# Alternating Tree Automata with Qualitative Semantics

We study alternating automata with qualitative semantics over infinite binary trees: alternation means that two opposing players construct a decoration of the input tree called a run, and the qualitative semantics says that a run of the automaton is accepting if almost all branches of the run are accepting. In this paper we prove a positive and a negative result for the emptiness problem of alternating automata with qualitative semantics. The positive result is the decidability of the emptiness problem for the case of Büchi acceptance condition. An interesting aspect of our approach is that we do not extend the classical solution for solving the emptiness problem of alternating automata, which first constructs an equivalent non-deterministic automaton. Instead, we directly construct an emptiness game making use of imperfect information. The negative result is the undecidability of the emptiness problem for the case of co-Büchi acceptance condition. This result has two direct consequences: the undecidability of monadic second-order logic extended with the qualitative path-measure quantifier, and the undecidability of the emptiness problem for alternating tree automata with non-zero semantics, a recently introduced probabilistic model of alternating tree automata.

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## 1. Introduction

The study of tree-automata models can be organised by distinguishing three semantic features.

The first feature is the operational mode: deterministic, non-deterministic, universal, probabilistic, and alternating, are the most studied notions. Intuitively, in each case, an automaton reading an input tree (with labels on the nodes) constructs a decoration of this tree called a run, which is itself a tree. The run labels nodes of the tree by states respecting the local constraints imposed by the transition relation of the automaton. In the deterministic case, a state and a letter uniquely determine the labels at the level below in the run, hence there is a unique run. In the non-deterministic, universal, and alternating case, there may be several valid transitions at each node, yielding possibly several runs on a single tree. In the non-deterministic case we say that the tree is accepted if there exists an accepting run, i.e. the choices are existential. In the universal case, we say that the tree is accepted if all runs are accepting, i.e. the choices are universal. The alternating case unifies both previous cases by introducing existential and universal transitions.

The second feature is the branching semantics. The classical one says that a run is accepting if all its branches satisfy a given acceptance condition. We are concerned in this paper with the qualitative semantics, which is an alternative branching semantics introduced by Carayol, Haddad, and Serre (Carayol et al., 2014). The qualitative semantics says that a run is accepting if almost all its branches satisfy a given acceptance condition, in other words if by picking a branch uniformly at random it almost-surely satisfies the condition. The paper (Carayol et al., 2014) showed that non-deterministic and probabilistic tree automata with qualitative semantics are both robust computational models with appealing algorithmic properties.

The third feature is the acceptance condition (on branches), with -regular conditions such as Büchi and parity conditions being the most important for their tight connections to logical formalisms; see, e.g., (Thomas, 1997).

One motivation for studying tree automata with qualitative semantics is to extend the deep connections between automata and monadic second-order logic (MSO) which hold for the classical semantics (Rabin, 1969). Indeed, the general goal is to construct decidable extensions of MSO over infinite trees; we review some of the efforts and results obtained in this direction. A (unary) generalised quantifier is of the form “the set of all sets that satisfy has the property ”, where is a property of sets. For instance, the ordinary existential quantifier corresponds to the property of being a non-empty set. More interestingly, the quantifier “there exist infinitely many such that ” corresponds to the property of being infinite. It turns out that certain cardinality quantifiers such as “there exist infinitely many ” and “there exist continuum many ” do not add expressive power to MSO over the infinite binary tree (in fact, they can be effectively eliminated) (Bárány et al., 2010). On the other hand, adding the generalised quantifier “the set of all sets satisfying has Lebesgue-measure one” results in an undecidable theory (Mio et al., 2018). A weaker version of this quantifier is “the set of paths of the tree that satisfy has Lebesgue-measure one”. Intuitively, this quantifier, written , means that a random path almost-surely satisfies , where a random path is generated by repeatedly flipping a coin to decide whether to go left or right. It was proved in (Bojańczyk, 2016; Bojańczyk et al., 2017) that adding the quantifier to a restriction of MSO called “thin MSO” yields a decidable logic, but the decidability of MSO+ was left open in (Michalewski and Mio, 2016; Mio et al., 2018). The emptiness problem for non-deterministic parity tree automata with qualitative semantics can easily be expressed using MSO+, as already observed in (Mio et al., 2018), and this is also the case for universal tree automata with qualitative semantics.

In this paper, we initiate the study of alternating automata with qualitative semantics, and focus on the emptiness problem. We present a positive result and a negative result that delimit a clear and sharp decidability frontier.

