q init ) and W = L(B) be the objective it recognizes. Our algorithm first verifies that the prefix preorder is total and recognizability by S ∼ , and then, under these first two assumptions, progress-consistency. For each condition, we sketch an algorithm to decide it, and we discuss the time complexity of this algorithm.
Total prefix preorder. To check that W has a total prefix preorder, it suffices to check that the states of B are totally preordered by ⪯ B . We start by computing, for each pair of states q, q ′ ∈ Q, whether q ⪯ B q ′ , q ′ ⪯ B q, or none of these. This can be rephrased as an inclusion problem for two DBA-recognizable objectives: if B q = (Q, C, q, ∆, α) and In particular, the prefix preorder is total if and only if for all q, Recognizability by the prefix-classifier. After all the relations ⪯ B and ∼ B between pairs of states are computed in the previous step, we can compute the states and transitions of the prefix-classifier S ∼ = (Q ∼ , C, q init , ∆ ∼ ) by merging all the equivalence classes for ∼ B . We assume for simplicity that Q ∼ = Q ∼ B .
We now wonder whether it is possible to recognize W by carefully selecting a set α ∼ of Büchi transitions in S ∼ . We simplify the search for such a set with the following result, which shows that it suffices to try with one specific set α ∼ . We can then simply check whether W = L((Q ∼ , C, q init , ∆ ∼ , α ∼ )), an equivalence query which, as discussed above, can be performed in time O(|C| 2 · |Q| 2 ).
▶ Lemma 24. We assume that B is saturated and that W is recognized by a DBA built on top of the prefix-classifier S ∼ = (Q ∼ , C, q init , ∆ ∼ ). We define Proof. We assume that W is recognized by a DBA built on top of S ∼ . We start by saturating this DBA, which yields a set of Büchi transitions α ′ such that W is also recognized by the saturated DBA B ′ = (Q ∼ , C, q init , ∆ ∼ , α ′ ) (Lemma 9). To prove the claim, we show that

Progress-consistency.
We assume that we have already checked that W is recognizable by a Büchi automaton built on top of S ∼ , and that we know the (total) ordering of the states. We show that checking progress-consistency, under these two hypotheses, can be done in polynomial time. We prove a lemma reducing the search for words witnessing that W is not progress-consistent to a problem computationally easier to investigate.
▶ Lemma 25. We assume that B is built on top of the prefix-classifier S ∼ and that the prefix preorder of W is total. Then, W is progress-consistent if and only if for all q, q ′ ∈ Q with q ≺ B q ′ , Proof. For the left-to-right implication, we assume by contrapositive that there exist q, q ′ ∈ Q with q ≺ B q ′ and w ∈ C + such that δ * (q, w) = q ′ and w ∈ SafeCycles B (q ′ ). Let w q ∈ C * be a word such that δ * (q init , w q ) = q. We have that w q ≺ w q w, but w q w ω is not accepted by B as w is a cycle on q ′ that does not see any Büchi transition. Hence, W is not progress-consistent. For the right-to-left implication, we assume by contrapositive that W is not progressconsistent. Thus, there exist w ′ ∈ C * and w ∈ C + such that w ′ ≺ w ′ w and w ′ w ω / ∈ W . Let q 1 = δ * (q init , w ′ ) and q 2 = δ * (q 1 , w) -we have q 1 ≺ q 2 . As q 1 ≺ q 2 , by Lemma 14, we have δ * (q 1 , w) = q 2 ⪯ δ * (q 2 , w). We distinguish two cases, using the fact that there is exactly one state per equivalence class of ∼ B . We represent what happens in Figure 6.
If q 2 = δ * (q 2 , w), we then have that w ∈ SafeCycles B (q 2 ), and we have what we want with q = q 1 and q ′ = q 2 .
If not, we have that q 2 ≺ δ * (q 2 , w). Let q 3 = δ * (q 2 , w). We can repeat the argument on q 2 and q 3 : either w ∈ SafeCycles B (q 3 ) and we are done, or q 3 ≺ δ * (q 3 , w). As there are finitely many states, this process necessarily ends with two states q = q n and q ′ = q n+1 such that δ * (q, w) = q ′ and w ∈ SafeCycles Figure 6 Situation in the proof of Lemma 25.
Notice that for each pair of states q, q ′ ∈ Q, the sets {w ∈ C + | δ(q, w) = q ′ } and SafeCycles B (q ′ ) are both regular languages recognized by deterministic finite automata with at most |Q| states. The emptiness of their intersection can be decided in time O(|C| 2 ·|Q| 2 ) (by solving a reachability problem in the product of the two automata) [55]. Thanks to Lemma 25, we can therefore decide whether B is progress-consistent in time O(|Q| 2 · (|C| 2 · |Q| 2 )) = O(|C| 2 · |Q| 4 ): for all |Q| 2 pairs of states q, q ′ ∈ Q, if q ≺ q ′ , we test the emptiness of the intersection of these two regular languages.

Necessity of the conditions
We prove that each of the three conditions introduced in Section 3.1 is necessary for halfpositionality over finite one-player arenas of DBA-recognizable objectives. Each condition gets its own devoted subsection. For the first two conditions (having a total prefix preorder and progress-consistency), we also show for completeness that they are necessary for halfpositionality of general objectives over countably infinite one-player arenas, and necessary for half-positionality of ω-regular objectives over finite one-player arenas. This distinction is worthwhile, as there exist half-positional objectives that are not ω-regular (see, e.g., finitary Büchi objectives [21]).

Total prefix preorder
Having a total prefix preorder is necessary in general for half-positionality over countably infinite arenas, and even over finite arenas for ω-regular objectives.
▶ Proposition 26. Let W ⊆ C ω be an objective. If W is half-positional over countably infinite one-player arenas, then its prefix preorder is total. If W is ω-regular and half-positional over finite one-player arenas, then its prefix preorder is total.
