Alternating Tree Automata with Qualitative Semantics
Raphaël Berthon, Nathanaël Fijalkow, Emmanuel Filiot, Shibashis Guha, Bastien Maubert, Aniello Murano, Laureline Pinault, Sophie Pinchinat, Sasha Rubin, Olivier Serre
aa r X i v : . [ c s . F L ] D ec Alternating Tree Automata with Qualitative Semantics
RAPHAËL BERTHON,
Université libre de Bruxelles, Belgium
NATHANAËL FIJALKOW,
CNRS & LaBRI, France
EMMANUEL FILIOT,
Université libre de Bruxelles, Belgium
SHIBASHIS GUHA,
Université libre de Bruxelles, Belgium
BASTIEN MAUBERT,
Università degli Studi di Napoli “Federico II”, Italy
ANIELLO MURANO,
Università degli Studi di Napoli “Federico II”, Italy
LAURELINE PINAULT,
Univ Lyon, CNRS, ENS de Lyon, UCB Lyon 1, LIP, France
SOPHIE PINCHINAT,
Univ Rennes, CNRS, IRISA, France
SASHA RUBIN,
University of Sydney, Australia
OLIVIER SERRE,
Université de Paris, IRIF, CNRS, FranceWe study alternating automata with qualitative semantics over infinite binary trees: alternation means thattwo opposing players construct a decoration of the input tree called a run, and the qualitative semantics saysthat a run of the automaton is accepting if almost all branches of the run are accepting. In this paper we provea 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 emptinessproblem 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 con-dition. This result has two direct consequences: the undecidability of monadic second-order logic extendedwith the qualitative path-measure quantifier, and the undecidability of the emptiness problem for alternat-ing tree automata with non-zero semantics, a recently introduced probabilistic model of alternating treeautomata.CCS Concepts: •
Theory of computation → Automata over infinite objects ; Tree languages ; Quanti-tative automata ; Logic ; Probabilistic computation .Additional Key Words and Phrases: tree automata, 𝜔 -regular conditions, almost-sure semantics Authors’ addresses: Raphaël Berthon, Université libre de Bruxelles, Bruxelles, Belgium, [email protected];Nathanaël Fijalkow, CNRS & LaBRI, Bordeaux, France, [email protected]; Emmanuel Filiot, Université libre deBruxelles, Bruxelles, Belgium, efi[email protected]; Shibashis Guha, Université libre de Bruxelles, Bruxelles, Belgium, [email protected]; Bastien Maubert, Università degli Studi di Napoli “Federico II”, DIETI, Naples, Italy, [email protected]; Aniello Murano, Università degli Studi di Napoli “Federico II”, DIETI, Naples, Italy, [email protected]; Lau-reline Pinault, Univ Lyon, CNRS, ENS de Lyon, UCB Lyon 1, LIP, Lyon, France, [email protected]; SophiePinchinat, Univ Rennes, CNRS, IRISA, Rennes, France, [email protected]; Sasha Rubin, University of Sydney, Syd-ney, Australia, [email protected]; Olivier Serre, Université de Paris, IRIF, CNRS, Bâtiment Sophie Germain, Case courrier7014, 8 Place Aurélie Nemours, Paris Cedex 13, 75205, France, [email protected] to make digital or hard copies of all or part of this work for personal or classroom use is granted without feeprovided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice andthe full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored.Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requiresprior specific permission and/or a fee. Request permissions from [email protected].© 2020 Association for Computing Machinery.1529-3785/2020/12-ART $15.00https://doi.org/10.1145/nnnnnnn.nnnnnnnACM Trans. Comput. Logic, Vol. 1, No. 1, Article . Publication date: December 2020.
R. Berthon et al.
ACM Reference Format:
Raphaël Berthon, Nathanaël Fijalkow, Emmanuel Filiot, Shibashis Guha, Bastien Maubert, Aniello Murano,Laureline Pinault, Sophie Pinchinat, Sasha Rubin, and Olivier Serre. 2020. Alternating Tree Automata withQualitative Semantics.
ACM Trans. Comput. Logic
1, 1 (December 2020), 24 pages. https://doi.org/10.1145/nnnnnnn.nnnnnnn
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, probabilis-tic, and alternating, are the most studied notions. Intuitively, in each case, an automaton reading aninput tree (with labels on the nodes) constructs a decoration of this tree called a run, which is itselfa tree. The run labels nodes of the tree by states respecting the local constraints imposed by thetransition relation of the automaton. In the deterministic case, a state and a letter uniquely deter-mine 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 pos-sibly several runs on a single tree. In the non-deterministic case we say that the tree is accepted ifthere exists an accepting run, i.e. the choices are existential. In the universal case, we say that thetree is accepted if all runs are accepting, i.e. the choices are universal. The alternating case unifiesboth previous cases by introducing existential and universal transitions.The second feature is the branching semantics . The classical one says that a run is acceptingif all its branches satisfy a given acceptance condition. We are concerned in this paper with thequalitative semantics, which is an alternative branching semantics introduced by Carayol, Had-dad, and Serre [8]. The qualitative semantics says that a run is accepting if almost all its branchessatisfy a given acceptance condition, in other words if by picking a branch uniformly at random italmost-surely satisfies the condition. The paper [8] showed that non-deterministic and probabilis-tic tree automata with qualitative semantics are both robust computational models with appealingalgorithmic properties.The third feature is the acceptance condition (on branches), with 𝜔 -regular conditions such asBüchi and parity conditions being the most important for their tight connections to logical for-malisms; see, e.g., [32].One motivation for studying tree automata with qualitative semantics is to extend the deepconnections between automata and monadic second-order logic (MSO) which hold for the classicalsemantics [29]. Indeed, the general goal is to construct decidable extensions of MSO over infinitetrees; we review some of the efforts and results obtained in this direction. A (unary) generalisedquantifier is of the form “the set of all sets 𝑋 that satisfy 𝜑 has the property 𝐶 ”, where 𝐶 is a propertyof sets. For instance, the ordinary existential quantifier ∃ 𝑋 .𝜑 corresponds to the property 𝐶 ofbeing a non-empty set. More interestingly, the quantifier “there exist infinitely many 𝑋 such that 𝜑 ”corresponds to the property 𝐶 of being infinite. It turns out that certain cardinality quantifiers suchas “there exist infinitely many 𝑋 ” and “there exist continuum many 𝑋 ” do not add expressive powerto MSO over the infinite binary tree (in fact, they can be effectively eliminated) [2]. On the otherhand, adding the generalised quantifier “the set of all sets 𝑋 satisfying 𝜑 has Lebesgue-measureone” results in an undecidable theory [25]. A weaker version of this quantifier is “the set of paths ofthe tree that satisfy 𝜑 has Lebesgue-measure one”. Intuitively, this quantifier, written ∀ = path , meansthat a random path almost-surely satisfies 𝜑 , where a random path is generated by repeatedlyflipping a coin to decide whether to go left or right. It was proved in [5, 6] that adding the quantifier ∀ = path to a restriction of MSO called “thin MSO” yields a decidable logic, but the decidability ofMSO+ ∀ = path was left open in [24, 25]. The emptiness problem for non-deterministic parity tree ACM Trans. Comput. Logic, Vol. 1, No. 1, Article . Publication date: December 2020. lternating Tree Automata with Qualitative Semantics 3 automata with qualitative semantics can easily be expressed using MSO+ ∀ = path , as already observedin [25], 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 focuson the emptiness problem. We present a positive result and a negative result that delimit a clearand sharp decidability frontier. Contributions
The positive result is the decidability of the emptiness problem for the case of the Büchi acceptancecondition (Theorem 3.7).The usual roadmap for solving the emptiness problem for alternating automata is to first con-struct an equivalent non-deterministic automaton, and then to construct an emptiness game for thenon-deterministic automaton, i.e., a game such that the first player wins if and only if the automa-ton is non-empty. This first step is an effective construction of an equivalent non-deterministicautomaton, which in some cases is not possible, unknown, or computationally too expensive. Inthe case at hand the second situation arises: we do not know whether alternating automata withqualitative semantics can effectively be turned into equivalent non-deterministic ones. We remarkthat our undecidability result shows that there is no such effective construction for co-Büchi con-ditions (but there might be one for the Büchi conditions).Here, instead, we develop a new approach which directly constructs an emptiness game for thealternating automaton. The emptiness game we construct uses imperfect information . Our con-struction extends the notion of blindfold games of Reif [30], used to check universality of non-deterministic automata over finite words. The key ingredient to proving the correctness of ourimperfect information emptiness game is a new positionality result for stochastic Büchi games oncertain infinite arenas (that we call chronological ). To the best of our knowledge, very few position-ality results are known in the literature that combine both stochastic features and infinite arenas;a notable exception is [23].The negative result is the undecidability of the emptiness problem for the case of the co-Büchiacceptance condition. In fact, our main technical contribution (Theorem 4.2) is to establish theundecidability 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 oninfinite words and trees). We initially resort to the known undecidability of the value 1 problem forprobabilistic automata on finite words [20] to deduce the undecidability of the emptiness problemfor simple probabilistic co-Büchi automata on infinite words (Proposition 4.1). Here, simple meansthat the transitions of the automaton only involve probabilities in { , , } . Then, we reduce thelatter problem to the original emptiness problem for universal co-Büchi tree automata with qual-itative semantics, hence proving our negative result. The correctness of this last reduction relieson particular properties of another class of automata, namely, probabilistic tree automata .Our negative result has two interesting consequences: the undecidability of MSO+ ∀ = path , and ofthe emptiness problem for alternating tree automata with non-zero semantics, a model combiningsure, almost-sure, and positive semantics and studied in [19]. Related Work
The study of automata with qualitative semantics was initiated in [8] with several decidabilityresults. The first result is a polynomial-time algorithm entailing the decidability of the empti-ness problem for non-deterministic parity tree automata with qualitative semantics [8], obtainedthrough a polynomial reduction to the almost-sure problem for Markov decision processes (for
ACM Trans. Comput. Logic, Vol. 1, No. 1, Article . Publication date: December 2020.