### Contributions

The positive result is the decidability of the emptiness problem for the case of the Büchi acceptance condition (Theorem 3.7).

The usual roadmap for solving the emptiness problem for alternating automata is to first construct an equivalent non-deterministic automaton, and then to construct an emptiness game for the non-deterministic automaton, i.e., a game such that the first player wins if and only if the automaton is non-empty. This first step is an effective construction of an equivalent non-deterministic automaton, which in some cases is not possible, unknown, or computationally too expensive. In the case at hand the second situation arises: we do not know whether alternating automata with qualitative semantics can effectively be turned into equivalent non-deterministic ones. We remark that our undecidability result shows that there is no such effective construction for co-Büchi conditions (but there might be one for the Büchi conditions).

Here, instead, we develop a new approach which directly constructs an emptiness game for the alternating automaton. The emptiness game we construct uses imperfect information. Our construction extends the notion of blindfold games of Reif (Reif, 1979), used to check universality of non-deterministic automata over finite words. The key ingredient to proving the correctness of our imperfect information emptiness game is a new positionality result for stochastic Büchi games on certain infinite arenas (that we call chronological). To the best of our knowledge, very few positionality results are known in the literature that combine both stochastic features and infinite arenas; a notable exception is (Kučera, 2011).

The negative result is the undecidability of the emptiness problem for the case of the co-Büchi acceptance condition. In fact, our main technical contribution (Theorem 4.2) is to establish the undecidability already for universal automata (a special subclass of alternating automata).

We establish this by a chain of reductions that consider various classes of automata (both on infinite words and trees). We initially resort to the known undecidability of the value 1 problem for probabilistic automata on finite words (Gimbert and Oualhadj, 2010) to deduce the undecidability of the emptiness problem for simple probabilistic co-Büchi automata on infinite words (Proposition 4.1). Here, simple

means that the transitions of the automaton only involve probabilities in

. Then, we reduce the latter problem to the original emptiness problem for universal co-Büchi tree automata with qualitative semantics, hence proving our negative result. The correctness of this last reduction relies on particular properties of another class of automata, namely, probabilistic tree automata.

Our negative result has two interesting consequences: the undecidability of MSO+, and of the emptiness problem for alternating tree automata with non-zero semantics, a model combining sure, almost-sure and positive semantics and studied in (Fournier and Gimbert, 2018).

### Related Work

The study of automata with qualitative semantics was initiated in (Carayol et al., 2014) with several decidability results. The first result is a polynomial-time algorithm entailing the decidability of the emptiness problem for non-deterministic parity tree automata with qualitative semantics (Carayol et al., 2014)

, obtained through a polynomial reduction to the almost-sure problem for Markov decision processes (for which a polynomial-time algorithm is known from

(Courcoubetis and Yannakakis, 1990)). This reduction extends to probabilistic tree automata with qualitative semantics, showing an equivalence with partial-observation Markov decision processes. It is then used to prove the decidability of the emptiness problem for probabilistic Büchi tree automata with qualitative semantics (Carayol et al., 2014).

Alternation was later considered by Fijalkow, Pinchinat and Serre in (Fijalkow et al., 2013a) where the focus was on designing a novel emptiness checking procedure working directly on alternating automata, i.e. directly building an emptiness imperfect-information game without making use of the intermediate transformation to a non-deterministic automaton: this was successfully applied to classical alternating parity tree automata as well as to alternating Büchi tree automata with qualitative semantics (see Theorem 3.7).

This line of work was pursued using the related model of non-zero automata. The first decidability result was obtained for the subclass of zero automata (Bojańczyk et al., 2017), yielding the decidability of the thin restriction of MSO+. A second decidability result concerned the class of alternating zero automata (Fournier and Gimbert, 2018), restricting the abilities of the second player. This latter result is applied to solve the satisfiability problem of a probabilistic extension of . The general case of non-zero automata was left open. We close it negatively (thanks to our negative result) since alternating tree automata with non-zero semantics subsume universal tree automata with qualitative semantics.

A side result in (Fijalkow et al., 2013a) states the undecidability of alternating co-Büchi automata with qualitative semantics. The proof, not given in the conference proceedings, is rather sketchy in the full version (Fijalkow et al., 2013b). The proof we give here (Theorem 4.2) follows the same lines but clarifies a technical loophole in the original proof. Indeed, the last reduction requires the undecidability of the emptiness problem for probabilistic co-Büchi simple automata over infinite words, where simple means that the transitions probabilities are either , , or . The undecidability result was known only for general automata, while we refine it for the simple ones, thus filling in the gap of the undecidability proof in the full version (Fijalkow et al., 2013b).