Proof. By contrapositive, we assume that the prefix preorder of W is not total. Then, there exist two finite words w 1 , w 2 ∈ C * such that w 1 ̸ ⪯ w 2 and w 2 ̸ ⪯ w 1 . We can find two infinite Using these four words, we build a countably infinite one-player arena depicted in Figure 7 (left) for which P 1 has no positional optimal strategy. Indeed, if the game started with w 1 , P 1 needs to reply with w ′ 1 in v 3 to win, but if the game started with w 2 , P 1 needs to reply with w ′ 2 in v 3 to win.
Moreover, if W is ω-regular, so are w −1 1 W and w −1 2 W . By Lemma 5, we can therefore assume w.l.o.g. that w ′ 1 = x 1 (y 1 ) ω and w ′ 2 = x 2 (y 2 ) ω are ultimately periodic, and we can carry out similar arguments with the finite arena depicted in Figure 7 (right).

Progress-consistency
Progress-consistency is necessary in general for half-positionality in countably infinite arenas, and even in finite arenas for ω-regular objectives.
▶ Proposition 27. Let W ⊆ C ω be an objective. If W is half-positional over countably infinite one-player arenas, then it is progress-consistent. If W is ω-regular and half-positional even over finite one-player arenas, then it is progress-consistent.
Proof. By contrapositive, we assume that W is not progress-consistent. Then there exist w 1 ∈ C * and w 2 ∈ C + such that w 1 ≺ w 1 w 2 , but w 1 (w 2 ) ω / ∈ W . As w 1 ≺ w 1 w 2 , there exists an infinite continuation w ′ ∈ C ω such that w 1 w ′ / ∈ W and w 1 w 2 w ′ ∈ W . Using these three words, we build a countably infinite one-player arena depicted in Figure 8 (left). In this arena, from vertex v 1 , a positional strategy can only achieve words w 1 (w 2 ) ω or w 1 w ′ , which are both losing. However, there is a (non-positional) winning strategy achieving word w 1 w 2 w ′ .
If W is additionally ω-regular, using Lemma 5, we can assume w.l.o.g. that w ′ = xy ω is ultimately periodic, and we can carry out similar arguments with the finite arena depicted in Figure 8

Recognizability by the prefix-classifier
We now prove that for a DBA-recognizable objective, being recognized by a Büchi automaton built on top of its prefix-classifier S ∼ is necessary for half-positionality. The rest of Section 4.3 is devoted to the proof of this result, which is more involved than the proofs in Sections 4.1 and 4.2. We fix an objective W ⊆ C ω recognized by a DBA B = (Q, C, q init , ∆, α). We make the assumption that W is half-positional over finite one-player arenas. Our goal is to show that W can be defined by a Büchi automaton built on top of S ∼ . We assume w.l.o.g. that B is saturated. Many upcoming arguments heavily rely on this assumption through the use of Lemma 10 (any safe word can be completed into a safe cycle).
Our proof first assumes in Section 4.3.1 that B recognizes a prefix-independent objective. We will then use build on this first case to conclude for the general case in Section 4.3.2. We provide a proof sketch at the start of each subsection.

Prefix-independent case
We assume that the objective W recognized by B is prefix-independent, so all the states of B are equivalent for ∼. We want to show that W can be recognized by a Büchi automaton built on top of S ∼ , and in this case, the automaton structure S ∼ has just one state. Therefore, we want to find F ⊆ C such that W = Büchi(F ). We start with a high level description of the proof technique.
Proof sketch. The goal is to find a suitable definition for F . To do so, we exhibit a state q max of B that is "the most rejecting state of the automaton": it verifies that the set of safe words from q max contains the safe words from all the other states (q max is then called a safe-maximum) and that the set of safe cycles on q max contains the safe cycles on all the other states (it is also a safe-cycle-maximum). We define F as the set of colors c such that (q max , c) ∈ α.
We first show that if a safe-maximum exists, we can assume w.l.o.g. that it is unique (Lemma 29). In Lemmas 31, 32 and 33, we show the existence of a safe-cycle-maximum. This part of the proof relies on the half-positionality over finite one-player arenas of W . Finally, defining F using q max as explained above, we prove that W = Büchi(F ) (Lemma 34). ◀ We call a state q max ∈ Q of B a safe-maximum (resp. a safe-cycle-maximum) if for all q ∈ Q, we have Safe B (q) ⊆ Safe B (q max ) (resp. SafeCycles B (q) ⊆ SafeCycles B (q max )). We remark that if B is saturated, a safe-cycle-maximum is also a safe-maximum (this can be shown using Lemma 10).
We first show that we can remove states from B, while still recognizing the same objective, until it has at most one safe-maximum.
▶ Lemma 29. There exists a DBA B ′ recognizing W with at most one safe-maximum.
Proof. Assume that q 1 max , q 2 max ∈ Q are distinct safe-maxima. In particular, Safe B (q 1 max ) = Safe B (q 2 max ). We show that in such a situation, the objective recognized by B can be recognized by an automaton with one less state, in which we discard one of the two safe-maxima. To simplify the upcoming arguments, we assume that q init = q 1 max (which is without loss of generality as all states of B are equivalent for ∼).
We also assume that states that are not reachable from q ′ init in B ′ are removed from Q ′ . We show that this automaton with (at least) one less state recognizes the same objective as B. Let w = c 1 c 2 . . . ∈ C ω be an infinite word. We show that w is accepted by B if and only if it is accepted by B ′ . Let } be the transitions of B ′ that were directed to q 2 max in B but are now redirected to q 1 max in B ′ . Let ϱ be the run of B on w, and ϱ ′ = (q ′ 0 , c 1 , q ′ 1 )(q ′ 1 , c 2 , q ′ 2 ) . . . be the run of B ′ on w. The two runs start by taking corresponding transitions, but differ once a transition in ∆ ̸ →q2 is taken.
We first assume that ϱ ′ uses transitions in ∆ ̸ →q2 only finitely many times. Then, there exists k ∈ N such that q ′ k = q 1 max and for all l ≥ k, (q ′ l , c l+1 , q ′ l+1 ) / ∈ ∆ ̸ →q2 . Let w >k = c k+1 c k+2 . . . be the infinite word consisting of the colors taken after the last occurrence of a transition in ∆ ̸ →q2 . We have that B accepts w ⇐⇒ B accepts w >k as B recognizes a prefix-independent objective ⇐⇒ B ′ accepts w >k as w >k visits exactly the same transitions as in B ⇐⇒ B ′ accepts w as c 1 . . . c k is a cycle on the initial state q 1 max of B ′ .