R. Berthon et al. which a polynomial-time algorithm is known from [12]). This reduction extends to probabilis-tic tree automata with qualitative semantics, showing an equivalence with partial-observationMarkov decision processes. It is then used to prove the decidability of the emptiness problem forprobabilistic Büchi tree automata with qualitative semantics [8].Alternation was later considered by Fijalkow, Pinchinat, and Serre in [17] where the focus wason 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 intermedi-ate transformation to a non-deterministic automaton: this was successfully applied to classicalalternating parity tree automata as well as to alternating Büchi tree automata with qualitativesemantics (see Theorem 3.7).This line of work was pursued using the related model of non-zero automata. The first decid-ability result was obtained for the subclass of zero automata [6], yielding the decidability of thethin restriction of MSO+ ∀ = path . A second decidability result concerned the class of alternating zeroautomata [19], restricting the abilities of the second player. This latter result is applied to solve thesatisfiability problem of a probabilistic extension of CTL ∗ . The general case of non-zero automatawas left open. We close it negatively (thanks to our negative result) since alternating tree automatawith non-zero semantics subsume universal tree automata with qualitative semantics.A side result in [17] states the undecidability of the emptiness problem for alternating co-Büchiautomata with qualitative semantics. The proof, not given in the conference proceedings, is rathersketchy in the full version [18]. The proof we give here (Theorem 4.2) follows the same lines butclarifies a technical loophole in the original proof. Indeed, the last reduction requires the undecid-ability of the emptiness problem for probabilistic co-Büchi simple automata over infinite words,where simple means that the transitions probabilities are either 0, , or 1. The undecidability resultwas known only for general automata, while we refine it for the simple ones, thus filling in thegap of the undecidability proof in the full version [18].More recently, Berthon et al. [3] proved the slightly weaker undecidability result that emptinessis undecidable for universal parity tree automata with qualitative semantics. Although their prooffollows the same lines as [18], the result is weaker because they need a stronger acceptance con-dition to obtain simple automata and prove the correctness of the original reduction. Still, theirresult is strong enough to entail the undecidability of MSO+ ∀ = path , the main contribution of theirwork.There is another proof of the undecidability of MSO+ ∀ = path , obtained independently and at thesame time as [3] by Bojańczyk, Kelmendi, and Skrzypczak [7]. Their proof technique is very differ-ent from ours: they obtain undecidability by a direct encoding of two-counter machines into thelogic. 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 ma-chines and to prove the correctness of the reduction . The proof of this lemma is involved: it mostlyrelies on tools (such as asymptotic behaviours of vector sequences) previously used to show un-decidability of MSO+U logic over infinite words. We remark that MSO+ ∀ = path is known as MSO+ ∇ in [7]. Organisation of the Paper
Section 2 presents the different classes of automata used for our main undecidability result, relyingon Markov chains as a unifying notion to define acceptance by these different automata. Section 3 More precisely, the lemma states that for a set 𝐷 of pairwise disjoint finite paths in the infinite binary tree called intervals ,there is an MSO+ ∀ = path formula that, when true at the root of the infinite binary tree, is equivalent to having with probability1, a branch 𝜋 and some integer ℓ such that with finitely many exceptions, if an interval intersects 𝜋 then it is of length ℓ .ACM Trans. Comput. Logic, Vol. 1, No. 1, Article . Publication date: December 2020. lternating Tree Automata with Qualitative Semantics 5 gives our decidability result for alternating Büchi tree automata. Section 4 is about our undecid-ability result for universal co-Büchi tree automata, while Section 5 presents its consequences forMSO+ ∀ = path (Section 5.1) and for alternating automata with non-zero semantics (Section 5.2). Throughout the paper we implicitly fix a finite alphabet Σ . We denote by Σ ∗ the set of finite words over Σ and by Σ 𝜔 the set of infinite words over Σ . We let 𝜀 denote the empty word, and for a word 𝑢 ∈ Σ ∗ , | 𝑢 | denotes its length. Finally, we write Σ 𝑘 for the set of words over Σ of length 𝑘 .The infinite binary tree is { , } ∗ , elements of { , } ∗ are called its nodes , and elements of { , } 𝜔 are called its (infinite) branches . For a finite alphabet Σ , a 𝚺 -tree is a function 𝑡 : { , } ∗ → Σ and we write Trees ( Σ ) for the set of Σ -trees. For a branch 𝑏 = 𝑏 𝑏 · · · ∈ { , } 𝜔 we denote by 𝑡 [ 𝑏 ] = 𝑡 ( 𝜀 ) 𝑡 ( 𝑏 ) 𝑡 ( 𝑏 𝑏 ) 𝑡 ( 𝑏 𝑏 𝑏 ) · · · ∈ Σ 𝜔 the infinite word read in 𝑡 along the branch 𝑏 .A distribution over a set 𝑄 is a function 𝛿 : 𝑄 → [ , ] such that Í 𝑞 ∈ 𝑄 𝛿 ( 𝑞 ) =
1. Any distribu-tion 𝛿 considered in the paper is implicitly assumed to have a finite support, i.e. { 𝑞 ∈ 𝑄 | 𝛿 ( 𝑞 ) ≠ } is finite. For 𝑄 ′ ⊆ 𝑄 , we write Í 𝑞 ∈ 𝑄 ′ 𝑝 𝑞 · 𝑞 for the distribution that assigns probability 𝑝 𝑞 to 𝑞 ∈ 𝑄 ′ and 0 to 𝑞 ∈ 𝑄 \ 𝑄 ′ . For example, 𝑞 + 𝑞 is the distribution 𝛿 such that 𝛿 ( 𝑞 ) = 𝛿 ( 𝑞 ) = ,unless 𝑞 = 𝑞 in which case 𝛿 ( 𝑞 ) = 𝛿 ( 𝑞 ) =
1, and 𝛿 ( 𝑞 ) = 𝑞 . The set ofdistributions over 𝑄 is denoted D ( 𝑄 ) .A Markov chain M = ( 𝑆, 𝑠 in ,𝑇 ) is given by a possibly infinite set of states 𝑆 , an initial state 𝑠 in ∈ 𝑆 , and a probability transition function 𝑇 : 𝑆 → D ( 𝑆 ) . An (infinite) path in M is an infinitesequence of states 𝑠 𝑠 𝑠 . . . ∈ 𝑆 𝜔 such that 𝑠 = 𝑠 in and 𝑇 ( 𝑠 𝑖 ) ( 𝑠 𝑖 + ) > 𝑖 ≥
0. A cone is aset of paths of the form 𝑢 · 𝑆 𝜔 for some 𝑢 ∈ 𝑆 ∗ . Now, consider the 𝜎 -algebra over paths in M builtfrom the set of cones. Then, a classical way to equip this 𝜎 -algebra with a probability measure 𝑃 is to recursively define it on the set of cones as follows: 𝑃 ( 𝑠 𝑠 · · · 𝑠 𝑘 · 𝑆 𝜔 ) = ( 𝑘 = 𝑃 ( 𝑠 · · · 𝑠 𝑘 − · 𝑆 𝜔 ) · 𝑇 ( 𝑠 𝑘 − ) ( 𝑠 𝑘 ) otherwiseand then to extend it (uniquely) to the 𝜎 -algebra thanks to Carathéodory’s extension theorem (werefer the reader to Reference [27] for more details on this classical construction).When needed, for a given length 𝑘 , we also see 𝑃 as a probability measure on paths of length 𝑘 (i.e. elements in 𝑆 𝑘 ) by defining the probability measure of 𝑢 ∈ 𝑆 𝑘 as the probability of the cone 𝑢 · 𝑆 𝜔 . A graph is a pair 𝐺 = ( 𝑉 , 𝐸 ) where 𝑉 is a (possibly infinite) set of vertices and 𝐸 ⊆ 𝑉 × 𝑉 is a setof edges . For every vertex 𝑣 , let 𝐸 ( 𝑣 ) = { 𝑤 | ( 𝑣, 𝑤 ) ∈ 𝐸 } , and say that 𝑣 is a dead-end if 𝐸 ( 𝑣 ) = ∅ .In the rest of the paper, we only consider graphs of finite out-degree, i.e. such that | 𝐸 ( 𝑣 )| is finitefor every vertex 𝑣 ∈ 𝑉 , and without dead-ends.A (turn-based) stochastic arena is a tuple G = ( 𝐺, 𝑉 𝐸 , 𝑉 𝐴 , 𝑉 𝑅 , 𝛿, 𝑣 in ) where 𝐺 = ( 𝑉 , 𝐸 ) is a graph, ( 𝑉 𝐸 , 𝑉 𝐴 , 𝑉 𝑅 ) is a partition of the vertices among two players, Éloïse and Abélard, and an extra playerRandom, 𝛿 : 𝑉 𝑅 → D ( 𝑉 ) is a map such that for all 𝑣 ∈ 𝑉 𝑅 the support of 𝛿 ( 𝑣 ) is included in 𝐸 ( 𝑣 ) ,and 𝑣 in ∈ 𝑉 is an initial vertex. In a vertex 𝑣 ∈ 𝑉 𝐸 ( resp. 𝑣 ∈ 𝑉 𝐴 ) Éloïse ( resp. Abélard) chooses asuccessor vertex from 𝐸 ( 𝑣 ) , and in a random vertex 𝑣 ∈ 𝑉 𝑅 , a successor vertex is chosen accordingto the probability distribution 𝛿 ( 𝑣 ) . A play 𝜆 = 𝑣 𝑣 𝑣 · · · is an infinite sequence of vertices startingfrom the initial vertex, i.e. 𝑣 = 𝑣 in , and such that, for every 𝑘 ≥ 𝑣 𝑘 + ∈ 𝐸 ( 𝑣 𝑘 ) if 𝑣 𝑘 ∈ 𝑉 𝐸 ∪ 𝑉 𝐴 and 𝛿 ( 𝑣 𝑘 ) ( 𝑣 𝑘 + ) > 𝑣 𝑘 ∈ 𝑉 𝑅 . A history is a finite prefix of a play. ACM Trans. Comput. Logic, Vol. 1, No. 1, Article . Publication date: December 2020.
R. Berthon et al.
A (pure ) strategy for Éloïse is a function 𝜎 𝐸 : 𝑉 ∗ · 𝑉 𝐸 → 𝑉 such that for every history 𝜆 · 𝑣 ∈ 𝑉 ∗ · 𝑉 𝐸 one has 𝜎 𝐸 ( 𝜆 · 𝑣 ) ∈ 𝐸 ( 𝑣 ) . Strategies of Abélard are defined likewise, and usually denoted 𝜎 𝐴 .A play 𝜆 = 𝑣 𝑣 𝑣 . . . is consistent with a pair of strategies ( 𝜎 𝐸 , 𝜎 𝐴 ) for Éloïse and Abélard if theplayers always choose their move according to their strategy. Formally, for all 𝑘 ≥ 𝑣 𝑘 is controlled by Éloïse then 𝑣 𝑘 + = 𝜎 𝐸 ( 𝑣 . . .𝑣 𝑘 ) and if it is controlled by Abélardthen 𝑣 𝑘 + = 𝜎 𝐴 ( 𝑣 . . .𝑣 𝑘 ) . The set of plays consistent with ( 𝜎 𝐸 , 𝜎 𝐴 ) is denoted Plays G 𝜎 𝐸 ,𝜎 𝐴 , and ahistory is consistent with ( 𝜎 𝐸 , 𝜎 𝐴 ) if it is the finite prefix of some play in Plays G 𝜎 𝐸 ,𝜎 𝐴 .In order to equip the set Plays G 𝜎 𝐸 ,𝜎 𝐴 with a probability measure, we define the following Markovchain M G 𝜎 𝐸 ,𝜎 𝐴 : its set of states is the set of histories consistent with ( 𝜎 𝐸 , 𝜎 𝐴 ) , its initial state is 𝑣 in ,and its probability transition function 𝑇 is defined by 𝑇 ( 𝜆 · 𝑣 ) = 𝜆 · 𝑣 · 𝜎 𝐸 ( 𝜆 · 𝑣 ) if 𝑣 ∈ 𝑉 𝐸 𝜆 · 𝑣 · 𝜎 𝐴 ( 𝜆 · 𝑣 ) if 𝑣 ∈ 𝑉 𝐴 Í 𝑣 ′ ∈ 𝐸 ( 𝑣 ) 𝛿 ( 𝑣 ) ( 𝑣 ′ ) 𝜆 · 𝑣 · 𝑣 ′ if 𝑣 ∈ 𝑉 𝑅 Then, the set Plays G 𝜎 𝐸 ,𝜎 𝐴 of those plays consistent with ( 𝜎 𝐸 , 𝜎 𝐴 ) is in bijection with the set ofinfinite paths in the Markov chain M G 𝜎 𝐸 ,𝜎 𝐴 . Hence, the associated probability measure 𝑃 G 𝜎 𝐸 ,𝜎 𝐴 canbe used as a probability measure for measurable subsets of Plays G 𝜎 𝐸 ,𝜎 𝐴 .When G is understood, we omit it and simply write 𝑃 𝜎 𝐸 ,𝜎 𝐴 and Plays 𝜎 𝐸 ,𝜎 𝐴 .A winning condition is a subset Ω ⊆ 𝑉 𝜔 and a (two-player perfect-information) stochasticgame is a pair G = (G , Ω ) .A strategy 𝜎 𝐸 for Éloïse is surely winning if Plays 𝜎 𝐸 ,𝜎 𝐴 ⊆ Ω for every strategy 𝜎 𝐴 of Abélard;it is almost-surely winning if 𝑃 𝜎 𝐸 ,𝜎 𝐴 ( Ω ) = 𝜎 𝐴 of Abélard. Similar notionsfor 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 𝑉 ∗ 𝐹𝑉 𝜔 forsome subset 𝐹 ⊆ 𝑉 , i.e. winning plays are those that eventually visit a vertex in 𝐹 . A Büchi game isa stochastic game whose winning condition is of the form Ñ 𝑖 ≥ 𝑉 𝑖 𝑉 ∗ 𝐹𝑉 𝜔 for some subset 𝐹 ⊆ 𝑉 ,i.e. winning plays are those that infinitely often visit a vertex in 𝐹 . Finally, a co-Büchi game isstochastic game whose winning condition is of the form 𝑉 ∗ ( 𝑉 \ 𝐹 ) 𝜔 for some subset 𝐹 ⊆ 𝑉 , i.e.winning plays are those that finitely often visit a vertex in 𝐹 . When it is clear from the context,we write G = (G , 𝐹 ) (i.e. write 𝐹 instead of Ω ) for the reachability ( resp. Büchi, co-Büchi) gamerelying on 𝐹 .A positional strategy 𝜎 is a strategy that does not require any memory, i.e. such that for anytwo histories of the form 𝜆 · 𝑣 and 𝜆 ′ · 𝑣 , one has 𝜎 ( 𝜆 · 𝑣 ) = 𝜎 ( 𝜆 ′ · 𝑣 ) . Positional strategies onlydepend on the current vertex, and for convenience they are written as functions from 𝑉 into 𝑉 .A game is deterministic whenever 𝑉 𝑅 = ∅ . It is well-known (see e.g. [33]) that positional strate-gies suffice to surely win in deterministic games with a parity winning condition , which we do notdefine but captures the reachability, Büchi, and co-Büchi winning conditions that we are interestedin. Theorem 2.1 (Positional determinacy [33]).