More recently, Berthon et al. (Berthon et al., 2019) proved the slightly weaker undecidability result that emptiness is undecidable for universal parity tree automata with qualitative semantics. Although their proof follows the same lines as (Fijalkow et al., 2013b), the result is weaker because they need a stronger acceptance condition to obtain simple automata and prove the correctness of the original reduction. Still, their result is strong enough to entail the undecidability of MSO+, the main contribution of their work.

There is another proof of the undecidability of MSO+, obtained independently and at the same time as (Berthon et al., 2019) by Bojańczyk, Kelmendi, and Skrzypczak (Bojańczyk et al., 2019). Their proof technique is very different from ours: they obtain undecidability by a direct encoding of two-counter machines into the logic. However, the core technical part of the paper is not the reduction from counter machines (which is nevertheless tricky), but a crucial technical lemma used to encode runs of counter machines and to prove the correctness of the reduction111More precisely, the lemma states that for a set of pairwise disjoint finite paths in the infinite binary tree called intervals, there is an MSO+ formula that, when true at the root of the infinite binary tree, is equivalent to having with probability , a branch and some integer such that with finitely many exceptions, if an interval intersects then it is of length .

. The proof of this lemma is involved: it mostly relies on tools (such as asymptotic behaviours of vector sequences) previously used to show undecidability of MSO+U logic over infinite words. We remark that MSO+

is known as MSO+ in (Bojańczyk et al., 2019).

### Organisation of the Paper

Section 2

presents the different classes of automata used for our main undecidability result, relying on Markov chains as a unifying notion to define acceptance by these different automata. Section

3 gives our decidability result for alternating Büchi tree automata. Section 4 is about our undecidability result for universal co-Büchi tree automata, while Section 5 presents its consequences for MSO+ (Section 5.1) and for alternating automata with non-zero semantics (Section 5.2).

## 2. Preliminaries

Throughout the paper we implicitly fix a finite alphabet . We denote by the set of finite words over and by the set of infinite words over . We let denote the empty word, and for a word , denotes its length. Finally, we write for the set of words over of length .

The infinite binary tree is , element of are called its nodes, and elements of are called its (infinite) branches. For a finite alphabet , a -tree is a function and we write for the set of -trees. For a branch we denote by the infinite word read in along the branch .

A distribution over a set is a function such that . Any distribution considered in the paper is implicitly assumed to have a finite support, i.e. is finite. For , we write for the distribution that assigns probability to and to . For example, is the distribution such that , unless in which case , and for any other element . The set of distributions over is denoted .

A Markov chain is given by a possibly infinite set of states , an initial state , and a probability transition function . An (infinite) path in is an infinite sequence of states, i.e. an element in . A cone is a set of paths of the form for some . Now, consider the -algebra over paths in built from the set of cones. Then, a classical way to equip this -algebra with a probability measure is to recursively define it on the set of cones as follows:

 P(s0s1⋯sk⋅Sω)=⎧⎪⎨⎪⎩1if k=0 and s0=sin0if k=0 and s0≠sinP(s0⋯sk−1⋅Sω)⋅T(sk−1)(sk)if k>0

and then to extend it (uniquely) to the -algebra thanks to Carathéodory’s extension theorem (we refer the reader to (Puterman, 1994) for more details on this classical construction).

When needed, for a given length , we also see as a probability measure on paths of length (i.e. elements in ) by defining the probability measure of as the probability of the cone .

### 2.1. Two-Player Perfect-Information Stochastic Games

A graph is a pair where is a (possibly infinite) set of vertices and is a set of edges. For every vertex , let , and say that is a dead-end if . In the rest of the paper, we only consider graphs of finite out-degree, i.e. such that is finite for every vertex , and without dead-ends.

A (turn-based) stochastic arena is a tuple where is a graph, is a partition of the vertices among two players, Éloïse and Abélard, and an extra player Random, is a map such that for all the support of is included in , and is an initial vertex. In a vertex (resp. ) Éloïse (resp. Abélard) chooses a successor vertex from , and in a random vertex

, a successor vertex is chosen according to the probability distribution

. A play is an infinite sequence of vertices starting from the initial vertex, i.e. , and such that, for every , if and if . A history is a finite prefix of a play.

A (pure222We only consider pure strategies, as these are sufficient for our purpose. However, our main results on positionality (Theorems 3.1 and 3.2) remain true for randomised strategies as later discussed in Remark 3.4.) strategy for Éloïse is a function such that for every history one has . Strategies of Abélard are defined likewise, and usually denoted .