We now assume that ϱ ′ uses transitions in ∆ ̸ →q2 infinitely many times. We decompose ϱ ′ into infinitely many finite runs ϱ ′ 1 , ϱ ′ 2 , . . . such that ϱ ′ = ϱ ′ 1 ϱ ′ 2 . . . and every run ϱ ′ i sees exactly one transition in ∆ ̸ →q2 as its last transition. This implies that all these finite runs start in state q 1 max . We represent run ϱ ′ in Figure 9. We define words w 1 , w 2 , . . . as the respective projection of runs ϱ ′ 1 , ϱ ′ 2 , . . . to their colors (we have w = w 1 w 2 . . .). Notice that as the transitions used by w i from q 1 max in B ′ correspond to the transitions used by w i from q 1 max in B (this property is not true for all words, but this is true for these words that read only one transition in ∆ ̸ →q2 as their last transition). We also have by construction that We distinguish whether w is accepted or rejected by B ′ . Assume w is accepted by B ′ . Then, we know that for infinitely many i ∈ N, w i / ∈ Safe B ′ (q 1 max ). This implies that for these indices i, w i / ∈ Safe B (q 1 max ) by Equation (1). As q 1 max is a safe-maximum, for all q ∈ Q, w i / ∈ Safe B (q) (this is simply the contrapositive of the definition of safe-maximum). Hence, for infinitely many i ∈ N, when w i is read in B (no matter from where), a Büchi transition is seen, so w is accepted by B.
Assume w is rejected by B ′ . Then there exists k ∈ N such that for all l ≥ k, w l ∈ Safe B ′ (q 1 max ). As B is prefix-independent, up to removing the start of w, we assume w.l.o.g. that k = 1. We show by induction that This is true for i = 1, as δ * (q 1 max , w 1 ) = q 2 max by Equation (2) and the fact that q 1 max and q 2 max are both safe-maxima. Assume Safe B (q 1 max ) = Safe B (δ * (q 1 max , w 1 . . . w i−1 )) for some i ≥ 2. Then, by Lemma 8, as w i ∈ Safe B (q 1 max ), we have Safe B (δ * (q 1 max , w i )) = Safe B (δ * (q 1 max , w 1 . . . w i−1 w i )). By Equation (2), we have Safe B (δ * (q 1 max , w i )) = Safe B (q 2 max ), which is itself equal to Safe B (q 1 max ). We now know that for all i ≥ 1, w i ∈ Safe B (q 1 max ) by Equation (1). Therefore, we conclude that for all i ≥ 1, w i ∈ Safe B (δ * (q 1 max , w 1 . . . w i−1 )). In particular, w sees no Büchi transition when read from q 1 max in B and is also rejected by B. We have shown that B ′ is a DBA with fewer states than B recognizing W . If B ′ still has two or more safe-maxima, we repeat our construction until there is at most one left. ◀ w2 w3 Figure 9 Features of run ϱ ′ when it takes infinitely many transitions in ∆ ̸ →q 2 .
Thanks to Lemma 29, we now assume w.l.o.g. that B has at most one safe-maximum. We intend to show that there exists a safe-cycle-maximum. To do so, we exhibit an (infinite) arena in which P 1 has no winning strategy, which we prove by using half-positionality of W over finite one-player arenas. We then prove that the non-existence of a safe-cycle-maximum would imply that P 1 has a winning strategy in this arena (Lemma 33).
Let A B be the infinite one-player arena of P 1 depicted in Figure 10. This arena consists of one vertex v with a choice to make among all non-empty words that are safe cycles from some state of B. Vertex v is the only vertex with multiple outgoing edges. The goal of the next three short lemmas is to show that in this arena, P 1 has no winning strategy. ▶ Lemma 30. If P 1 has a winning strategy in A B , then P 1 has a positional winning strategy.
Proof. Suppose that there is a winning strategy of P 1 in A B . Let w = w 1 w 2 . . . be an infinite winning word such that for i ≥ 1, w i ∈ q∈Q SafeCycles B (q) \ {ε}. Let q 0 = q init , and for i ≥ 1, let q i = δ * (q init , w 1 . . . w i ) be the current automaton state after reading the first i finite words composing w. As there are only finitely many automaton states and w is winning, there are k, l ≥ 1 with k < l such that q k = q l and w k+1 . . . w l / ∈ Safe B (q k ). Word w 1 . . . w k (w k+1 . . . w l ) ω is also a winning word and uses only finitely many different words in Hence, there is a finite restriction ("subarena") A ′ B of the arena A B with at most l choices in v in which P 1 has a winning strategy. Arena A ′ B being finite and one-player, half-positionality of W over finite one-player arenas implies that P 1 has a positional winning strategy in A ′ B . This positional winning strategy can also be played in A B (as every choice available in A ′ B is also available in A B ). ◀ ▶ Lemma 31. No positional strategy of P 1 is winning in A B .
Proof. Any positional strategy of P 1 generates a word w ω , where w ∈ SafeCycles B (q) \ {ε} for some q ∈ Q. In particular, word w ω is rejected when it is read from state q. As all the states in Q are equivalent for ∼ (as we assume that W is prefix-independent), we have q ∼ q init , so w ω is also rejected when read from the initial state q init of the automaton. We use the statement of Lemma 32 to show the existence of a safe-cycle-maximum.
▶ Lemma 33. There exists a safe-maximum q max in B that is a moreover a safe-cyclemaximum: for all q ∈ Q, SafeCycles B (q) ⊆ SafeCycles B (q max ).