Let G be a deterministic parity game. Then, eitherÉloïse or Abélard has a positional surely winning strategy. We 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. Formally, one needs to require that Ω is measurable for the probability measure 𝑃 𝜎 𝐸 ,𝜎 𝐴 , which is always trivially true inthis paper.ACM Trans. Comput. Logic, Vol. 1, No. 1, Article . Publication date: December 2020. lternating Tree Automata with Qualitative Semantics 7 For stochastic games, the following result is well-known (see e.g. [21] for a slightly more generalresult).
Theorem 2.2.
Let G be a stochastic parity game played on a finite arena. If Éloïse almost-surelywins then she has an a positional almost-surely winning strategy. Note that dropping the assumption that the arena is finite substantially changes the situation. In-deed, for infinite arenas, even with a reachability condition and assuming finite out-degree, almost-surely winning strategies for Éloïse may require infinite memory [23, Proposition 5.7]. However,imposing a natural structural restriction on the (possibly infinite) arena, namely to be chronologi-cal, yields a result like Theorem 2.2 for Büchi games, see Theorem 3.1.
We now introduce a subclass of the usual games with imperfect information which is essentially astochastic version of the model in [11]. Our model of imperfect-information games is quite restric-tive compared to general models developed in [4, 9, 10, 22], as in our setting Abélard is perfectlyinformed. However, it turns out to be expressive enough to be used as a central tool to checkemptiness for alternating Büchi tree automata with qualitative semantics.A stochastic arena of imperfect information is a tuple G = ( 𝑉 , 𝐴,𝑇 , ∼ , 𝑣 in ) where 𝑉 is a finite set of vertices, 𝑣 in ∈ 𝑉 is an initial vertex, 𝐴 is the finite alphabet of Éloïse’s actions, 𝑇 ⊆ 𝑉 × 𝐴 ×D ( 𝑉 ) is a stochastic transition relation and ∼ is an equivalence relation over 𝑉 that denotesthe observational capabilities of Éloïse and therefore imposes restrictions on legitimate strategiesfor her (see further). We additionally require that for all ( 𝑣, 𝑎 ) ∈ 𝑉 × 𝐴 there is at least one 𝛿 ∈ D ( 𝑉 ) such that ( 𝑣, 𝑎, 𝛿 ) ∈ 𝑇 .A play starts from the initial vertex 𝑣 in and proceeds as follows: Éloïse plays an action 𝑎 ∈ 𝐴 ,then Abélard resolves the non-determinism by choosing a distribution 𝛿 such that ( 𝑣 in , 𝑎 , 𝛿 ) ∈ 𝑇 and finally a new vertex is randomly chosen according to 𝛿 . 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 𝑣 in 𝑎 𝛿 𝑣 𝑎 𝛿 𝑣 · · · ∈ ( 𝑉 · 𝐴 · D ( 𝑉 )) 𝜔 . A history is a prefix of aplay ending in a vertex in 𝑉 .An imperfect-information stochastic Büchi game is a pair G = (G , 𝐹 ) where G is a stochasticarena of imperfect information with a subset of states 𝐹 ⊂ 𝑉 used to define the Büchi winningcondition as follows: a play 𝜆 = 𝑣 𝑎 𝛿 𝑣 𝑎 𝛿 𝑣 · · · in G is won by Éloïse if, and only if, the set { 𝑖 ≥ | 𝑣 𝑖 ∈ 𝐹 } is infinite, i.e. winning plays are those that infinitely often visit a vertex in 𝐹 .The imperfect-information of the game is modelled by the equivalence relation ∼ that conveyswhich vertices Éloïse cannot distinguish, namely those that are ∼ -equivalent. We will write 𝑉 / ∼ forthe set of equivalence classes of ∼ in 𝑉 , and for every 𝑣 ∈ 𝑉 , we will write [ 𝑣 ] ∼ for its ∼ -equivalenceclass.Relation ∼ plays a crucial role when defining strategies for Éloïse. Intuitively, Éloïse should notplay differently in two indistinguishable plays, where the indistinguishability of Éloïse is based on perfect recall [14]: Éloïse cannot distinguish two histories 𝑣 in 𝑎 𝛿 𝑣 𝑎 𝛿 · · · 𝑣 ℓ and 𝑣 ′ in 𝑎 ′ 𝛿 ′ 𝑣 ′ 𝑎 ′ 𝛿 ′ · · · 𝑣 ′ ℓ whenever 𝑣 𝑖 ∼ 𝑣 ′ 𝑖 for all 𝑖 ≤ ℓ and 𝑎 𝑖 = 𝑎 ′ 𝑖 for all 𝑖 < ℓ . Note that in particular, Éloïse does not observeAbélard’s choices for the distributions along a play. Hence, a (pure ) strategy for Éloïse is a func-tion 𝜎 𝐸 : ( 𝑉 / ∼ · 𝐴 ) ∗ · ( 𝑉 / ∼ ) → 𝐴 assigning an action to every set of indistinguishable histories. Éloïse respects a strategy 𝜎 𝐸 during a play 𝜆 = 𝑣 in 𝑎 𝛿 𝑣 𝑎 𝛿 · · · if 𝑎 𝑖 + = 𝜎 𝐸 ( [ 𝑣 in ] ∼ 𝑎 [ 𝑣 ] ∼ · · · [ 𝑣 𝑖 ] ∼ ) , forall 𝑖 ≥ Again, as for perfect information games, we do not consider randomised strategies as pure strategies are the right modelfor our purpose. ACM Trans. Comput. Logic, Vol. 1, No. 1, Article . Publication date: December 2020.
R. Berthon et al.
A strategy for Abélard is defined as a function 𝜎 𝐴 : ( 𝑉 · 𝐴 · D ( 𝑉 )) ∗ ( 𝑉 · 𝐴 ) → D ( 𝑉 ) such that ( 𝑣, 𝑎, 𝜎 𝐴 ( 𝜆 · 𝑣 · 𝑎 )) ∈ 𝑇 for every 𝜆 ∈ ( 𝑉 · 𝐴 · D ( 𝑉 )) ∗ . Abélard respects a strategy 𝜎 𝐴 during a play 𝜆 = 𝑣 in 𝑎 𝛿 𝑣 𝑎 𝛿 · · · if 𝛿 𝑖 = 𝜎 𝐴 ( 𝑣 in 𝑎 𝛿 𝑣 𝑎 𝛿 · · · 𝑣 𝑖 𝑎 𝑖 ) , for all 𝑖 ≥ ( 𝜎 𝐸 , 𝜎 𝐴 ) the set Plays G 𝜎 𝐸 ,𝜎 𝐴 of those plays where Éloïse ( resp. Abélard) respects 𝜎 𝐸 ( resp. 𝜎 𝐴 ), and equip itwith a probability measure.Finally, a strategy 𝜎 𝐸 for Éloïse is almost-surely winning if, against any strategy 𝜎 𝐴 for Abélard,the set of winning plays for Éloïse has measure 1 for the probability measure on Plays G 𝜎 𝐸 ,𝜎 𝐴 . Remark 2.3.
It is important to note that Éloïse may not observe whether a vertex belongs to 𝐹 aswe do not require that 𝑣 ∼ 𝑣 ′ ⇒ ( 𝑣 ∈ 𝐹 ⇔ 𝑣 ′ ∈ 𝐹 ) . In particular, this has to be taken into accountwhen eventually solving the game.The following decidability result will be crucial in Section 3.2. Theorem 2.4 ([9, 10]).
Let G be an imperfect-information stochastic Büchi game. One can decidein exponential time whether Éloïse has an almost-surely winning strategy in G . Probabilistic automata on finite words generalize non-deterministic automata by letting the tran-sition function map a state and a letter to a distribution over states [28]. The reference book forearly developments on probabilistic automata is due to Paz [26].A probabilistic word automaton is a tuple A = ( 𝑄, 𝑞 in , 𝛿 ) , where 𝑄 is the finite set of states, 𝑞 in is the initial state, and 𝛿 : 𝑄 × Σ → D ( 𝑄 ) is the transition function. We say that a probabilisticautomaton is simple when the distribution 𝛿 ( 𝑞, 𝑎 ) is always of the form 𝑞 + 𝑞 (possibly with 𝑞 = 𝑞 ).Intuitively, a finite word 𝑢 = 𝑢 . . . 𝑢 𝑘 ∈ Σ ∗ induces a set of runs of A each of which comes witha probability of being realised; if one fixes a set of final states, the acceptance probability of 𝑢 by A is the mere sum of the probabilities of those runs of A over 𝑢 that end in a final state. Toformally define acceptance probability (and extend it further to richer settings) we associate with A and 𝑢 a Markov chain M 𝑢 A as follows.The Markov chain M 𝑢 A has the (finite) set of states 𝑄 × { , . . . , 𝑘 } , the initial state ( 𝑞 in , ) , andthe probability transition function 𝑇 𝑢 A defined for every ( 𝑝, 𝑖 ) ∈ 𝑄 ×{ , . . . , 𝑘 − } (we do not defineit for states of the form ( 𝑝, 𝑘 ) that will be useless) by 𝑇 𝑢 A (( 𝑝, 𝑖 )) = Õ 𝑞 ∈ 𝑄 𝛿 ( 𝑝, 𝑢 𝑖 ) ( 𝑞 ) · ( 𝑞, 𝑖 + ) Call a finite path of length 𝑘 + M 𝑢 A a run of A on 𝑢 and let 𝑃 𝑢 A be the probability measureon runs induced by M 𝑢 A . Given a subset of (final) states 𝐹 ⊆ 𝑄 , call Last ( 𝐹 ) the set of runs whose(first coordinate of the) last state is in 𝐹 . We then define the acceptance probability of A over 𝑢 as 𝑃 𝑢 A ( Last ( 𝐹 )) .A classic decision problem for probabilistic word automata is the value problem . INPUT:
A probabilistic word automaton A and a subset 𝐹 ⊆ 𝑄 QUESTION: ∀ 𝜀 > , ∃ 𝑢 ∈ Σ ∗ , 𝑃 𝑢 A ( Last ( 𝐹 )) ≥ − 𝜀 ?Informally, the value 1 problem asks for the existence of words with acceptance probabilities thatare arbitrarily close to 1. In this case, we say that A has value 1. The undecidability of the value 1problem for simple probabilistic automata was first established in [20] (see also [16] and [15] fora simple proof). ACM Trans. Comput. Logic, Vol. 1, No. 1, Article . Publication date: December 2020. lternating Tree Automata with Qualitative Semantics 9
Theorem 2.5 ([20]).
The value problem for simple probabilistic word automata is undecidable. Baier, Größer, and Bertrand conducted an in-depth study of probabilistic automata over infinitewords [1]. To define the semantics of a probabilistic word automaton A = ( 𝑄, 𝑞 in , 𝛿 ) over aninfinite word 𝑤 = 𝑤 𝑤 · · · , we proceed as for finite words and construct a Markov chain M 𝑤 A whose set of states is 𝑄 × N . The initial state is again ( 𝑞 in , ) , and the probability transition function 𝑇 𝑤 A is still defined by 𝑇 𝑤 A (( 𝑝, 𝑖 )) = Õ 𝑞 ∈ 𝑄 𝛿 ( 𝑝, 𝑤 𝑖 ) ( 𝑞 ) · ( 𝑞, 𝑖 + ) A run of A on 𝑤 is now an infinite path in M 𝑤 A and the Markov chain yields a probabilitymeasure 𝑃 𝑤 A on runs.For probabilistic automata on infinite words we mostly focus on the co-Büchi acceptance condi-tion that is defined as follows. Given a subset of states 𝐹 ⊆ 𝑄 , we let co-Büchi ( 𝐹 ) = ( 𝑄 × N ) ∗ (( 𝑄 \ 𝐹 ) × N ) 𝜔 be the (measurable) set of runs that visit 𝐹 only finitely often, and, when this set of runshas measure 1, we say that 𝑤 is almost-surely accepted by A for the co-Büchi condition 𝐹 , written 𝑤 ∈ 𝐿 = ( 𝐹 ) (A) . Formally, 𝐿 = ( 𝐹 ) (A) = (cid:8) 𝑤 ∈ Σ 𝜔 : 𝑃 𝑤 A ( co-Büchi ( 𝐹 )) = (cid:9) . Example 2.6.
Let Σ be an alphabet and ♯ ∉ Σ be a fresh symbol. Let C be the simple probabilisticco-Büchi automaton with set { 𝑝 , 𝑝 } of states, initial state 𝑝 , and transition function given by:- 𝛿 ( 𝑝 , 𝑎 ) = 𝑝 for any 𝑎 ∈ Σ \ (cid:8) ♯ (cid:9) ;- 𝛿 ( 𝑝 , ♯ ) = 𝑝 + 𝑝 ; and- 𝛿 ( 𝑝 , 𝑎 ) = 𝑝 for any 𝑎 ∈ Σ ∪ (cid:8) ♯ (cid:9) .As 𝑝 is absorbing and as moving from 𝑝 to 𝑝 may only happen when reading ♯ , the language 𝐿 = ({ 𝑝 }) (C) consists of those infinite words over Σ ∪ (cid:8) ♯ (cid:9) that contain infinitely many occur-rences of ♯ . Note that we will later use this example as a gadget in the proof of Proposition 4.1The emptiness problem for probabilistic co-Büchi word automata with almost-sure semanticsis the following decision problem: INPUT:
A probabilistic word automaton A and a set 𝐹 ⊆ 𝑄 QUESTION: Is 𝐿 = ( 𝐹 ) (A) = ∅ ?It was shown in [1] that this problem is undecidable. Proposition 2.7 ([1]).