A play is consistent with a pair of strategies for Éloïse and Abélard if the players always choose their move according to their strategy. Formally, for all the following should hold: if is controlled by Éloïse then and if it is controlled by Abélard then . The set of plays consistent with is denoted , and a history is consistent with if it is the finite prefix of some play in .

In order to equip the set with a probability measure, we define the following Markov chain : its set of states is the set of histories consistent with , its initial state is , and its probability transition function is defined (according to our notation for distributions) by

 T(λ⋅v)=⎧⎪⎨⎪⎩λ⋅v⋅σE(λ⋅v)if v∈VEλ⋅v⋅σA(λ⋅v)if v∈VA∑v′∈E(v)δ(v)(v′) λ⋅v⋅v′if v∈VR

Then, the set of those plays consistent with is in bijection with the set of infinite paths in the Markov chain . Hence, the associated probability measure can be used as a probability measure for measurable subsets of .
When is understood, we omit it and simply write and .

A winning condition is a subset333Formally one needs to require that is measurable for the probability measure , which is always trivially true in this paper. and a (two-player perfect-information) stochastic game is a pair .

A strategy for Éloïse is surely winning if for every strategy of Abélard; it is almost-surely winning if for every strategy of Abélard. Similar notions for Abélard are defined dually. Éloïse surely (resp. almost-surely) wins if she has a surely (resp. almost-surely) winning strategy.

A reachability game is a stochastic game whose winning condition is of the form for some subset , i.e. winning plays are those that eventually visit a vertex in . A Büchi game is a stochastic game whose winning condition is of the form for some subset , i.e. winning plays are those that infinitely often visit a vertex in . Finally, a co-Büchi game is stochastic game whose winning condition is of the form for some subset , i.e. winning plays are those that finitely often visit a vertex in . When it is clear from the context, we write (i.e. write instead of ) for the reachability (resp. Büchi, co-Büchi) game relying on .

A positional strategy is a strategy that does not require any memory, i.e. such that for any two histories of the form and , one has . Positional strategies only depend on the current vertex, and for convenience they are written as functions from into .

A game is deterministic whenever . It is well-known (see e.g. (Zielonka, 1998)) that positional strategies suffice to surely win in deterministic games with a parity winning condition, that we do not define but that captures the reachability, Büchi and co-Büchi winning conditions that we are interested in.

###### Theorem 2.1 (Positional determinacy  (Zielonka, 1998)).

Let be a deterministic parity game. Then, either Éloïse or Abélard has a positional surely winning strategy.

For stochastic games, the following result is well-known (see e.g. (Gimbert and Zielonka, 2007) for a slightly more general result).

###### Theorem 2.2 ().

Let be a stochastic parity game played on a finite arena. If Éloïse almost-surely wins then she has an a positional almost-surely winning strategy.

Note that dropping the assumption that the arena is finite substantially changes the situation. Indeed, for infinite arenas, even with a reachability condition and assuming finite out-degree, almost-surely winning strategies for Éloïse may require infinite memory (Kučera, 2011, Proposition 5.7). However, imposing a natural structural restriction on the (possibly infinite) arena, namely to be chronological, yields a result like Theorem 2.2 for Büchi games, see Theorem 3.1.

### 2.2. Two-Player Imperfect-Information Stochastic Büchi Games

We now introduce a subclass of the usual games with imperfect information which is essentially a stochastic version of the model in (Chatterjee et al., 2007). Our model of imperfect-information games is quite restrictive compared to general models developed in (Gripon and Serre, 2009; Bertrand et al., 2009; Chatterjee and Doyen, 2014; Carayol et al., 2018), as in our setting Abélard is perfectly informed. However, it turns out to be expressive enough to be used as a central tool to check emptiness for alternating Büchi tree automata with qualitative semantics.

A stochastic arena of imperfect information is a tuple where is a finite set of vertices, is an initial vertex, is the finite alphabet of Éloïse’s actions, is a stochastic transition relation and is an equivalence relation over that denotes the observational capabilities of Éloïse and therefore imposes restrictions on legitimate strategies for her (see further). We additionally require that for all there is at least one such that .

A play starts from the initial vertex and proceeds as follows: Éloïse plays an action , then Abélard resolves the non-determinism by choosing a distribution such that and finally a new vertex is randomly chosen according to . Then, Éloïse plays a new action, Abélard resolves the non-determinism and a new vertex is randomly chosen, and so on forever. Hence, a play is an infinite word . A history is a prefix of a play ending in a vertex in .