Proof. Let us assume by contradiction that there is no safe-maximum, or if there is one, that it is not a safe-cycle-maximum. We show how to build a winning strategy of P 1 in A B , contradicting Lemma 32. To do so, we build an infinite word accepted by B by combining finite words that are safe cycles from some state. We claim that for q ∈ Q which is not a safe-maximum, there exists Let q ′ ∈ Q be such that Safe B (q ′ ) ̸ ⊆ Safe B (q), which exists as q is not a safe-maximum. Let w 1 ∈ Safe B (q ′ ) \ Safe B (q). By Lemma 10, there exists w 2 ∈ C * such that w 1 w 2 ∈ SafeCycles B (q ′ ) (this holds as we have assumed w.l.o.g. that B is saturated). As w 1 / ∈ Safe B (q), we also have w 1 w 2 / ∈ Safe B (q). Taking w q = w 1 w 2 proves the claim. For q ∈ Q not a safemaximum, we fix w q ∈ C + such that w q ∈ q ′ ∈Q SafeCycles B (q ′ ) \ {ε} and w q / ∈ Safe B (q). Let q max ∈ Q be a safe-maximum (that we suppose to be unique by Lemma 29), if it exists. We suppose by contradiction that q max is not a safe-cycle-maximum: there is q ∈ Q such that SafeCycles B (q) ̸ ⊆ SafeCycles B (q max ). Let w max ∈ SafeCycles B (q) \ SafeCycles B (q max ). Notice that as w max ∈ Safe B (q) and q max is a safe-maximum, w max ∈ Safe B (q max ). Therefore, w max cannot be a cycle on q max , i.e., δ * (q max , w max ) ̸ = q max .
We build iteratively an infinite winning word that can be played by P 1 in A B . As P 1 plays, we keep track in parallel of the current automaton state. The game starts in v, with current automaton state q 0 = q init . Let n ≥ 0. We distinguish two cases.
If q n is not a safe-maximum, then P 1 plays word w qn . As w qn / ∈ Safe B (q n ), a Büchi transition is seen along the way. The current automaton state becomes q n+1 = δ * (q n , w qn ). If q n = q max is the safe-maximum, then P 1 plays w max . The current automaton state becomes q n+1 = δ(q max , w max ), which is not equal to q max . For infinitely many i ≥ 0, the automaton state q i is not a safe-maximum (as there is at most one safe-maximum in B, and it cannot appear twice in a row). Therefore, we have described a winning strategy for P 1 , since the corresponding run over B visits infinitely often a Büchi transition. ◀ We now know that there exists a unique safe-maximum q max ∈ Q, and moreover, that it is a safe-cycle-maximum. We show how to use the outgoing transitions of q max in order to realize W as an objective of the kind Büchi(F ) for some F ⊆ C, which is the goal of the current subsection.
Proof. Let F = {c ∈ C | (q max , c) ∈ α} -equivalently, if we consider colors as words with one letter, F is the set of colors c such that c / ∈ Safe B (q max ). We first show that Büchi(F ) ⊆ W . Let c ∈ F . Then, c / ∈ Safe B (q max ). As q max is a safe-maximum, for all q ∈ Q, c / ∈ Safe B (q). Therefore, any word seeing infinitely many colors in F sees infinitely many Büchi transitions and is accepted by B.
We now show that W ⊆ Büchi(F ). By contrapositive, let w = c 1 c 2 . . . / ∈ Büchi(F ) be an infinite word with only finitely many colors in F . We show that w / ∈ W . As W is prefix-independent, we may assume w.l.o.g. that w has no color in F , i.e., that for all i ≥ 1, c i ∈ C \ F . We claim that when read from q max , word w sees no Büchi transition and is thus rejected. This implies that w / ∈ W as q max ∼ q init .
Assume by contradiction that there is some Büchi transition when reading w from q max , i.e., there exists k ≥ 0 such that for w ≤k = c 1 . . . c k , w ≤k ∈ Safe B (q max ), but w ≤k c k+1 / ∈ Safe B (q max ). We will deduce that (q max , c k+1 ) is a Büchi transition, contradicting that c k+1 ∈ C \ F .
We depict the situation in Figure 11. Let q 1 = δ * (q max , w ≤k ) (whether q 1 equals q max or not does not matter). By Lemma 10, there exists w ′ ∈ C * such that w ≤k w ′ ∈ SafeCycles B (q max ). By construction, we have w ′ w ≤k ∈ SafeCycles B (q 1 ). As q max is a safe-cycle-maximum, we have SafeCycles B (q 1 ) ⊆ SafeCycles B (q max ), so we also have w ′ w ≤k ∈ SafeCycles B (q max ). Let q 2 = δ * (q max , w ′ ). Notice that q max = δ * (q 2 , w ≤k ) and w ≤k ∈ Safe B (q 2 ). As q max is a safemaximum and w ≤k c k+1 / ∈ Safe B (q max ), we also have that w ≤k c k+1 / ∈ Safe B (q 2 ). Therefore, transition (q max , c k+1 ) must be a Büchi transition, which contradicts that Figure 11 Situation in the proof of Lemma 34, with w ≤k ∈ SafeB(qmax) but w ≤k c k+1 / ∈ SafeB(qmax).

General case
We now relax the prefix-independence assumption on W . We still assume that W is halfpositional over finite one-player arenas, and show that W can be recognized by a Büchi automaton built on top of S ∼ . If B has exactly one state per equivalence class of ∼, it means that it is built on top of S ∼ , and we are done. If not, let q ∼ ∈ Q be a state such that We briefly sketch the proof technique for this section.
Proof sketch. Our proof will show how to modify B by "merging" all states in equivalence class [q ∼ ] into a single state, while still recognizing the same objective W . The main technical argument is to build a variant W [q∼] of objective W on a new set of colors C [q∼] , that turns out to also be half-positional over finite one-player arenas and DBA-recognizable, but which is prefix-independent. We can therefore use Lemma This set contains all the finite words that, read from q ∼ , come back to a state in [q ∼ ]. By Lemma 7, for all q ∈ [q ∼ ], for all w ∈ C [q∼] , we also have that δ * (q, w) ∼ δ * (q ∼ , w) ∼ q ∼ .