The emptiness problem for probabilistic co-Büchi word automata withalmost-sure semantics is undecidable.
The proof in [1] 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 [1] fordefinitions) can be effectively complemented into probabilistic co-Büchi word automata with thealmost-sure semantics, and whose universality problem is proved to be undecidable. As the com-plementation procedure does not preserve the property of being simple, we will later argue (seeProposition 4.1) that Proposition 2.7 still holds for simple probabilistic co-Büchi word automatawith almost-sure semantics.
ACM Trans. Comput. Logic, Vol. 1, No. 1, Article . Publication date: December 2020.
The qualitative semantics for tree automata was introduced by Carayol, Haddad, and Serre in [8]and was studied for non-deterministic [8], alternating [17], and probabilistic automata [8].In this section, we define universal tree automata with qualitative semantics and then extendthis concept to alternating tree automata with qualitative semantics in the next section.A tree automaton is a tuple A = ( 𝑄, 𝑞 in , Δ ) , where 𝑄 is a finite set of states, 𝑞 in is the initialstate, and Δ ⊆ 𝑄 × Σ × 𝑄 × 𝑄 is the transition relation. A run of A over a Σ -tree 𝑡 is a 𝑄 -tree 𝜌 : { , } ∗ → 𝑄 such that 𝜌 ( 𝜀 ) = 𝑞 in and, for all 𝑢 ∈ { , } ∗ , we have ( 𝜌 ( 𝑢 ) , 𝑡 ( 𝑢 ) , 𝜌 ( 𝑢 ) , 𝜌 ( 𝑢 )) ∈ Δ .We let Runs A ( 𝑡 ) denote the set of runs of A over 𝑡 .A tree automaton A and a run 𝜌 induce a Markov chain M 𝜌 A as follows. The set of states is 𝑄 × { , } ∗ , the initial state is ( 𝑞 in , 𝜀 ) , and the probability transition function 𝑇 𝜌 A is given by 𝑇 𝜌 A (( 𝜌 ( 𝑢 ) , 𝑢 )) = ( 𝜌 ( 𝑢 ) ,𝑢 ) + ( 𝜌 ( 𝑢 ) ,𝑢 ) yielding the probability measure 𝑃 𝜌 A on branches of the run 𝜌 .Given a subset of states 𝐹 ⊆ 𝑄 , we let co-Büchi ( 𝐹 ) = ( 𝑄 × { , } ∗ ) ∗ (( 𝑄 \ 𝐹 ) × { , } ∗ ) 𝜔 be the(measurable) set of infinite paths in M 𝜌 A that visit 𝐹 only finitely often, and we say that the run 𝜌 is qualitatively accepting for the co-Büchi condition 𝐹 if 𝑃 𝜌 A ( co-Büchi ( 𝐹 )) =
1. Equivalently, a run 𝜌 is qualitatively accepting for the co-Büchi condition if and only if the set of branches in 𝜌 thatcontain finitely many nodes labelled by a state in 𝐹 has measure 1 for the classical coin-flipping measure 𝜇 on branches: 𝜇 is the unique complete probability measure such that 𝜇 ( 𝑢 · { , } 𝜔 ) = −| 𝑢 | .The universal semantics yields the following definition: 𝐿 ∀ Qual , co-Büchi ( 𝐹 ) (A) = (cid:8) 𝑡 ∈ Trees ( Σ ) : ∀ 𝜌 ∈ Runs A ( 𝑡 ) , 𝑃 𝜌 A ( co-Büchi ( 𝐹 )) = (cid:9) . In words, a tree 𝑡 belongs to 𝐿 ∀ Qual , co-Büchi ( 𝐹 ) (A) if every run of A over 𝑡 is such that almost allits branches contain finitely many states in 𝐹 .The emptiness problem for universal co-Büchi tree automata with qualitative semantics is thefollowing decision problem: INPUT:
A tree automaton A and a set 𝐹 ⊆ 𝑄 QUESTION: Is 𝐿 ∀ Qual , co-Büchi ( 𝐹 ) (A) = ∅ ?We will prove in Theorem 4.2 that this problem is undecidable. An alternating tree automaton is a tuple A = ( 𝑄, 𝑞 in , 𝑄 𝐸 , 𝑄 𝐴 , Δ ) , where 𝑄 is the finite set ofstates, 𝑞 in is the initial state, ( 𝑄 𝐸 , 𝑄 𝐴 ) is a partition of 𝑄 into Éloïse’s and Abélard’s states and Δ ⊆ 𝑄 × Σ × 𝑄 × 𝑄 is the transition relation.The input of such an automaton is a Σ -tree 𝑡 and acceptance is defined by means of the followingtwo-player perfect-information stochastic game G = A ,𝑡 . Intuitively, a play in this game consists inmoving a pebble along a branch of 𝑡 starting from the root: the pebble is attached to a state and ina node 𝑢 with state 𝑞 , Éloïse (if 𝑞 ∈ 𝑄 𝐸 ) or Abélard (if 𝑞 ∈ 𝑄 𝐴 ) picks a transition ( 𝑞, 𝑡 ( 𝑢 ) , 𝑞 , 𝑞 ) ∈ Δ ,and then Random chooses to move down the pebble either to node 𝑢 𝑞 ) or to node 𝑢 𝑞 ). ACM Trans. Comput. Logic, Vol. 1, No. 1, Article . Publication date: December 2020. lternating Tree Automata with Qualitative Semantics 11
Formally, let 𝐺 = ( 𝑉 𝐸 ∪ 𝑉 𝐴 ∪ 𝑉 𝑅 , 𝐸 ) with 𝑉 𝐸 = 𝑄 𝐸 × { , } ∗ , 𝑉 𝐴 = 𝑄 𝐴 × { , } ∗ and 𝑉 𝑅 = {( 𝑞, 𝑢, 𝑞 , 𝑞 ) | 𝑢 ∈ { , } ∗ and ( 𝑞, 𝑡 ( 𝑢 ) , 𝑞 , 𝑞 ) ∈ Δ } , and 𝐸 = {(( 𝑞, 𝑢 ) , ( 𝑞, 𝑢, 𝑞 , 𝑞 )) | ( 𝑞, 𝑢, 𝑞 , 𝑞 ) ∈ 𝑉 𝑅 } ∪{(( 𝑞, 𝑢, 𝑞 , 𝑞 ) , ( 𝑞 𝑥 , 𝑢 · 𝑥 )) | 𝑥 ∈ { , } and ( 𝑞, 𝑢, 𝑞 , 𝑞 ) ∈ 𝑉 𝑅 } Then, we define G = A ,𝑡 = ( 𝐺, 𝑉 𝐸 , 𝑉 𝐴 , 𝑉 𝑅 , 𝛿, ( 𝑞 in , 𝜀 )) where 𝛿 (( 𝑞, 𝑢, 𝑞 , 𝑞 )) = ( 𝑞 , 𝑢 ) + ( 𝑞 , 𝑢 ) .Given a subset of states 𝐹 ⊆ 𝑄 , we say that 𝑡 is qualitatively accepted by A for the Büchi ( resp. co-Büchi) condition 𝐹 if Éloïse has an almost-surely winning strategy in the Büchi ( resp. co-Büchi)game G = A ,𝑡 = (G = A ,𝑡 , 𝐹 × { , } ∗ ) .For an alternating tree automaton A and a subset of states 𝐹 , we denote by 𝐿 AltQual , Büchi ( 𝐹 ) (A) ( resp. 𝐿 AltQual , co-Büchi ( 𝐹 ) (A) ) the set of trees qualitatively accepted by A for the Büchi ( resp. co-Büchi)condition 𝐹 . Remark 2.8.
Any positional strategy for Éloïse in G = A ,𝑡 can be described as a function 𝜎 : 𝑄 𝐸 ×{ , } ∗ → 𝑄 × 𝑄 that satisfies the following property: ∀ 𝑢 ∈ { , } ∗ , if 𝜎 ( 𝑞, 𝑢 ) = ( 𝑞 , 𝑞 ) then ( 𝑞, 𝑡 ( 𝑢 ) , 𝑞 , 𝑞 ) ∈ Δ . Equivalently, in a curried form, 𝜎 is a map { , } ∗ → ( 𝑄 𝐸 → 𝑄 × 𝑄 ) . Hence, ifone lets T be the set of functions from 𝑄 𝐸 into 𝑄 × 𝑄 , Éloïse’s positional strategies are in bijectionwith T -labelled binary trees.It is easily seen that universal tree automata with qualitative semantics are subsumed by al-ternating 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 otheracceptance condition). Proposition 2.9.
Let A = ( 𝑄, 𝑞 in , Δ ) be a tree automaton and let 𝐹 ⊆ 𝑄 . Consider the alternatingtree automaton B = ( 𝑄, 𝑞 in , ∅ , 𝑄, Δ ) , meaning that all states of A are interpreted as Abélard’s. Thenthe following holds. 𝐿 ∀ Qual , co-Büchi ( 𝐹 ) (A) = 𝐿 AltQual , co-Büchi ( 𝐹 ) (B) Proof.
For a fixed tree 𝑡 , runs of A over 𝑡 are in bijection with strategies of Abélard in theco-Büchi game G = B ,𝑡 (where Éloïse is making no choice), and moreover a run is qualitatively ac-cepting for A if and only if Éloïse almost-surely wins in G = B ,𝑡 when Abélard uses the correspond-ing strategy. Hence, all runs of A over 𝑡 are qualitatively accepting if and only if Éloïse almost-surely wins against every strategy of Abélard in G = B ,𝑡 , which means that 𝐿 ∀ Qual , co-Büchi ( 𝐹 ) (A) = 𝐿 AltQual , co-Büchi ( 𝐹 ) (B) . (cid:3) The emptiness problem for alternating Büchi tree automata with qualitative semantics is thefollowing decision problem:
INPUT:
An alternating tree automaton A and a set 𝐹 ⊆ 𝑄 QUESTION: Is 𝐿 AltQual , Büchi ( 𝐹 ) (A) = ∅ ?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 au-tomata with qualitative semantics. However, this problem is undecidable as a corollary of Proposi-tion 2.9 together with the forthcoming Theorem 4.2, proving the undecidability of the emptinessproblem for universal co-Büchi tree automata with qualitative semantics.
ACM Trans. Comput. Logic, Vol. 1, No. 1, Article . Publication date: December 2020.
Probabilistic tree automata with qualitative semantics were defined in [8] with the intention oflifting the definition of probabilistic automata on infinite words to the case of infinite trees. Inparticular, an input tree induces a probability distribution over runs and acceptance is defined byrequiring 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 containingfinitely many states from 𝐹 . Contrary to the authors of [8] who define a probability measure onruns, we follow another approach (still yielding an equivalent notion [8, Proposition 45]) based onMarkov chains.A probabilistic tree automaton is a tuple A = ( 𝑄, 𝑞 in , 𝛿 ) , where 𝑄 is the finite set of states, 𝑞 in is the initial state, and 𝛿 : 𝑄 × Σ → D ( 𝑄 × 𝑄 ) is the transition function.A probabilistic tree automaton A and a tree 𝑡 induce a Markov chain M 𝑡 A as follows. The set ofstates is 𝑄 × { , } ∗ , the initial state is ( 𝑞 in , 𝜀 ) , and the probability transition function 𝑇 𝑡 A is givenby (where · distributes over + ) 𝑇 𝑡 A (( 𝑞, 𝑢 )) = Õ 𝑞 ,𝑞 ∈ 𝑄 𝛿 ( 𝑞, 𝑡 ( 𝑢 )) ( 𝑞 , 𝑞 ) · (cid:18) ·( 𝑞 , 𝑢 ) + ·( 𝑞 , 𝑢 ) (cid:19) , Given a subset of states 𝐹 ⊆ 𝑄 , we again let co-Büchi ( 𝐹 ) = ( 𝑄 × { , } ∗ ) ∗ (( 𝑄 \ 𝐹 ) × { , } ∗ ) 𝜔 bethe (measurable) set of infinite paths in M 𝑡 A that visit 𝐹 only finitely often. Then the probabilitymeasure 𝑃 𝑡 A induced by M 𝑡 A yields the following definition of the set of trees almost-surelyqualitatively accepted by A : 𝐿 ∀ = Qual , co-Büchi ( 𝐹 ) (A) = (cid:8) 𝑡 ∈ Trees ( Σ ) : 𝑃 𝑡 A ( co-Büchi ( 𝐹 )) = (cid:9) . We now turn to our main decidability result about emptiness of alternating Büchi tree automatawith qualitative semantics.