An imperfect-information stochastic Büchi game is a pair where is a stochastic arena of imperfect information with a subset of states used to define the Büchi winning condition as follows: a play in is won by Éloïse if, and only if, the set is infinite, i.e. winning plays are those that infinitely often visit a vertex in .

The imperfect-information of the game is modelled by the equivalence relation that conveys which vertices Éloïse cannot distinguish, namely those that are -equivalent. We will write for the set of equivalence classes of in , and for every , we shall write for its -equivalence class.

Relation plays a crucial role when defining strategies for Éloïse. Intuitively, Éloïse should not play differently in two indistinguishable plays, where the indistinguishability of Éloïse is based on perfect recall (Fagin et al., 1995): Éloïse cannot distinguish two histories and whenever for all and for all . Note that in particular, Éloïse does not observe Abélard’s choices for the distributions along a play. Hence, a (pure444Again, as for perfect information games, we do not consider randomised strategies as pure strategies are the right model for our purpose.) strategy for Éloïse is a function assigning an action to every set of indistinguishable histories. Éloïse respects a strategy during a play if , for all .

A strategy for Abélard is defined as a function such that for every . Abélard respects a strategy during a play if , for all .

Exactly as in the perfect-information setting, one associates with a pair of strategies the set of those plays where Éloïse (resp. Abélard) respects (resp. ), and equip it with a probability measure.

Finally, a strategy for Éloïse is almost-surely winning if, against any strategy for Abélard, the set of winning plays for Éloïse has measure for the probability measure on .

###### Remark 2.3 ().

It is important to note that Éloïse may not observe whether a vertex belongs to as we do not require that . In particular, this has to be taken into account when eventually solving the game.

The following decidability result will be crucial in Section 3.2.

###### Theorem 2.4 ((Chatterjee and Doyen, 2014; Carayol et al., 2018)).

Let be an imperfect-information stochastic Büchi game. One can decide in exponential time whether Éloïse has an almost-surely winning strategy in .

### 2.3. Probabilistic Automata on Finite Words

Probabilistic automata on finite words generalize non-deterministic automata by letting the transition function map a state and a letter to a distribution over states (Rabin, 1963). The reference book for early developments on probabilistic automata is due to Paz (Paz, 1971).

A probabilistic word automaton is a tuple , where is the finite set of states, is the initial state, and is the transition function. We say that a probabilistic automaton is simple when the distribution is always of the form (possibly with ).

Intuitively, a finite word induces a set of runs of each of which comes with a probability of being realised; if one fixes a set of final states, the acceptance probability of by is the mere sum of the probabilities of those runs of over that end in a final state. To formally define acceptance probability (and extend it further to richer settings) we associate with and a Markov chain as follows.

The Markov chain has the (finite) set of states , the initial state , and the probability transition function defined for every (we do not define it for states of the form that will be useless) by

 TuA((p,i))=∑q∈Qδ(p,ui)(q)⋅(q,i+1)

Call a finite path of length of a run of on and let be the probability measure on runs induced by . Given a subset of (final) states , call the set of runs whose (first coordinate of the) last state is in . We then define the acceptance probability of over as .

A classic decision problem for probabilistic word automata is the value problem.

INPUT: A probabilistic word automaton and a subset QUESTION: ?

Informally, the value problem asks for the existence of words with acceptance probabilities that are arbitrarily close to . In this case, we say that has value . The undecidability of the value  problem for simple probabilistic automata was first established in (Gimbert and Oualhadj, 2010) (see also (Fijalkow et al., 2015) and (Fijalkow, 2017) for a simple proof).

###### Theorem 2.5 ((Gimbert and Oualhadj, 2010)).

The value problem for simple probabilistic word automata is undecidable.

### 2.4. Probabilistic Automata on Infinite Words

Baier, Größer, and Bertrand conducted an in-depth study of probabilistic automata over infinite words (Baier et al., 2012). To define the semantics of a probabilistic word automaton over an infinite word , we proceed as for finite words and construct a Markov chain whose set of states is . The initial state is again , and the probability transition function is still defined by

 TwA((p,i))=∑q∈Qδ(p,wi)(q)⋅(q,i+1)

A run of on is now an infinite path in and the Markov chain yields a probability measure on runs.

For probabilistic automata on infinite words we mostly focus on the co-Büchi acceptance condition that is defined as follows. Given a subset of states , we let be the (measurable) set of runs that visit only finitely often, and, when this set of runs has measure , we say that is almost-surely accepted by for the co-Büchi condition , written . Formally,

 L=1co-Büchi(F)(A)={w∈Σω:PwA(co-Büchi(F))=1}.
###### Example 2.6 ().