The set C [q∼] therefore corresponds to the set of words with the seemingly stronger property that, when read from any state in [q ∼ ], come back to a state in [q ∼ ]. We define an objective W [q∼] of infinite words on this new set of colors such that We show that W [q∼] has the three conditions allowing us to apply Lemma 34 to it.
Objective W [q∼] is DBA-recognizable: we consider the DBA is prefix-independent, as adding or removing a finite number of cycles on q ∼ does not affect the accepted status of a word in q −1 ∼ W . Half-positionality of W [q∼] over finite one-player arenas is implied by half-positionality of W over finite one-player arenas. Indeed, every (finite one-player) arena A [q∼] using colors in C [q∼] can be transformed into a (finite one-player) arena A with similar properties using colors in C. Two transformations are applied: (i) we replace every C [q∼] -colored edge in A [q∼] by a corresponding finite chain of C-colored edges, and (ii) for every vertex v of A [q∼] , we prefix it with a chain of C-colored edges starting from a vertex v ′ reading a word w q∼ such that δ * (q init , w q∼ ) = q ∼ . We then have that P 1 has a winning strategy from a vertex v in A [q∼] if and only if P 1 has a winning strategy from v ′ in A, and a positional winning strategy from v ′ in A can be transformed into a positional winning strategy from v in A [q∼] . By Lemma 34, there exists a set We now show links between safe words from states of [q ∼ ] and the words in F [q∼] . The arguments once again rely on the saturation of B.
Proof. For the first item, we assume by contrapositive that there exists a state q ∈ [q ∼ ] such that w ∈ Safe B (q). By Lemma 10, there is w ′ ∈ C * such that ww ′ ∈ SafeCycles B (q). In particular, (ww ′ ) ω / ∈ q −1 W = q −1 ∼ W . We can assume w.l.o.g. that w ′ ∈ C + (if w ′ = ε, then we can simply replace it with w ′ = w). Therefore, (ww ′ ) ω is also an infinite word on C [q∼] , and we have ( , which ends the proof of the first item. For the second item, assume by contradiction that for all q ∈ [q ∼ ], there exists w q ∈ C [q∼] \ F [q∼] such that w q / ∈ Safe B (q). As there are only finitely many states in [q ∼ ], it is then possible to build a word w = w q1 . . . w qn such that for all 1 ≤ i < n, δ * (q i , w qi ) = q i+1 , δ * (q n , w qn ) = q 1 , and for all 1 ≤ i ≤ n, w qi ∈ C [q∼] \ F [q∼] and w qi / ∈ Safe B (q i ). Word w ω is accepted from q 1 as it sees infinitely many Büchi transitions, so it is in (q 1 ) −1 W = q −1 ∼ W . However, if we consider w ω as an infinite word on C [q∼] , then it is not in as every letter of the word is in C [q∼] \ F [q∼] . This yields a contradiction. ◀ We use the previous result in a straightforward way to exhibit a state q max whose non-safe words in C [q∼] are exactly the words in F [q∼] . The reader may notice that, echoing the proof of the prefix-independent case (Section 4.3.1), the state q max given by Corollary 36 is actually a safe-maximum among states in [q ∼ ].
▶ Corollary 36. There exists q max ∈ [q ∼ ] such that for all w ∈ C [q∼] , w ∈ F [q∼] if and only if w / ∈ Safe B (q max ).
Proof. By the second item of Lemma 35, we take q max ∈ [q ∼ ] such that, for all w ∈ C [q∼] \F [q∼] , w ∈ Safe B (q max ). Let w ∈ C [q∼] . The property we already have on q max gives us by contrapositive that w / ∈ Safe B (q max ) implies that w ∈ F [q∼] . Reciprocally, the first item of Lemma 35 gives us that if w ∈ F [q∼] , then w / ∈ Safe B (q max ). ◀ From now on, we assume that q max ∈ [q ∼ ] is a state having the property of Corollary 36. We show that W can be recognized by a smaller DBA consisting of DBA B in which all the states in [q ∼ ] have been merged into the single state q max , by redirecting all incoming transitions of [q ∼ ] to q max . We assume w.l.o.g. that if q init ∈ [q ∼ ], then q init = q max (this does not change the objective recognized by B, and will be convenient in the upcoming construction). We consider DBA We also assume that states that are not reachable from q ′ init in B ′ are removed from Q ′ .
▶ Lemma 37. The objective recognized by B ′ is also W .
. . . be the run of B ′ on w. We have that q 0 = q ′ 0 , but states in both runs may not coincide after a state in [q ∼ ] has been reached. Yet, we show inductively that It is true for i = 0 (we even have equality in this case), and if q n ∼ B q ′ n , then by construction of the transitions of B ′ and Lemma 7, we still have q n+1 ∼ B q ′ n+1 . We want to show that w is accepted by B if and only if it is accepted by B ′ . This is clear if w never goes through a state in [q ∼ ] (as the same transitions are then taken in B and B ′ ).
We first assume that run ϱ visits [q ∼ ] finitely many times, and that the last visit to [q ∼ ] happens in q n for some n ≥ 0. Notice that run ϱ ′ also visits [q ∼ ] for the last time in q ′ n by Equation (3). Therefore, word c n+1 c n+2 . . . is accepted from q ′ n in B ′ if and only if it is accepted from q ′ n in B: all the subsequent transitions coincide. As q n ∼ B q ′ n , we have moreover that c n+1 c n+2 . . . is accepted from q ′ n in B ′ if and only if it is accepted from q n in B. This implies that w is accepted by B if and only if it is accepted by B ′ .
We now assume that run ϱ visits [q ∼ ] infinitely many times. We decompose w inductively into a prefix w q∼ reaching [q ∼ ] followed by cycles w 1 , w 2 , . . . on [q ∼ ]. Formally, let w q∼ be any finite prefix of w such that δ * (q init , w q∼ ) ∈ [q ∼ ]. By induction hypothesis, assume w q∼ w 1 . . . w n is a prefix of w such that δ * (q init , w q∼ w 1 . . . w n ) ∈ [q ∼ ]. We define w n+1 ∈ C + as the shortest non-empty word such that w q∼ w 1 . . . w n w n+1 is a prefix of w and δ * (q init , w q∼ w 1 . . . w n w n+1 ) ∈ [q ∼ ]. We have w = w q∼ w 1 w 2 . . . by construction. By Equation (3), we also have that for all n ≥ 0, . . w n ) = q max as this is the only state left in that class in B ′ . If w is accepted by B, then for infinitely many i ≥ 1, word w i is in F [q∼] . By Corollary 36, all these infinitely many words are not in Safe B (q max ) and therefore see a Büchi transition when read from q max , so w is also accepted by B ′ .