In this section, we prove Theorem 3.7 that states the decidability of the emptiness problem foralternating Büchi tree automata with qualitative semantics, which contrasts with the forthcomingresult that the emptiness problem for universal co-Büchi tree automata with qualitative semanticsis undecidable (Theorem 4.2 of Section 4).Our approach for checking emptiness of an alternating Büchi tree automaton A with qualitativesemantics 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 A is non-empty. As for this classof games, one can decide whether Éloïse has an almost-surely winning strategy, the announceddecidability result follows.We establish in Section 3.1 a preliminary general positionality result to be used in Section 3.2for proving the equivalence between Éloïse almost-surely winning in the game and A acceptingsome tree. For the rest of this section, we fix a stochastic arena G = ( 𝐺, 𝑉 𝐸 , 𝑉 𝐴 , 𝑉 𝑅 , 𝛿, 𝑣 in ) with 𝐺 = ( 𝑉 , 𝐸 ) .Moreover, we assume that the game is chronological in the sense that there exists a function rank : 𝑉 𝐸 ∪ 𝑉 𝐴 ∪ 𝑉 𝑅 → N such that rank − ( ) = { 𝑣 in } and for ( 𝑣, 𝑣 ′ ) ∈ 𝐸 , rank ( 𝑣 ′ ) = rank ( 𝑣 ) + G = A ,𝑡 used in Section 2.6 to define acceptance of a tree 𝑡 by an alternating tree ACM Trans. Comput. Logic, Vol. 1, No. 1, Article . Publication date: December 2020. lternating Tree Automata with Qualitative Semantics 13 automaton with qualitative semantics A is chronological. Note also that a chronological arenawith finite out-degree has a countable set of vertices. Theorem 3.1.
In a two-player perfect-information stochastic Büchi game played on a chronologicalarena with finite out-degree, Éloïse has an almost-surely winning strategy if, and only if, she has apositional 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 chrono-logical arena with finite out-degree, Éloïse has an almost-surely winning strategy if, and only if, shehas a positional almost-surely winning strategy.
Proof.
The direction from right to left is immediate. For the other direction, the key steps arethe following. First, we establish (Lemma 3.3) that if Éloïse can ensure to reach 𝐹 with probability 1from some initial vertex, then there exists a bound 𝑘 such that she can ensure to reach 𝐹 withprobability at least half within 𝑘 steps. Second, we exploit Lemma 3.3 to “slice” the arena intoinfinitely many disjoint finite arenas: in each slice Éloïse plays to reach 𝐹 with probability at leasthalf. 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 𝐹 inthe long run.Let G = (G , 𝐹 ) be a two-player perfect-information stochastic reachability game played on achronological arena with finite out-degree. In the following, a strategy in G from a vertex 𝑣 is astrategy in the game obtained from G by changing the initial vertex of the arena G to 𝑣 .The following lemma allows us to decompose the infinite arena G into infinitely many finitearenas. Lemma 3.3.
Let 𝜎 𝐸 be an almost-surely winning strategy for Éloïse in G from some vertex 𝑣 . Then,there exists an integer 𝑘 such that for any strategy 𝜎 𝐴 of Abélard, we have Pr 𝜎 𝐸 ,𝜎 𝐴 ( 𝑉 ≤ 𝑘 𝐹𝑉 𝜔 ) ≥ . Proof of Lemma 3.3.
Toward a contradiction, assume that such a 𝑘 does not exist. Hence, foreach 𝑘 there exists a strategy 𝜎 𝐴,𝑘 such that Pr 𝜎 𝐸 ,𝜎 𝐴,𝑘 ( 𝑉 ≤ 𝑘 𝐹𝑉 𝜔 ) < .Without loss of generality, we can assume that 𝜎 𝐴,𝑘 is positional. Indeed, one can pick for 𝜎 𝐴,𝑘 astrategy for Abélard that minimises the probability of winning for Éloïse in the reachabililty gameobtained by restricting G to vertices of rank at most 𝑘 . This game has a finite arena since G hasfinite out-degree, and by e.g. [21] such a strategy for Abélard can be chosen positional.From the sequence of strategies ( 𝜎 𝐴,𝑘 ) 𝑘 ≥ , we now extract a strategy 𝜎 𝐴, ∞ (designed to contradictthe assumption that Éloïse has an almost-surely winning strategy) that for every 𝑘 ≥
0, agrees withinfinitely many 𝜎 𝐴,ℎ on its first 𝑘 moves. Since G has countably many vertices, fix an (arbitrary)enumeration 𝑣 , 𝑣 , · · · of the vertices in 𝑉 .We define 𝜎 𝐴, ∞ step-wise inductively on 𝑖 : at step 𝑖 , 𝜎 𝐴, ∞ is defined on 𝑣 , · · · , 𝑣 𝑖 and on thesevertices agrees with all those strategies 𝜎 𝐴,ℎ with ℎ ∈ 𝐼 𝑖 where the sequence 𝐼 ⊇ 𝐼 ⊇ 𝐼 ⊇ 𝐼 ⊇ · · · is also defined inductively on 𝑖 and is such that each 𝐼 𝑖 is infinite.We let 𝐼 = N be the set of all positive integers.For 𝐼 𝑖 where 𝑖 ≥
1, consider the values of 𝜎 𝐴,ℎ ( 𝑣 𝑖 ) for all ℎ ∈ 𝐼 𝑖 − . Because 𝐺 has finite out-degree,there is some 𝑣 such that 𝜎 𝐴,ℎ ( 𝑣 𝑖 ) = 𝑣 , for infinitely many ℎ ∈ 𝐼 𝑖 − . We define 𝜎 𝐴, ∞ ( 𝑣 𝑖 ) = 𝑣 and welet 𝐼 𝑖 = (cid:8) ℎ ∈ 𝐼 𝑖 − | 𝜎 𝐴,ℎ ( 𝑣 𝑖 ) = 𝑣 (cid:9) ; note that 𝐼 𝑖 is infinite.Now, for 𝑘 ≥
0, it is easy to see that by choosing 𝑖 big enough so that all vertices of rank atmost 𝑘 belong to { 𝑣 , . . . , 𝑣 𝑖 } , strategy 𝜎 𝐴, ∞ agrees on its 𝑘 first moves with the infinitely many 𝜎 𝐴,ℎ where ℎ ∈ 𝐼 𝑖 . ACM Trans. Comput. Logic, Vol. 1, No. 1, Article . Publication date: December 2020.
As a consequence, for every 𝑘 there is some ℎ ≥ 𝑘 such thatPr 𝜎 𝐸 ,𝜎 𝐴, ∞ ( 𝑉 ≤ 𝑘 𝐹𝑉 𝜔 ) = Pr 𝜎 𝐸 ,𝜎 𝐴,ℎ ( 𝑉 ≤ 𝑘 𝐹𝑉 𝜔 ) ≤ Pr 𝜎 𝐸 ,𝜎 𝐴,ℎ ( 𝑉 ≤ ℎ 𝐹𝑉 𝜔 ) < 𝑉 ∗ 𝐹𝑉 𝜔 = Ð 𝑘 ≥ 𝑉 ≤ 𝑘 𝐹𝑉 𝜔 and as the sequence ( 𝑉 ≤ 𝑘 𝐹𝑉 𝜔 ) 𝑘 ≥ is increasing for set inclusion, oneconcludes that Pr 𝜎 𝐸 ,𝜎 𝐴, ∞ ( 𝑉 ∗ 𝐹𝑉 𝜔 ) = lim 𝑘 →∞ Pr 𝜎 𝐸 ,𝜎 𝐴, ∞ ( 𝑉 ≤ 𝑘 𝐹𝑉 𝜔 ) ≤ < 𝜎 𝐸 being almost-surely winning, and concludes the proof ofLemma 3.3. (cid:3) Keeping on with the proof of Theorem 3.2, assume that Éloïse has an almost-surely winingstrategy 𝜎 𝐸 in G . Without loss of generality, we can assume that she has an almost-surely win-ning strategy from everywhere, by restricting the arena to vertices reachable by an almost-surelywinning strategy.For 𝑘 < 𝑘 ′ , we define the reachability game G [ 𝑘,𝑘 ′ ] induced by restricting the arena G = ( 𝐺, 𝑉 𝐸 , 𝑉 𝐴 , 𝑉 𝑅 , 𝛿, 𝑣 in ) to vertices of rank in [ 𝑘, 𝑘 ′ ] where we add self-loops on vertices of rank 𝑘 ′ to avoid having dead-end vertices. Since 𝐺 has finite out-degree, there are finitely many verticesof rank in [ 𝑘, 𝑘 ′ ] , hence G [ 𝑘,𝑘 ′ ] is finite.We define inductively an increasing sequence of ranks ( 𝑘 𝑖 ) 𝑖 ≥ together with a sequence of strate-gies ( 𝜎 𝐸, [ 𝑘 𝑖 ,𝑘 𝑖 + [ ) 𝑖 ≥ such that for all 𝑖 ≥ 𝜎 𝐸, [ 𝑘 𝑖 ,𝑘 𝑖 + [ is a positional strategy, defined on all verticesof rank in [ 𝑘 𝑖 , 𝑘 𝑖 + [ , and such that from all vertices of rank 𝑘 𝑖 , for all strategies 𝜎 𝐴 , we havePr 𝜎 𝐸, [ 𝑘𝑖,𝑘𝑖 + [ ,𝜎 𝐴 ( 𝑉 ≤ ℓ 𝐹𝑉 𝜔 ) ≥ , where ℓ = 𝑘 𝑖 + − 𝑘 𝑖 .Assume the first 𝑖 ranks and strategies are defined. For each vertex of rank 𝑘 𝑖 , Lemma 3.3 givesthe existence of some bound 𝑘 ; since there are finitely many such vertices, we can consider themaximum of those bounds that we call ℓ , and we let 𝑘 𝑖 + = 𝑘 𝑖 + ℓ . By construction and Lemma 3.3,from all vertices of rank 𝑘 𝑖 , for all strategies 𝜎 𝐴 , we havePr 𝜎 𝐸 ,𝜎 𝐴 ( 𝑉 ≤ ℓ 𝐹𝑉 𝜔 ) ≥ , where ℓ = 𝑘 𝑖 + − 𝑘 𝑖 . In other words, Éloïse wins the reachability game G [ 𝑘 𝑖 ,𝑘 𝑖 + ] with probabilityat least half, so, relying on a generalisation of Theorem 2.2 (see e.g. [21, 23]), there exists anoptimal uniform (i.e. working from any initial vertex) positional strategy, that we call 𝜎 𝐸, [ 𝑘 𝑖 ,𝑘 𝑖 + [ .This concludes the inductive construction.Now, define 𝜎 𝐸, ∞ as the disjoint union of the strategies 𝜎 𝐸, [ 𝑘 𝑖 ,𝑘 𝑖 + [ . This is a positional strategy;we argue that it is almost-surely winning. Assume, towards a contradiction, that this is not thecase. Then, there exists 𝜀 > 𝜎 𝐴 such thatPr 𝜎 𝐸, ∞ ,𝜎 𝐴 ( 𝑉 ∗ 𝐹𝑉 𝜔 ) ≤ − 𝜀 . Observe that playing consistently with the first 𝑝 strategies 𝜎 𝐸, [ 𝑘 𝑖 ,𝑘 𝑖 + [ ensures to reach 𝐹 withprobability at least 1 − 𝑝 . Since playing consistently with 𝜎 𝐸, ∞ implies playing consistently withthe first 𝑝 strategies 𝜎 𝐸, [ 𝑘 𝑖 ,𝑘 𝑖 + [ , we reach a contradiction by considering 𝑝 large enough so that 𝑝 < 𝜀 . (cid:3) More precisely, when playing a reachability game on a finite arena, Éloïse always has an optimal positional strategy, where 𝜎 𝐸 being optimal means that inf 𝜎 𝐴 Pr 𝜎 𝐸 ,𝜎 𝐴 ( 𝑉 ≤ ℓ 𝐹𝑉 𝜔 ) = sup 𝜎 ′ 𝐸 inf 𝜎 𝐴 Pr 𝜎 𝐸 ,𝜎 ′ 𝐴 ( 𝑉 ≤ ℓ 𝐹𝑉 𝜔 ) .ACM Trans. Comput. Logic, Vol. 1, No. 1, Article . Publication date: December 2020. lternating Tree Automata with Qualitative Semantics 15 Theorem 3.1 is an easy consequence of Theorem 3.2 thanks to a simple and neat reductionfrom [10, Remark 2.3] (also see [1, Lemma 8.3]). Roughly speaking, to turn a Büchi game into areachability game equivalent with respect to almost-sure winning, one adds a unique final vertexand replaces every Büchi vertex by a fresh random vertex which either reaches the final vertexor proceeds in the game, each with probability half. Then, visiting infinitely many Büchi verticesensures to almost-surely reach the final vertex, and conversely, reaching almost-surely the finalvertex requires to almost-surely visit infinitely many Büchi vertices.We make all this more formal.