Let be an alphabet and be a fresh symbol. Let be the simple probabilistic co-Büchi automaton with set of states, initial state , and transition function given by:

• for any ;

• ; and

• for any .

As is absorbing and as moving from to may only happen when reading , the language consists of those infinite words over that contain infinitely many occurrences of .

The emptiness problem for probabilistic co-Büchi word automata with almost-sure semantics is the following decision problem:

INPUT: A probabilistic word automaton and a set QUESTION: Is ?

It was shown in (Baier et al., 2012) that this problem is undecidable.

###### Proposition 2.7 ((Baier et al., 2012)).

The emptiness problem for probabilistic co-Büchi word automata with almost-sure semantics is undecidable.

The proof in (Baier et al., 2012) is obtained by reducing the universality problem for simple probabilistic Büchi word automata with the positive semantics: Indeed, automata in this class (we refer to (Baier et al., 2012) for definitions) can be effectively complemented into probabilistic co-Büchi word automata with the almost-sure semantics, and whose universality problem is proved to be undecidable. As the complementation procedure does not preserve the property of being simple, we will later argue (see Proposition 4.1) that Proposition 2.7 still holds for simple probabilistic co-Büchi word automata with almost-sure semantics.

### 2.5. Universal Automata on Infinite Trees with Qualitative Semantics

The qualitative semantics for tree automata was introduced by Carayol, Haddad, and Serre in (Carayol et al., 2014) and was studied for non-deterministic (Carayol et al., 2014), alternating (Fijalkow et al., 2013a), and probabilistic automata (Carayol et al., 2014).

In this section we define universal tree automata with qualitative semantics and then extend this concept to alternating tree automata with qualitative semantics in the next section.

A tree automaton is a tuple , where is a finite set of states, is the initial state, and is the transition relation. A run of over a -tree is a -tree such that and for all we have . We let denote the set of runs of over .

A tree automaton and a run induce a Markov chain as follows. The set of states is , the initial state is , and the probability transition function is given by

 TρA((ρ(u),u))=12(ρ(u0),u0)+12(ρ(u1),u1)

yielding the probability measure on branches of the run .

Given a subset of states , we let be the (measurable) set of infinite paths in that visit only finitely often, and we say that the run is qualitatively accepting for the co-Büchi condition if . Equivalently, a run is qualitatively accepting for the co-Büchi condition if and only if the set of branches in that contain finitely many nodes labelled by a state in has measure for the classical coin-flipping measure on branches: is the unique complete probability measure such that .

The universal semantics yields the following definition:

 L∀Qual,co-Büchi(F)(A)={t∈% Trees(Σ):∀ρ∈RunsA(t),PρA(co-Büchi(F))=1}.

In words, a tree belongs to if every run of over is such that almost all its branches contain finitely many states in .

The emptiness problem for universal co-Büchi tree automata with qualitative semantics is the following decision problem:

INPUT: A tree automaton and a set QUESTION: Is ?

We will prove in Theorem 4.2 that this problem is undecidable.

### 2.6. Alternating Automata on Infinite Trees with Qualitative Semantics

An alternating tree automaton is a tuple , where is the finite set of states, is the initial state, is a partition of into Éloïse’s and Abélard’s states and is the transition relation.

The input of such an automaton is a -tree and acceptance is defined by means of the following two-player perfect-information stochastic game . Intuitively, a play in this game consists in moving a pebble along a branch of starting from the root: the pebble is attached to a state and in a node with state , Éloïse (if ) or Abélard (if ) picks a transition , and then Random chooses to move down the pebble either to node (and then updates the state to ) or to node (and then updates the state to ).

Formally, let with , and , and

 E={((q,u),(q,u,q0,q1))∣(q,u,q0,q1)∈VR}∪{((q,u,q0,q1),(qx,u⋅x))∣x∈{0,1} and (q,u,q0,q1)∈VR}

Then, we define where .

Given a subset of states , we say that is qualitatively accepted by for the Büchi (resp. co-Büchi) condition if Éloïse has an almost-surely winning strategy in the Büchi (resp. co-Büchi) game .

For an alternating tree automaton and a subset of states , we denote by (resp. ) the set of trees qualitatively accepted by for the Büchi (resp. co-Büchi) condition .