If w is rejected by B, then there exists n ≥ 1 such that for all n ′ ≥ n, w n ′ ∈ C [q∼] \ F [q∼] . By Corollary 36, for all n ′ ≥ n, w n ′ is in Safe B (q max ) and therefore does not see a Büchi transition when read from q max , so w is also rejected by B ′ . ◀ We have all the arguments to show our goal for the section (Proposition 28), that is, to show that W can be recognized by a Büchi automaton built on top of S ∼ .
Proof of Proposition 28. We have shown in Lemma 37 how to merge an equivalence class of B into a single state, while still recognizing the same objective. Repeating this construction for each equivalence class with two or more states, we end up with a DBA with exactly one state per equivalence class of ∼ still recognizing W . By definition of S ∼ , this DBA is necessarily built on top of (some automaton isomorphic to) S ∼ . ◀

Sufficiency of the conditions
We show that a DBA B with the three conditions from Section 3.1 (recognizing a progressconsistent objective having a total prefix preorder and being recognizable by a Büchi automaton built on top of S ∼ ) recognizes a half-positional objective. As these three conditions have been shown to be necessary for the half-positionality of objectives recognized by a DBA, this will imply a characterization of half-positionality.
Our main technical tool is to construct, thanks to these three conditions, a family of completely well-monotonic universal graphs. The existence of such objects implies thanks to recent result [50] that P 1 has positional optimal strategies, even in two-player arenas of arbitrary cardinality.

Completely well-monotonic universal graphs
We fix extra terminology about graphs only used in Section 5, and recall the relevant results from [50].
Extra preliminaries on graphs. Let G = (V, E) be a graph and W ⊆ C ω be an objective.
We consider a graph G = (V, E) along with a total order ≤ on its vertex set V . We say there is also an edge with color c from v to all states smaller than v ′ for ≤, and (ii) whenever v ≥ v ′ , then v has at least the same outgoing edges as v ′ . Graph G is well-monotonic if it is monotonic and the total order ≤ is a well-order (i.e., any set of vertices has a minimum). Graph G is completely well-monotonic if it is well-monotonic and there exists a vertex ⊤ ∈ V maximum for ≤ such that for all v ∈ V , ▶ Example 38. We provide an example illustrating these notions with C = {a, b} and W = Büchi({a}). This is a special case of our upcoming construction, and it is already discussed in more depth in [50,Chapter 2]. For θ an ordinal, we define a graph U θ with vertices U θ = θ ∪ {⊤}, and such that for all λ, λ ′ < θ, λ a − → λ ′ , and λ b − → λ ′ if and only if λ ′ < λ.
Moreover, for all v ∈ U θ , we define edges ⊤ a − → v and ⊤ b − → v. We order vertices using the natural order on the ordinals, and with λ < ⊤ for all λ < θ.
Vertex ⊤ does not satisfy W , as there is an infinite path ⊤ . ., and b ω / ∈ W . All other vertices satisfy W by construction: they cannot reach ⊤, and as reading b decreases the current ordinal, a path cannot have an infinite suffix using only color b (there is no infinite decreasing sequence of ordinals).
This graph is (κ, W )-universal for κ < |θ|. Intuitively, for any graph G with at most κ vertices, a W -preserving morphism from G to U θ can be defined by mapping vertices not satisfying W to ⊤, and vertices satisfying W to an ordinal λ that depends on how long it may take to guarantee seeing an a from them.
Graph U θ is monotonic (which can be quickly checked with the definition), and as ≤ is a well-order and there is a vertex ⊤ with the right properties, U θ is even completely well-monotonic. The properties of U θ imply half-positionality of Büchi({a}), thanks to the following theorem. ⌟ We state an important result linking half-positionality and completely well-monotonic universal graphs from [50]. The exact result [50, Theorem 1.1] can actually be instantiated on more precise classes of arenas. However, we use it to prove here half-positionality of a family of objectives over all arenas, so the above result turns out to be sufficient.
▶ Remark 40. Our approach also implies the half-positionality of W over arenas with εedges (cf. Remark 1). Indeed, we can add a fresh color e to C and define an objective W e ⊆ (C ∪ {e}) ω such that for w ∈ (C ∪ {e}) ω , w ∈ W e ⇐⇒ the word obtained by removing e is infinite and belongs to W , or w = w ′ e ω for some w ′ ∈ (C ∪ {e}) * .
Then, the existence of a family of completely well-monotonic (κ, W )-universal graphs for W implies that W e is half-positional [50] (in the terminology of [50], e is a strongly neutral letter). Therefore, we can label the ε-edges of any arena with e and obtain an equivalent game with objective W e . ⌟

Universal graphs for Büchi automata
We show that for a DBA-recognizable objective, the three conditions that were shown to be necessary for half-positionality in Section 4 are actually sufficient.
▶ Proposition 41. Let W ⊆ C ω be an objective that has a total prefix preorder, is progressconsistent, and is recognizable by a Büchi automaton built on top of S ∼ . Then, W is half-positional.
The rest of the section is devoted to the proof of this result, using Theorem 39. Let W ⊆ C ω be an objective with a total prefix preorder, that is progress-consistent, and that is recognized by a DBA B = (Q, C, q init , ∆, α) built on top of S ∼ for the rest of this section. We assume as in previous sections that B is saturated (in particular, by Lemma 10, any safe path can be extended to a safe cycle). An implication of the fact that B is built on top of S ∼ that we will use numerous times in the upcoming arguments is that for q, q ′ ∈ Q, q ∼ q ′ if and only if q = q ′ .