Proof of Theorem 3.1.
Recall that we denote by G = ( 𝐺, 𝑉 𝐸 , 𝑉 𝐴 , 𝑉 𝑅 , 𝛿, 𝑣 in ) , with 𝐺 = ( 𝑉 , 𝐸 ) , theunderlying arena of G and denote by 𝐹 ⊆ 𝑉 the set of vertices defining the Büchi condition. Wenow build an arena G ′ = ( 𝐺 ′ , 𝑉 ′ 𝐸 , 𝑉 ′ 𝐴 , 𝑉 ′ 𝑅 , 𝛿 ′ , 𝑣 ′ in ) , with 𝐺 ′ = ( 𝑉 ′ , 𝐸 ′ ) , and a set 𝐹 ′ ⊆ 𝑉 ′ of verticessuch that Éloïse almost-surely wins in the Büchi game G = (G , 𝐹 ) if and only if she almost-surelywins in the reachability game G ′ = (G ′ , 𝐹 ′ ) , and in addition, if she has a positional almost-surelywinning strategy in one game, she has one in the other. This permits to deduce Theorem 3.1 fromTheorem 3.2.We formally explain how to construct G ’, taking care that it is chronological. The set of vertices 𝑉 ′ consists of 𝑉 augmented with a countable set of vertices { 𝑓 𝑖 | 𝑖 ≥ } , and with extra randomvertices 𝐹 𝑅 = { 𝑣 𝑠 | 𝑠 ∈ 𝐹 } , one per vertex in 𝐹 . The vertex 𝑓 has a unique outgoing transitionto 𝑓 and it can be reached only from vertices in 𝐹 𝑅 . For every 𝑖 ≥
1, the vertex 𝑓 𝑖 has a uniqueoutgoing transition to 𝑓 𝑖 + and it can be reached only from 𝑓 𝑖 − . From a vertex 𝑣 𝑠 ∈ 𝐹 𝑅 there aretwo outgoing edges: one to 𝑓 and one to 𝑠 and both can be chosen with the same probability half,i.e. 𝛿 ′ ( 𝑣 𝑠 ) = 𝑓 + 𝑠 . Any edge in 𝐺 going from a vertex 𝑣 ∈ 𝑉 to a vertex 𝑠 ∈ 𝐹 is replaced by anedge from 𝑣 to 𝑣 𝑠 , and if 𝑣 ∈ 𝑉 𝑅 we let 𝛿 ′ ( 𝑣 ) ( 𝑠 ) = 𝛿 ′ ( 𝑣 ) ( 𝑣 𝑠 ) = 𝛿 ( 𝑣 ) ( 𝑠 ) . All other edges areleft untouched: for every 𝑣 ∈ 𝑉 𝑅 and 𝑠 ∉ 𝐹 ∪ 𝐹 𝑅 ∪ { 𝑓 𝑖 | 𝑖 ≥ } , we let 𝛿 ′ ( 𝑣 ) ( 𝑠 ) = 𝛿 ( 𝑣 ) ( 𝑠 ) . Finally welet 𝑉 ′ 𝐸 = 𝑉 𝐸 ∪ { 𝑓 𝑖 | 𝑖 ≥ } , 𝑉 ′ 𝐴 = 𝑉 𝐴 , 𝑉 ′ 𝑅 = 𝑉 𝑅 ∪ 𝐹 𝑅 and 𝐹 ′ = { 𝑓 } . Note that G ′ is chronological byconstruction and G being chronological.There is an obvious correspondence between strategies (of both Éloïse and Abélard) in G andstrategies in G ′ , and it preserves positionality. Moreover, Éloïse almost-surely reaches the finalstate 𝑓 in G ′ with strategy 𝜎 ′ 𝐸 if and only if she almost-surely visits infinitely often 𝐹 in G withthe corresponding strategy 𝜎 𝐸 . Indeed, if she almost-surely visits 𝐹 in G using 𝜎 𝐸 , due to positivetransition probability to 𝑓 from states in 𝐹 , she almost-surely reaches 𝑓 in G ′ using 𝜎 ′ 𝐸 . Conversely,if against any strategy 𝜎 𝐸 of Éloïse in G , Abélard has a strategy 𝜎 𝐴 that ensures that 𝐹 is visitedfinitely often with some positive probability 𝜀 >
0, then in G ′ , when Éloïse and Abélard use thecorresponding pair of strategies ( 𝜎 ′ 𝐸 , 𝜎 ′ 𝐴 ) , there is a positive probability 𝜀 ′ that 𝑓 is never reached,as the only way of reaching 𝑓 is by going through 𝐹 ; hence, in G ′ , against any strategy 𝜎 ′ 𝐸 of Éloïse,Abélard has a strategy 𝜎 ′ 𝐴 that avoids reaching 𝑓 with positive probability. (cid:3) Remark 3.4.
As already announced, in this paper we only considered pure (i.e. non-randomised)strategies. Hence, “Éloïse has an almost-surely winning strategy” should be understood in bothTheorem 3.2 and Theorem 3.1 as “Éloïse has an almost-surely winning pure strategy”. However,our proof directly carries over to the more general case of randomised strategies.
Fix an alternating tree automaton A = ( 𝑄, 𝑞 in , 𝑄 𝐸 , 𝑄 𝐴 , Δ ) and a subset 𝐹 ⊆ 𝑄 of final states. Inorder to check whether 𝐿 AltQual , Büchi ( 𝐹 ) (A) = ∅ , we design an imperfect-information stochastic Büchigame G ∅A in which Éloïse has an almost-surely winning strategy if and only if 𝐿 AltQual , Büchi ( 𝐹 ) (A) ≠ ∅ . ACM Trans. Comput. Logic, Vol. 1, No. 1, Article . Publication date: December 2020.
The equivalence is proved by applying the positionality result established in Theorem 3.1 to theacceptance game for A .In the game, Éloïse describes both a tree 𝑡 and a positional strategy 𝜎 𝑡 for her in the game G = A ,𝑡 .Following Remark 2.8, the positional strategy 𝜎 𝑡 is described as a T -labelled tree, where T denotesthe set of functions from 𝑄 𝐸 into 𝑄 × 𝑄 . As the plays are of 𝜔 -length, Éloïse actually does not fullydescribe 𝑡 and 𝜎 𝑡 but only a branch: this branch is chosen by Random while Abélard takes care ofcomputing the sequence of states along it (either by updating an existential state according to 𝜎 𝑡 or, when the state is universal, by choosing an arbitrary valid transition of the automaton). In thisgame, Éloïse observes the directions, but not the actual control state of the automaton. Remark 3.5.
The fact that Éloïse does not observe the control state of the automaton is crucialhere, as it avoids her to cheat when describing the input tree. Indeed, consider an alternatingtree automaton whose initial state belongs to Abélard and from which there are two possibletransitions: one that makes the automaton check that both subtrees only contain nodes labelledby 𝑎 , and one that makes the automaton check that both subtrees only contain nodes labelled by 𝑏 . Trivially, no tree is accepted by such an automaton. However, if one plays a modified versionof the previous game where Éloïse observes the control state she can surely win in this game byproducing a tree with all nodes labeled by 𝑎 ( resp. by 𝑏 ) depending on the initial choice by Abélard.Formally, we let G ∅A = ( 𝑉 , 𝐴,𝑇 , ∼ , 𝑣 in ) where • 𝑉 = ( 𝑄 × { , }) ∪ {( 𝑞 in , 𝜀 )} ; • 𝑣 in = ( 𝑞 in , 𝜀 ) ; • 𝐴 ⊆ Σ × T is the set {( 𝑎, 𝜏 ) | ∀ 𝑞 ∈ 𝑄 𝐸 , ( 𝑞, 𝑎, 𝑞 , 𝑞 ) ∈ Δ where ( 𝑞 , 𝑞 ) = 𝜏 ( 𝑞 )} ; • 𝑇 = {(( 𝑞, 𝑖 ) , ( 𝑎, 𝜏 ) , 𝑑 𝑞 ,𝑞 ) | 𝑞 ∈ 𝑄 𝐸 and 𝜏 ( 𝑞 ) = ( 𝑞 , 𝑞 )} ∪{(( 𝑞, 𝑖 ) , ( 𝑎, 𝜏 ) , 𝑑 𝑞 ,𝑞 ) | 𝑞 ∈ 𝑄 𝐴 and ( 𝑞, 𝑎, 𝑞 , 𝑞 ) ∈ Δ } where 𝑑 𝑞 ,𝑞 = ( 𝑞 , ) + ( 𝑞 , ) ; and • ( 𝑞, 𝑖 ) ∼ ( 𝑞 ′ , 𝑖 ) for all 𝑞, 𝑞 ′ ∈ 𝑄 and 𝑖 ∈ { , } .Finally we let G ∅A = (G ∅A , 𝐹 × { , }) .The following theorem relates G ∅A and 𝐿 AltQual , Büchi ( 𝐹 ) (A) . Theorem 3.6.
Éloïse almost-surely wins in G ∅A iff 𝐿 AltQual , Büchi ( 𝐹 ) (A) ≠ ∅ . Proof.
Due to how ∼ is defined, a strategy for Éloïse in G ∅A can also be viewed as a map 𝜎 : { , } ∗ → 𝐴 . As 𝐴 ⊆ Σ × T , one can see 𝜎 as a pair ( 𝑡, 𝜎 𝑡 ) where 𝑡 is an infinite Σ -labelled binarytree, and 𝜎 𝑡 is a positional strategy for Éloïse in the acceptance game G = A ,𝑡 . Now, once such astrategy 𝜎 is fixed, the set of plays in G ∅A where Éloïse respects 𝜎 is in one-to-one correspondencewith the set of plays in G = A ,𝑡 where she respects 𝜎 𝑡 , and this correspondence preserves the propertyof being a winning play. Therefore, 𝜎 = ( 𝑡, 𝜎 𝑡 ) is almost-surely winning in G ∅A iff 𝜎 𝑡 is an almost-surely winning positional strategy in G = A ,𝑡 iff 𝑡 ∈ 𝐿 AltQual , Büchi ( 𝐹 ) (A) . The last equivalence holdsbecause, thanks to Theorem 3.1, we can restrict our attention to positional strategies for Éloïsein the perfect-information game G = A ,𝑡 which, we recall, is chronological and of course has finiteout-degree. Finally, Éloïse has an almost-surely winning strategy in G ∅A iff there exists some tree 𝑡 ∈ 𝐿 AltQual , Büchi ( 𝐹 ) (A) . (cid:3) Combining Theorem 3.6 with Theorem 2.4 directly implies decidability of the emptiness problemfor alternating Büchi tree automata with qualitative semantics.
Theorem 3.7.
The emptiness problem for alternating Büchi tree automata with qualitative seman-tics is decidable in exponential time.
ACM Trans. Comput. Logic, Vol. 1, No. 1, Article . Publication date: December 2020. lternating Tree Automata with Qualitative Semantics 17
Regarding lower bound, following the same ideas as in the undecidability proof in Theorem 4.2,one can reduce the emptiness problem for simple probabilistic
Büchi automata with almost-suresemantics to the emptiness problem for universal Büchi tree automata with qualitative semantics.
Theorem 3.8.
The emptiness problem for universal Büchi tree automata with qualitative semanticsis hard for ExpTime.
Proof.