###### Remark 2.8 ().

Any positional strategy for Éloïse in can be described as a function that satisfies the following property: , if then . Equivalently, in a curried form, is a map . Hence, if one lets be the set of functions from into , Éloïse’s positional strategies are in bijection with -labelled binary trees.

It is easily seen that universal tree automata with qualitative semantics are subsumed by alternating tree automata with qualitative semantics. Indeed we have the following classical result (that we state here only for co-Büchi acceptance condition but that works similarly for any other acceptance condition).

###### Proposition 2.9 ().

Let be a tree automaton and let . Consider the alternating tree automaton , meaning that all states of are interpreted as Abélard’s. Then the following holds.

 L∀Qual,co-Büchi(F)(A)=LAltQual,co-Büchi(F)(B)
###### Proof.

For a fixed tree , runs of over are in bijection with strategies of Abélard in the co-Büchi game (where Éloïse is making no choice), and moreover a run is qualitatively accepting for if and only if Éloïse almost-surely wins in when Abélard uses the corresponding strategy. Hence, all runs of over are qualitatively accepting if and only if Éloïse almost-surely wins against every strategy of Abélard in , which means that . ∎

The emptiness problem for alternating Büchi tree automata with qualitative semantics is the following decision problem:

INPUT: An alternating tree automaton and a set QUESTION: Is ?

We will prove in Theorem 3.7 that this problem is decidable in exponential time.

###### Remark 2.10 ().

The emptiness problem can be similarly defined for alternating co-Büchi tree automata with qualitative semantics. However, this problem is undecidable as a corollary of Proposition 2.9 together with forthcoming Theorem 4.2, proving the undecidability of the emptiness problem for universal co-Büchi tree automata with qualitative semantics.

### 2.7. Probabilistic Automata on Infinite Trees with Qualitative Semantics

Probabilistic tree automata with qualitative semantics were defined in (Carayol et al., 2014) with the intention of lifting the definition of probabilistic automata on infinite words to the case of infinite trees. In particular, an input tree induces a probability distribution over runs and acceptance is defined by requiring that almost all runs should be accepting. Mixed with the qualitative co-Büchi semantics, this means that a tree is accepted if almost all runs have almost all their branches containing finitely many states from . Contrary to the authors of (Carayol et al., 2014) who define a probability measure on runs, we follow another approach (still yielding an equivalent notion (Carayol et al., 2014, Proposition 45)) based on Markov chains.

A probabilistic tree automaton is a tuple , where is the finite set of states, is the initial state, and is the transition function.

A probabilistic tree automaton and a tree induce a Markov chain as follows. The set of states is , the initial state is , and the probability transition function is given by (where distributes over )

 TtA((q,u))=∑q0,q1∈Qδ(q,t(u))(q0,q1)⋅(12(q0,u0)+12(q1,u1)),

Given a subset of states , we again let be the (measurable) set of infinite paths in that visit only finitely often. Then the probability measure induced by yields the following definition of the set of trees almost-surely qualitatively accepted by :

 L∀=1Qual,co-Büchi(F)(A)={t∈Trees(Σ):PtA(co-Büchi(F))=1}.

We now turn to our main decidability result about emptiness of alternating Büchi tree automata with qualitative semantics.

## 3. Decidability of the Emptiness Problem for Alternating Büchi Tree Automata with Qualitative Semantics

In this section we prove Theorem 3.7 that states the decidability of the emptiness problem for alternating Büchi tree automata with qualitative semantics, which contrasts with the forthcoming result that the emptiness problem for universal co-Büchi tree automata with qualitative semantics is undecidable (Theorem 4.2 of Section 4).

Our approach for checking emptiness of an alternating Büchi tree automaton with qualitative semantics relies on a two-player imperfect-information stochastic finite Büchi game. In this game, Éloïse almost-surely wins if, and only if, the language accepted by is non-empty. As for this class of games, one can decide whether Éloïse has an almost-surely winning strategy, the announced decidability result follows.

We establish in Section 3.1 a preliminary general positionality result to be used in Section 3.2 for proving the equivalence between Éloïse almost-surely winning in the game and accepting some tree.

### 3.1. A Positionality Result for Chronological Games

For the rest of this section, we fix a stochastic arena with . Moreover, we assume that the game is chronological in the sense that there exists a function such that and for , . Note that the arena used in Section 2.6 to define acceptance of a tree by an alternating tree automaton with qualitative semantics is chronological. Note also that a chronological arena with finite out-degree has a countable set of vertices.

###### Theorem 3.1 ().

In a two-player perfect-information stochastic Büchi game played on a chronological arena with finite out-degree, Éloïse has an almost-surely winning strategy if, and only if, she has a positional almost-surely winning strategy.