Graph U B,θ is built such that on the one hand, it is sufficiently large and has sufficiently many edges so that there is a morphism from any graph G (of cardinality smaller than some function of |θ|) to U B,θ . On the other hand, for the morphism to be W -preserving, at least some vertices of U B,θ need to satisfy W , which imposes a restriction on the infinite paths from vertices. Graph U B,θ is actually built so that for any automaton state q ∈ Q and ordinal λ < θ, the vertex (q, λ) satisfies q −1 W (see Lemma 46). The intuitive idea is that for a non-Büchi transition (q, c) / ∈ α of the automaton such that δ(q, c) = q ′ , a c-colored edge from a vertex (q, λ) in the graph either (i) reaches a vertex with first component q ′ , in which case the ordinal must decrease on the second component, or (ii) reaches a vertex with first component q ′′ ≺ q ′ , with no restriction on the second component, but therefore with fewer winning continuations. Using progress-consistency and the fact that there is no infinitely decreasing sequence of ordinals, we can show that this implies that no infinite path in U B,θ corresponds to an infinite run in the automaton visiting only non-Büchi transitions.
We state two properties that directly follow from the definition of U B,θ : ▶ Example 42. We consider again the DBA B from Example 16, recognizing the words seeing a infinitely many times, or a twice in a row at some point. We represent the graph U B,θ , with θ = ω in Figure 12. In order to use Theorem 39, we show that the graph U B,θ is completely well-monotonic (Lemma 43) and, for all cardinals κ, is (κ, W )-universal for sufficiently large θ (Proposition 47). ▶ Lemma 43. Graph U B,θ is completely well-monotonic.
Proof. The order ≤ on the vertices is a well-order, and there exists a vertex ⊤ ∈ U B,θ maximum for ≤ such that for all v ∈ U B,θ , c ∈ C, ⊤ c − → v. To show that U B,θ is completely well-monotonic, it now suffices to show that U B,θ is monotonic.
It is left to discuss the case δ(q, c) = q ′′ and (q, c) / ∈ α. By the above inequalities, this implies that we also have δ(q ′ , c) = q ′′ .
If q = q ′ , then λ ≥ λ ′ . Moreover, as (q ′ , c) = (q, c) / ∈ α, the existence of edge (q ′ , λ ′ ) We show that q ′ ≺ q is not possible -we assume it holds and draw a contradiction. As (q, c) / ∈ α, we have c ∈ Safe B (q). By Lemma 10, there is w ∈ C * such that cw ∈ SafeCycles B (q). As δ(q, c) = δ(q ′ , c) and δ * (q, cw) = q, we have δ * (q ′ , cw) = q. As q ′ ≺ q, by progress-consistency, the word (cw) ω must be accepted by B when read from q ′ . It must therefore also be accepted when read from q (as q ′ ≺ q), which contradicts that cw ∈ SafeCycles B (q). ◀ We now intend to show (κ, W )-universality of some U B,θ for all cardinals κ. Lemmas 44 and 45 give insight on properties of the paths of U B,θ , to then establish which vertices of U B,θ satisfy W (Lemma 46). Understanding which vertices satisfy W is useful to later define a W -preserving morphism into U B,θ . We first show that paths in this graph "underapproximate" corresponding runs in the automaton: a finite path γ = (q 0 , λ 0 ) c1 − → . . . cn −→ (q n , λ n ) in U B,θ visits vertices labeled by automaton states at most as large (for ⪯) as the corresponding states visited by the run from q 0 on c 1 . . . c n .
Proof. Let ϱ = B(q 0 , w) = (q ′ 0 , c 1 , q ′ 1 ) . . . (q ′ n−1 , c n , q ′ n ) be the finite run of B obtained by reading w from q 0 . States q 0 , . . . , q n correspond to the first component of the vertices visited by γ in U B,θ , whereas q ′ 0 , . . . , q ′ n are the states visited by the finite word w in B. We have that q 0 = q ′ 0 , but the subsequent states may or may not correspond. By Lemma 44, we still know that for all i, 0 ≤ i ≤ n, q i ⪯ q ′ i . In particular, q 0 = q n ⪯ δ * (q 0 , w) = q ′ n . We distinguish three cases, depending on whether q 0 ≺ δ * (q 0 , w) or q 0 ∼ δ * (q 0 , w) (which implies q 0 = δ * (q 0 , w) as B is built on top of its prefix-classifier), and depending on whether q i = q ′ i for all 0 ≤ i ≤ n or not.
If q 0 ≺ δ * (q 0 , w), by progress-consistency, w ω ∈ q −1 0 W . If q 0 = δ * (q 0 , w) and for all i, 0 ≤ i ≤ n, q i = q ′ i (i.e., γ only uses edges that directly correspond to transitions of the automaton B), then for the ordinal to be greater than or equal to its starting value, some Büchi transition has to be taken since non-Büchi transitions strictly decrease the ordinal on the second component. Hence, w is not a safe cycle from q 0 , so w ω ∈ q −1 0 W . If q 0 = δ * (q 0 , w) and for some index i, 1 ≤ i < n, we have q i ≺ q ′ i (i.e., γ takes at least one edge that does not correspond to a transition of the automaton). We represent the situation in Figure 13. We know that δ * (q ′ i , c i+1 . . . c n ) = q ′ n ∼ q 0 . By Lemma 14, as q i ≺ q ′ i , this implies that δ * (q i , c i+1 . . . c n ) ⪯ q 0 . Also, in the graph, there is a path from q i to q 0 with colors c i+1 . . . c n . Therefore, by Lemma 44,q  We show that every vertex (q 0 , λ 0 ) such that q 0 ⪯ q init satisfies W .
. . be an infinite path of U B,θ from (q 0 , λ 0 ) and w = c 1 c 2 . . . be the sequence of colors along its edges. We show that w ∈ q −1 0 W . Let q ′ i = δ * (q 0 , c 1 . . . c i ) be the state of B reached after reading the first i colors of w. We claim that there are two states q, q ′ ∈ Q occurring infinitely often in the sequences (q i ) i≥0 , (q ′ i ) i≥0 , respectively, and an increasing sequence of indices (i k ) k≥1 verifying that for all k ≥ 1, q i k = q, q ′ i k = q ′ , and λ i k ≤ λ i k+1 .