Similarly to what was done in Section 2.4 for the co-Büchi acceptance condition, wedefine a probabilistic
Büchi automaton with almost-sure semantics on infinite words: for a proba-bilistic automaton A = ( 𝑄, 𝑞 in , 𝛿 ) and a subset of states 𝐹 ⊆ 𝑄 , we let Büchi ( 𝐹 ) = Ñ 𝑖 ≥ 𝑄 𝑖 𝑄 ∗ 𝐹𝑄 𝜔 be the (measurable) set of runs that visit 𝐹 infinitely often. We then let: 𝐿 = ( 𝐹 ) (A) = (cid:8) 𝑤 ∈ Σ 𝜔 : 𝑃 𝑤 A ( Büchi ( 𝐹 )) = (cid:9) . The emptiness problem for probabilistic Büchi word automata with almost-sure semantics isthe following decision problem:
INPUT:
A probabilistic word automaton A and a set 𝐹 ⊆ 𝑄 QUESTION: Is 𝐿 = ( 𝐹 ) (A) = ∅ ?It is proved in [1] that this problem is complete for ExpTime. Moreover, this result still holds withthe extra requirement that the automata are simple. Indeed, the lower bound in [1] is by reductionof the almost-sure repeated reachability for partial-observation Markov decision processes. Thislatter problem was shown to be ExpTime-complete by de Alfaro [13]. The hardness proof in [13],based on the concept of blindfold games as defined by Reif in his seminal paper [31], survives (withthe same proof) if the branching in the partial-observation Markov decision process has at mosttwo states. Consequently, hardness for ExpTime already holds for probabilistic automata whosedistributions involved in the transition function have a support of at most two states. Finally, asobserved in [1, Remark 8.9], emptiness is not affected by changing the probabilities in the distribu-tions as long as the support is unchanged: therefore, one can always reduce to the case of simpleautomata.Now, following exactly the same path as in Theorem 4.2 one proves that the emptiness problemfor simple probabilistic Büchi automata with almost-sure semantics can be polynomially reducedto the emptiness problem for universal Büchi tree automata with qualitative semantics, whichimplies the announced lower-bound. (cid:3) In this section we prove our main undecidability result on the emptiness problem for universal co-Büchi tree automata with qualitative semantics, from which we will then derive the undecidabilityof MSO+ ∀ = path in Section 5. We prove this result by reduction from the emptiness problem forsimple probabilistic co-Büchi word automata with almost-sure semantics. As already mentioned(Proposition 2.7) it was shown in [1] that this problem is undecidable for general probabilistic wordautomata, but in our reduction to probabilistic tree automata it will be crucial to work with simpleones. We thus start by giving a proof of this slightly stronger result. Proposition 4.1.
The emptiness problem for simple probabilistic co-Büchi word automata withalmost-sure semantics is undecidable. Following Proposition 2.9, we call universal an alternating Büchi tree automata whose set of states belonging to Éloïse isempty. ACM Trans. Comput. Logic, Vol. 1, No. 1, Article . Publication date: December 2020.
Proof.
The proof is by reduction from the value 1 problem for simple probabilistic automata,which is undecidable (Theorem 2.5).Let A = ( 𝑄, 𝑞 in , 𝛿 ) be a simple probabilistic word automaton over some alphabet Σ , and let 𝐹 ⊆ 𝑄 . Let ♯ ∉ Σ be a fresh symbol and let A ′ = ( 𝑄 ∪ { 𝑞 ′ in } , 𝑞 in , 𝛿 ′ ) be the simple probabilisticautomaton over Σ ∪ { ♯ } obtained from A as follows:- 𝑞 ′ in is a new state with 𝛿 ′ ( 𝑞 ′ in , 𝑎 ) = 𝛿 ( 𝑞 in , 𝑎 ) , for any letter 𝑎 ≠ ♯ , and 𝛿 ′ ( 𝑞 in , ♯ ) = 𝑞 ′ in ;- 𝛿 ′ ( 𝑞, 𝑎 ) = 𝛿 ( 𝑞, 𝑎 ) , for any state 𝑞 ∈ 𝑄 and any letter 𝑎 ≠ ♯ ;- 𝛿 ′ ( 𝑞, ♯ ) = 𝑞 in if 𝑞 ∈ 𝐹 and 𝛿 ′ ( 𝑞, ♯ ) = 𝑞 ′ in otherwise, for any state 𝑞 ∈ 𝑄 .We equip A ′ with the co-Büchi condition { 𝑞 ′ in } . Note that A ′ is simple.For a sequence of words ( 𝑢 𝑖 ) 𝑖 ≥ over Σ we let 𝑥 𝑖 be the acceptance probability of A over 𝑢 𝑖 ,for every 𝑖 ≥
1. Now consider an infinite word of the form 𝑤 = ♯ 𝑢 ♯ 𝑢 ♯ 𝑢 · · · , and let 𝐸 𝑖 be theevent: “ A ′ ends in 𝑞 ′ in when it reads 𝑢 𝑖 ♯ from 𝑞 in or 𝑞 ′ in ”. Each 𝐸 𝑖 has probability 1 − 𝑥 𝑖 , and they aremutually independent. Also, 𝑤 is almost-surely accepted by A ′ if and only if the probability thatinfinitely many of the events 𝐸 𝑖 occur is zero. It is then a direct consequence of the Borel-CantelliLemma (and its converse) that 𝑤 is almost-surely accepted by A ′ if and only if Í ∞ 𝑖 = − 𝑥 𝑖 < ∞ .It follows that A has value 1 if and only if A ′ almost-surely accepts a word of the form 𝑤 = ♯ 𝑢 ♯ 𝑢 ♯ 𝑢 · · · . Indeed, if A has value 1 then there is a sequence of words ( 𝑢 𝑖 ) 𝑖 ≥ such that 𝑥 𝑖 ≥ − 𝑖 and therefore such that Í ∞ 𝑖 = − 𝑥 𝑖 < ∞ ; conversely, if a sequence of words ( 𝑢 𝑖 ) 𝑖 ≥ is such that Í ∞ 𝑖 = − 𝑥 𝑖 < ∞ , one must have lim 𝑥 →∞ 𝑥 𝑖 = B that almost-surely accepts only those words that are almost-surely accepted by A ′ , starting witha ♯ and containing infinitely many ♯ .Consider the automaton C from Example 2.6 and recall that, when equipped with the acceptancecondition co-Büchi ({ 𝑝 }) , it accepts those infinite words over Σ ∪ { ♯ } that contain infinitely manyoccurrences of ♯ .Now, define B as the simple probabilistic automaton consisting of a fresh initial state 𝑞 ′′ in togetherwith a copy of A ′ and a copy of C . From 𝑞 ′′ in the only possible action is to read a ♯ and go either tothe initial state of B with probability or to the initial state of C with probability .Then it is immediate that 𝐿 = ({ 𝑞 ′ in ,𝑝 }) (B) is empty if and only if A does not have value 1. (cid:3) Our main undecidability result of Theorem 4.2 contrasts with two decidability results, for prob-abilistic Büchi tree automata [8] and for alternating Büchi tree automata [17] (Theorem 3.7), bothwith qualitative semantics.
Theorem 4.2.
The emptiness problem for universal co-Büchi tree automata with qualitative se-mantics is undecidable.
To prove Theorem 4.2 we construct a reduction from the emptiness problem for simple proba-bilistic co-Büchi word automata with almost-sure semantics to the emptiness problem for universalco-Büchi tree automata with qualitative semantics. The correctness of the reduction relies on thetwo following results (Lemma 4.3 and Lemma 4.4).Let A = ( 𝑄, 𝑞 in , 𝛿 ) be a simple probabilistic word automaton and 𝐹 ⊆ 𝑄 . Define the followingprobabilistic tree automata: • A = ( 𝑄, 𝑞 in , 𝛿 ′ ) where 𝛿 ′ ( 𝑝, 𝑎 ) = ( 𝑞 , 𝑞 ) + ( 𝑞 , 𝑞 ) if 𝛿 ( 𝑝, 𝑎 ) = 𝑞 + 𝑞 . • A = ( 𝑄, 𝑞 in , 𝛿 ′′ ) where 𝛿 ′′ ( 𝑝, 𝑎 ) = ( 𝑞 , 𝑞 ) + ( 𝑞 , 𝑞 ) if 𝛿 ( 𝑝, 𝑎 ) = 𝑞 + 𝑞 .Lemma 4.3 relates A and A , where 𝜇 denotes the coin-flipping measure on branches definedin Section 2.5, while Lemma 4.4 relates A and A . ACM Trans. Comput. Logic, Vol. 1, No. 1, Article . Publication date: December 2020. lternating Tree Automata with Qualitative Semantics 19
Lemma 4.3 ([8, Proposition 43]).
The following holds: 𝐿 ∀ = Qual , co-Büchi ( 𝐹 ) (A ) = n 𝑡 ∈ Trees ( Σ ) : 𝜇 (cid:16)n 𝑏 ∈ { , } 𝜔 : 𝑡 [ 𝑏 ] ∈ 𝐿 = co-Büchi ( 𝐹 ) (A) o(cid:17) = o . Now, for a fixed tree 𝑡 , the Markov chains M 𝑡 A and M 𝑡 A associated with A and A respec-tively are equal: indeed, they have the same states 𝑄 × { , } ∗ , the same initial state ( 𝑞 in , 𝜀 ) and thesame probability transition function 𝑇 given by 𝑇 (( 𝑞, 𝑢 )) = ( 𝑞 , 𝑢 ) + ( 𝑞 , 𝑢 ) + ( 𝑞 , 𝑢 ) + ( 𝑞 , 𝑢 ) where 𝛿 ( 𝑞, 𝑡 ( 𝑢 )) = 𝑞 + 𝑞 in A . As a consequence, A and A have the same qualitative co-Büchi semantics. Lemma 4.4. 𝐿 ∀ = Qual , co-Büchi ( 𝐹 ) (A ) = 𝐿 ∀ = Qual , co-Büchi ( 𝐹 ) (A ) . We are now ready to prove Theorem 4.2.
Proof of Theorem 4.2.
Let A = ( 𝑄, 𝑞 in , 𝛿 ) be a simple probabilistic word automaton and 𝐹 ⊆ 𝑄 .We define the tree automaton A 𝑈 = ( 𝑄, 𝑞 in , Δ ) where Δ = (cid:26) ( 𝑞, 𝑎, 𝑞 , 𝑞 ) , ( 𝑞, 𝑎, 𝑞 , 𝑞 ) | 𝛿 ( 𝑞, 𝑎 ) = 𝑞 + 𝑞 (cid:27) . Now, we establish that 𝐿 = ( 𝐹 ) (A) ≠ ∅ if, and only if, 𝐿 ∀ Qual , co-Büchi ( 𝐹 ) (A 𝑈 ) ≠ ∅ , whichconcludes the proof of Theorem 4.2.Assume that there is some 𝑤 = 𝑤 𝑤 · · · ∈ 𝐿 = ( 𝐹 ) (A) , that is such that 𝑃 𝑤 A ( co-Büchi ( 𝐹 )) =
1. We construct a tree 𝑡 𝑤 whose branches are all equal to 𝑤 , i.e. 𝑡 𝑤 ( 𝑢 ) = 𝑤 | 𝑢 | for every 𝑢 ∈ { , } ∗ .For a fixed run 𝜌 of A 𝑈 over 𝑡 𝑤 , there is a bijection between the infinite paths of M 𝑤 A and M 𝜌 A 𝑈 that preserves the measure (it suffices to notice that the measure is preserved for cones) andalso the property of visiting finitely many states in 𝐹 . As a result, 𝑃 𝑤 A ( co-Büchi ( 𝐹 )) = 𝑃 𝜌 A 𝑈 ( co-Büchi ( 𝐹 )) =
1, for all runs 𝜌 . Thus 𝑡 𝑤 ∈ 𝐿 ∀ Qual , co-Büchi ( 𝐹 ) (A 𝑈 ) .The converse implication is not immediate because a tree 𝑡 ∈ 𝐿 ∀ Qual , co-Büchi ( 𝐹 ) (A 𝑈 ) may notnecessarily be of the form 𝑡 𝑤 for some word 𝑤 ∈ Σ 𝜔 .In Section 2.7, we informally said that an equivalent definition of almost-sure acceptance forprobabilistic tree automata can be obtained by associating a probability measure on the set of allruns induced by a tree, and by requiring the measure of the set of qualitatively accepting runs to beequal to 1; in this approach the notion of a run is the same as for (non-probabilistic) tree automata(see [8] for details).Now, consider the probabilistic tree automaton A used in Lemma 4.4: for a fixed tree 𝑡 , theset of runs of A 𝑈 over 𝑡 is the same as the set of runs of A over 𝑡 . Since all runs of A 𝑈 over 𝑡 are qualitatively accepted, then all runs of A over 𝑡 are qualitatively accepted too, so the set ofqualitatively accepting runs of A over 𝑡 has measure 1. In other words, 𝑡 ∈ 𝐿 ∀ = Qual , co-Büchi ( 𝐹 ) (A ) .Hence, by Lemma 4.4, 𝑡 ∈ 𝐿 ∀ = Qual , co-Büchi ( 𝐹 ) (A ) . Finally, using Lemma 4.3, almost all branches of 𝑡 are in 𝐿 = ( 𝐹 ) (A) , entailing 𝐿 = ( 𝐹 ) (A) ≠ ∅ . (cid:3) In this section we derive two corollaries from Theorem 4.2: the undecidability of the MSO+ ∀ = path theory of the infinite binary tree (Theorem 5.1), and the undecidability of the emptiness problemfor alternating probabilistic automata with non-zero semantics (Theorem 5.2). ACM Trans. Comput. Logic, Vol. 1, No. 1, Article . Publication date: December 2020. ∀ = path Before stating the problem, we refer the reader to [32] for definitions and basic properties onMonadic Second Order logic (MSO) for trees.The logic MSO+ ∀ = , introduced and studied in [24, 25], extends MSO with a probabilistic opera-tor ∀ = 𝑋 .𝜑 stating that the set of all sets 𝑋 satisfying 𝜑 contains a subset of Lebesgue-measure one.Michalewski, Mio and Skrzypczak proved in these papers that the MSO+ ∀ = -theory of the infinitebinary tree is undecidable. They also considered a variant of this logic, denoted by MSO+ ∀ = path ,in which the quantification in the probabilistic operator is restricted to sets of nodes that forma path. They proved that, in terms of expressiveness, MSO+ ∀ = path is between MSO and MSO+ ∀ = ,with a strict gain in expressiveness compared to MSO. However, they left open the question of thedecidability of the MSO+ ∀ = path theory of the infinite binary tree [25, Problem 4].In this section, we establish that in fact MSO+ ∀ = path is undecidable over the infinite binary tree,as a direct consequence of Theorem 4.2.The syntax of MSO+ ∀ = path is given by the following grammar: 𝜑 :: = succ ( 𝑥, 𝑦 ) | succ ( 𝑥, 𝑦 ) | 𝑥 ∈ 𝑋 | ¬ 𝜑 | 𝜑 ∧ 𝜑 | ∀ 𝑥 .𝜑 | ∀ 𝑋 .𝜑 | ∀ = path 𝑋 .𝜑 where 𝑥 ranges over a countable set of first-order variables , and 𝑋 ranges over a countable set of monadic second-order variables (also called set variables ). The quantifier ∀ = path is called the path-measure quantifier .The semantics of MSO on the infinite binary tree is defined by interpreting the first-order vari-ables 𝑥 as nodes, and the set variables 𝑋 as subsets of nodes. Ordinary quantification and theBoolean operations are defined as usual, 𝑥 ∈ 𝑋 is interpreted as the membership relation, andsucc 𝑖 (for 𝑖 = ,
1) is interpreted as the binary relation {( 𝑥, 𝑥 · 𝑖 ) | 𝑥 ∈ { , } ∗ } . We now describehow to interpret quantified formulas of the form ∀ = path 𝑋 .𝜑 . A path is a prefix-closed non-emptyset 𝑋 ⊆ { , } ∗ such that for any node 𝑣 ∈ 𝑋 either 𝑣 ∈ 𝑋 or 𝑣 ∈ 𝑋 , but not both. We let Paths denote the set of all paths. Note that there is a one-to-one correspondence between
Paths and theset { , } 𝜔 of branches. Thus, the coin-flipping measure 𝜇 , defined over { , } 𝜔 (see Section 2.5),induces a measure over Paths , which we write 𝜇 . We let 𝑡 | = ∀ = path 𝑋 .𝜑 if there exists a measurablesubset of paths Π ⊆ Paths with 𝜇 ( Π ) = 𝜋 ∈ Π one has 𝑡, 𝜋 | = 𝜑 .A sentence is a formula without free variables. The MSO+ ∀ = path -theory of the infinite binary treeis the set of all MSO+ ∀ = path -sentences 𝜑 that hold in the infinite binary tree.We identify a { , } 𝑛 -tree 𝑡 with a tuple of 𝑛 subsets of nodes setTuple ( 𝑡 ) = ( 𝑋 , . . . , 𝑋 𝑛 ) wherea node 𝑥 belongs to 𝑋 𝑖 if and only if the 𝑖 -th element of 𝑡 ( 𝑥 ) is 1. This immediately permits tointerpret an MSO+ ∀ = path formula with 𝑛 free set variables on { , } 𝑛 -trees.The following result is an easy consequence of Theorem 4.2. Theorem 5.1.