Actually, the core difficulty lies in proving Theorem 3.1 for the simple case of reachability games.

###### Theorem 3.2 ().

In a two-player perfect-information stochastic reachability game played on a chronological arena with finite out-degree, Éloïse has an almost-surely winning strategy if, and only if, she has a positional almost-surely winning strategy.

###### Proof.

The direction from right to left is immediate. For the other direction, the key steps are the following. First, we establish (Lemma 3.3)that if Éloïse can ensure to reach with probability  from some initial vertex, then there exists a bound such that she can ensure to reach with probability at least half within steps. Second, we exploit Lemma 3.3 to “slice” the arena into infinitely many disjoint finite arenas: in each slice Éloïse plays to reach with probability at least half. Since each slice forms a finite sub-arena, optimal positional strategies always exist. Finally, the strategy that plays in turns the latter positional strategies ensures to almost-surely reach in the long run.

Let be a two-player perfect-information stochastic reachability game played on a chronological arena with finite out-degree. In the following, a strategy in from a vertex is a strategy in the game obtained from by changing the initial vertex of the arena to .

The following lemma allows us to decompose the infinite arena into infinitely many finite arenas.

###### Lemma 3.3 ().

Let be an almost-surely winning strategy for Éloïse in from some vertex . Then, there exists an integer such that for any strategy of Abélard, we have

 PrσE,σA(V≤kFVω)≥12 .
###### Proof of Lemma 3.3.

Toward a contradiction, assume that such a does not exist. Hence, for each there exists a strategy such that .

Without loss of generality, we can assume that is positional. Indeed, one can pick for a strategy for Abélard that minimises the probability of winning for Éloïse in the reachabililty game obtained by restricting to vertices of rank at most . This game has a finite arena since has finite out-degree, and by e.g. (Gimbert and Zielonka, 2007) such a strategy for Abélard can be chosen positional.

From the sequence of strategies , we now extract a strategy that for every , agrees with infinitely many on its first moves. Since has countably many vertices, fix an (arbitrary) enumeration of the vertices in .

We define step-wise inductively on : at step , is defined on and on these vertices agrees with all those strategies with where the sequence is also defined inductively on and is such that each is infinite.

We let be the set of all positive integers.

For where , consider the values of for all . Because has finite out-degree, there is some such that , for infinitely many . We define and we let ; note that is infinite.

Now, for , it is easy to see that by choosing big enough so that all vertices of rank at most belong to , strategy agrees on its first moves with the infinitely many where .

As a consequence, for every there is some such that

 PrσE,σA,∞(V≤kFVω)=PrσE,σA,h(V≤kFVω)≤PrσE,σA,h(V≤hFVω)<12

As and as the sequence is increasing for set inclusion, one concludes that

 PrσE,σA,∞(V∗FVω)=limk→∞PrσE,σA,∞(V≤kFVω)≤12<1

which leads to a contradiction with being almost-surely winning, and concludes the proof of Lemma 3.3. ∎

Keeping on with the proof of Theorem 3.2, assume that Éloïse has an almost-surely wining strategy in . Without loss of generality, we can assume that she has an almost-surely winning strategy from everywhere, by restricting the arena to vertices reachable by an almost-surely winning strategy.

For , we define the reachability game induced by restricting the arena to vertices of rank in where we add self-loops on vertices of rank to avoid having dead-end vertices. Since has finite out-degree, there are finitely many vertices of rank in , hence is finite.

We define inductively an increasing sequence of ranks together with a sequence of strategies such that for all , is a positional strategy, defined on all vertices of rank in , and such that from all vertices of rank , for all strategies , we have

 PrσE,[ki,ki+1[,σA(V≤ℓFVω)≥12 ,

where .

Assume the first ranks and strategies are defined. For each vertex of rank , Lemma 3.3 gives the existence of some bound ; since there are finitely many such vertices, we can consider the maximum of those bounds that we call , and we let . By construction and Lemma 3.3, from all vertices of rank , for all strategies , we have

 PrσE,σA(V≤ℓFVω)≥12 ,

where . In other words, Éloïse wins the reachability game with probability at least half, so, relying on a generalisation555More precisely, when playing a reachability game on a finite arena, Éloïse always has an optimal positional strategy, where being optimal means that . of Theorem 2.2 (see e.g. (Gimbert and Zielonka, 2007; Kučera, 2011)), there exists an optimal uniform (i.e. working from any initial vertex) positional strategy, that we call