Indeed, we can first choose a state q appearing infinitely often in (q i ) i≥0 and pick a sequence (i j ) j≥1 such that q ij = q and (λ ij ) j≥1 is not decreasing (this is possible since there is no infinite decreasing sequence of ordinals). Then, we can just pick q ′ appearing infinitely often in (q ′ ij ) j≥1 and extract the subsequence corresponding to its occurrences. Let q, q ′ ∈ Q, (i k ) k≥1 verifying the above properties. Let w 0 = c 1 . . . c i1 and for k ≥ 1, let w k = c i k +1 . . . c i k+1 ∈ C + be the colors over the edges in π from (q i k , λ i k ) to (q i k+1 , λ i k+1 ).
By Lemma 44, it is verified that q ⪯ q ′ = δ * (q 0 , w 0 ). Moreover, since λ i k ≤ λ i k+1 , by Lemma 45 we have that w ω k ∈ q −1 W ⊆ (q ′ ) −1 W for all k ≥ 1. We conclude that for all k ≥ 1, the word w k labels a cycle over q ′ in B visiting some Büchi transition, and therefore w = w 0 w 1 w 2 . . . ∈ q −1 0 W . ◀ We now have all the tools to show, for all cardinals κ, (κ, W )-universality of U B,θ for sufficiently large θ.
Proof. Let G = (V, E) be a graph such that |V | ≤ κ. For v ∈ V , let q v ∈ Q ∪ {⊤} be the smallest automaton state (for ⪯) such that v satisfies q −1 v W , or ⊤ if it satisfies none of them. We remark that q v ⪯ q if and only if v satisfies q −1 W . To show that there is a W -preserving morphism from G to U B,θ , we follow the six steps outlined in [50,Lemma 2.4].
(i) In this first step, we classify and order vertices of G in an inductive way, which will later be used to map them to vertices of U B,θ . For q ∈ Q and λ an ordinal, we define by transfinite induction Intuitively, for v to be in V q λ , it has to satisfy q −1 W and to guarantee that a Büchi transition is seen "soon" when colors of paths from v are read from q in B (how soon depends on the value of λ). We remark that for each state q ∈ Q, the sequence (V q λ ) λ is non-decreasing: for λ ≤ λ ′ , V q λ ⊆ V q λ ′ . We illustrate this induction on a concrete case in Example 48. The subsequent steps mostly follow from this definition. (ii) Let V q = λ V q λ . We show that if v satisfies q −1 W , then it is in V q . Assume that v / ∈ V q . If q ≺ q v , then we immediately have that v does not satisfy q −1 W . If q v ⪯ q, then v has an outgoing edge v c − → v ′ such that (q, c) / ∈ α and v ′ / ∈ λ V δ(q,c) λ . By induction, we build an infinite path from v whose projection in B only sees non-Büchi transitions, so v does not satisfy q −1 W . (iii) In this step and the next one, we show that there is no use in considering ordinals beyond θ in our construction. We first show that if for all q ∈ Q, V q λ = V q λ+1 , then for all q ∈ Q and all λ ′ ≥ λ, V q λ = V q λ ′ . For λ ≤ λ ′ , we always have V q λ ⊆ V q λ ′ . For the other inclusion, we assume by transfinite induction that V q ′ λ = V q ′ η for all q ′ ∈ Q and for all λ ≤ η < λ ′ . Let v ∈ V q λ ′ . Every edge v c − → v ′ either satisfies (q, c) ∈ α, or there exists η < λ ′ such that v ′ ∈ V δ(q,c) η . Since V δ(q,c) η ⊆ V δ(q,c) λ by induction hypothesis, we have v ′ ∈ V δ(q,c) λ . Hence, v ∈ V q λ+1 = V q λ . (iv) We prove that there exists λ < θ such that for all q ∈ Q, V q λ = V q λ+1 . If not, using the axiom of choice, we can build a map ψ : θ → Q × V such that for λ < θ, ψ(λ) = (q, v) for some q ∈ Q and v ∈ V q λ+1 \ V q λ . This map is injective, as any pair (q, v) can be chosen at most once (as (V q λ ) λ is non-decreasing). This implies that |θ| = |Q| · |θ ′ | ≤ |Q| · |V |, a contradiction since |V | < |θ ′ |. Using additionally Item (iii), we deduce that there exists λ < θ such that for all λ ′ ≥ λ, V q λ = V q λ ′ . (v) Let ϕ : V → U B,θ be such that By Item (ii), for all v ∈ V , there exists λ such that v ∈ V qv λ , so {λ | v ∈ V qv λ } is non-empty. By Item (iv), we have that if ϕ(v) = (q, λ), then λ < θ, so the image of ϕ is indeed in U B,θ . We show that ϕ is W -preserving: if v satisfies W , then q v ⪯ q init , so by Lemma 46, ϕ(v) also satisfies W . (vi) We show that ϕ is a graph morphism. Let v c − → v ′ be an edge of G -we need to show that ϕ(v) c − → ϕ(v ′ ) is an edge of U B,θ . If ϕ(v) = ⊤, this is clear as there are all possible outgoing edges from ⊤. If not, we denote ϕ(v) = (q, λ) and ϕ(v ′ ) = (q ′ , λ ′ ). We have that v satisfies q −1 W . Thus, v ′ must satisfy δ(q, c) −1 W . This implies that q ′ ⪯ δ(q, c). We distinguish two cases.

Conclusion and future work
We have provided a characterization of half-positionality for DBA-recognizable objectives.
While half-positionality of ω-regular objectives is still not completely understood, this is a novel step in this direction.
Another interesting extension is to characterize the memory requirements of DBArecognizable objectives. An intermediate, and already seemingly difficult step would be to characterize the memory requirements of objectives recognized by deterministic weak automata [61,58,45], generalizing the characterization for safety specifications [25].