The MSO+ ∀ = path -theory of the infinite binary tree is undecidable. Proof.
We reduce the emptiness problem for co-Büchi tree automata with qualitative semantics,that we proved undecidable (Theorem 4.2), to the MSO+ ∀ = path -theory of the infinite binary tree.Let A be a co-Büchi tree automaton over the alphabet Σ . Without loss of generality, we assumethat Σ ⊆ { , } 𝑛 for some 𝑛 . Note that, as MSO+ ∀ = path -formulas are interpreted over the (unla-belled) infinite binary tree, we use tuples of subsets of nodes to encode Σ -trees. We construct anMSO+ ∀ = path formula 𝜑 ( ® 𝑋 ) , with ® 𝑋 = ( 𝑋 , . . . , 𝑋 𝑛 ) , such that 𝐿 ∀ Qual , co-Büchi ( 𝐹 ) (A) = { 𝑡 | setTuple ( 𝑡 ) | = 𝜑 } . ACM Trans. Comput. Logic, Vol. 1, No. 1, Article . Publication date: December 2020. lternating Tree Automata with Qualitative Semantics 21
The formula 𝜑 mimics the definition of 𝐿 ∀ Qual , co-Büchi ( 𝐹 ) (A) : ∀ ® 𝑌 . ( “ ® 𝑌 is a run of A on ® 𝑋 ” ⇒ ∀ = path 𝑍 . ( “ 𝑍 is an accepting path of ® 𝑌 ” )) , where “ ® 𝑌 is a run of A on ® 𝑋 ” and “ 𝑍 is an accepting path of ® 𝑌 ” are expressed in first-order logic(we refer to [29] for this classical encoding). The desired formula is then ¬∃ ® 𝑋𝜑 , which achievesthe proof. (cid:3) The non-zero semantics for tree automata was introduced by Bojańczyk, Gimbert and Kelmendi [6].In a recent paper, Fournier and Gimbert initiated the study of alternating tree automata with non-zero semantics [19]. Their main result is the decidability of the emptiness problem for a subclass ofthese automata, called limited choice for Abélard , and this is used to solve the satisfiability problemof CTL ∗ +pCTL ∗ ; however the decidability of emptiness for the full class of alternating automatawith non-zero semantics was left open. Since this class easily subsumes universal tree automatawith qualitative semantics, Theorem 4.2 directly implies that this problem is undecidable.An alternating non-zero automaton on alphabet Σ is a tuple A = (( 𝑄, ≺) , 𝑞 in , 𝑄 𝐸 , 𝑄 𝐴 , Δ , 𝐹 ∀ , 𝐹 , 𝐹 > ) where 𝑄 is a finite set of states equipped with a total order ≺ , 𝑞 in ∈ 𝑄 is the initial state, ( 𝑄 𝐸 , 𝑄 𝐴 ) isa partition of 𝑄 into Éloïse’s and Abélard’s states, Δ is a set of transitions made of local transitions (elements of 𝑄 × Σ × 𝑄 ) and split transitions (elements of 𝑄 × Σ × 𝑄 × 𝑄 ), and 𝐹 ∀ , 𝐹 , 𝐹 > ⊆ 𝑄 aresubsets of 𝑄 defining the semantics of the acceptance game G n-z A ,𝑡 , to be defined later.The input of such an automaton is a Σ -tree 𝑡 and acceptance is defined thanks to a two-playerperfect-information stochastic game. The arena is quite similar to the arena G = A ,𝑡 defined in Sec-tion 2.6 for alternating tree automata with qualitative semantics (simply ignore the total order ≺ and subsets 𝐹 ∀ , 𝐹 , 𝐹 > ), except that local transitions are handled without interacting with the Ran-dom player (i.e. when Éloïse or Abélard simulates a local transition the state is simply updated andthe pebble stays in the same node).Formally one lets 𝐺 = ( 𝑉 𝐸 ∪ 𝑉 𝐴 ∪ 𝑉 𝑅 , 𝐸 ) with 𝑉 𝐸 = 𝑄 𝐸 × { , } ∗ , 𝑉 𝐴 = 𝑄 𝐴 × { , } ∗ and 𝑉 𝑅 = {( 𝑞, 𝑢, 𝑞 , 𝑞 ) | 𝑢 ∈ { , } ∗ and ( 𝑞, 𝑡 ( 𝑢 ) , 𝑞 , 𝑞 ) ∈ Δ } , and 𝐸 = {(( 𝑞, 𝑢 ) , ( 𝑞 ′ , 𝑢 )) | ( 𝑞, 𝑡 ( 𝑢 ) , 𝑞 ′ ) ∈ Δ )} ∪{(( 𝑞, 𝑢 ) , ( 𝑞, 𝑢, 𝑞 , 𝑞 )) | ( 𝑞, 𝑡 ( 𝑢 ) , 𝑞 , 𝑞 ) ∈ Δ )} ∪{(( 𝑞, 𝑢, 𝑞 , 𝑞 ) , ( 𝑞 𝑥 , 𝑢 · 𝑥 )) | 𝑥 ∈ { , } and ( 𝑞, 𝑢, 𝑞 , 𝑞 ) ∈ 𝑉 𝑅 )} Then, we define G n-z A ,𝑡 = ( 𝐺, 𝑉 𝐸 , 𝑉 𝐴 , 𝑉 𝑅 , 𝛿, ( 𝑞 in , 𝜀 )) where 𝛿 (( 𝑞, 𝑢, 𝑞 , 𝑞 )) = ( 𝑞 , 𝑢 ) + ( 𝑞 , 𝑢 ) .A strategy 𝜎 𝐸 for Éloïse beats a strategy for Abélard 𝜎 𝐴 if all the following conditions are satis-fied:(i) Sure winning : in every play consistent with ( 𝜎 𝐸 , 𝜎 𝐴 ) the largest (with respect to ≺ ) stateappearing infinitely often belongs to 𝐹 ∀ .(ii) Almost-sure winning : the (measurable) set of plays consistent with ( 𝜎 𝐸 , 𝜎 𝐴 ) where the largeststate (with respect to ≺ ) appearing infinitely often belongs to 𝐹 has measure 1.(iii) Positive winning : for every history consistent with ( 𝜎 𝐸 , 𝜎 𝐴 ) that ends with a state in 𝐹 > , the(measurable) set of infinite continuations of this history that contain only states in 𝐹 > andare consistent with ( 𝜎 𝐸 , 𝜎 𝐴 ) , has non-zero measure. ACM Trans. Comput. Logic, Vol. 1, No. 1, Article . Publication date: December 2020.
Finally, a tree 𝑡 is accepted by A if, and only if, Éloïse has a strategy that beats any strategy ofAbélard. The emptiness problem asks for a given alternating non-zero automaton whether the setof accepted trees is empty.It is easily seen that alternating automata with non-zero semantics subsume universal co-Büchitree automata with qualitative semantics. Indeed, consider a universal co-Büchi tree automaton A with qualitative semantics having a set of states 𝑄 and a set of states 𝐹 ⊆ 𝑄 defining the co-Büchicondition. Then, universality is captured by alternation (see Proposition 2.9) and the co-Büchiqualitative acceptance condition of A can be expressed by part (ii) of the beating condition: itis enough to rank the states in 𝐹 higher than those in 𝑄 \ 𝐹 in the total order on 𝑄 , and to let 𝐹 = 𝑄 \ 𝐹 .Together with Theorem 4.2 this yields the following undecidability result. Theorem 5.2.
The emptiness problem for alternating tree automata with non-zero semantics isundecidable.
Proof.
Consider a co-Büchi universal tree automaton A = ( 𝑄, 𝑞 in , Δ ) whose acceptance con-dition is given by a subset 𝐹 ⊆ 𝑄 . Without loss of generality, we can safely assume that 𝑄 = { 𝑞 , . . . 𝑞 𝑛 } where 𝑛 = | 𝑄 | and that 𝐹 = { 𝑞 𝑘 , . . . , 𝑞 𝑛 } for some 𝑘 ≤ 𝑛 +
1. We construct an alternat-ing non-zero automaton B = (( 𝑄, ≺) , 𝑞 in , ∅ , 𝑄, Δ , 𝑄, 𝐹 , 𝑄 ) , where the total order ≺ on 𝑄 is definedby 𝑞 𝑖 ≺ 𝑞 𝑗 if and only if 𝑖 < 𝑗 and 𝐹 = 𝑄 \ 𝐹 .Note that since A has only split transitions, the arenas G = A ,𝑡 and G n-z B ,𝑡 are the same for anytree 𝑡 , and so are the strategies for Éloïse and Abélard. Moreover, it is immediate that an Éloïse’sstrategy 𝜎 𝐸 beats an Abélard’s strategy 𝜎 𝐴 in G n-z B ,𝑡 if, and only if, almost all plays in G = A ,𝑡 consistentwith ( 𝜎 𝐸 , 𝜎 𝐴 ) satisfy the co-Büchi condition. Hence, Éloïse has a strategy that beats any strategyof Abélard in G n-z B ,𝑡 if and only if she has an almost-surely winning strategy in the co-Büchi game (G = A ,𝑡 , 𝐹 × { , } ∗ ) . Otherwise said, a tree is accepted by B if, and only if, it is accepted by A .Applying Theorem 4.2, concludes the proof. (cid:3) CONCLUSIONS
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