Ambiguity Hierarchy of Regular Infinite Tree Languages
aa r X i v : . [ c s . L O ] S e p AMBIGUITY HIERARCHY OF REGULAR INFINITE TREELANGUAGES
ALEXANDER RABINOVICH AND DORON TIFERETTel Aviv University, Israel e-mail address : [email protected]
URL : Tel Aviv University, Israel e-mail address : [email protected]
Abstract.
An automaton is unambiguous if for every input it has at most one acceptingcomputation. An automaton is k -ambiguous (for k >
0) if for every input it has at most k accepting computations. An automaton is boundedly ambiguous if there is k ∈ N , suchthat for every input it has at most k accepting computations. An automaton is finitely(respectively, countably) ambiguous if for every input it has at most finitely (respectively,countably) many accepting computations.The degree of ambiguity of a regular language is defined in a natural way. A languageis k -ambiguous (respectively, boundedly, finitely, countably ambiguous) if it is accepted bya k -ambiguous (respectively, boundedly, finitely, countably ambiguous) automaton. Overfinite words every regular language is accepted by a deterministic automaton. Over finitetrees every regular language is accepted by an unambiguous automaton. Over ω -wordsevery regular language is accepted by an unambiguous B¨uchi automaton and by a de-terministic parity automaton. Over infinite trees Carayol et al. showed that there areambiguous languages.We show that over infinite trees there is a hierarchy of degrees of ambiguity: For every k > k -ambiguous languages which are not k − Introduction
Degrees of Ambiguity. The relationship between deterministic and nondeterministic ma-chines plays a central role in computer science. An important topic is a comparison ofexpressiveness, succinctness and complexity of deterministic and nondeterministic models.Various restricted forms of nondeterminism were suggested and investigated (see [6, 10] forrecent surveys).Probably, the oldest restricted form of nondeterminism is unambiguity. An automatonis unambiguous if for every input there is at most one accepting run. For automata over
Key words and phrases: regular tree languages, parity tree automata, automata ambiguity.
Preprint submitted toLogical Methods in Computer Science c (cid:13)
A. Rabinovich and D. Tiferet CC (cid:13) Creative Commons
A. RABINOVICH AND D. TIFERET finite words there is a rich and well-developed theory on the relationship between determin-istic, unambiguous and nondeterministic automata [10]. All three models have the sameexpressive power. Unambiguous automata are exponentially more succinct than determinis-tic ones, and nondeterministic automata are exponentially more succinct than unambiguousones [12, 13].Some problems are easier for unambiguous than for nondeterministic automata. Asshown by Stearns and Hunt [19], the equivalence and inclusion problems for unambiguousautomata are in polynomial time, while these problems are PSPACE-complete for nonde-terministic automata.The complexity of basic regular operations on languages represented by unambiguousfinite automata was investigated in [11], and tight upper bounds on state complexity ofintersection, concatenation and many other operations on languages represented by unam-biguous automata were established.It is well-known that the tight bound on the state complexity of the complementation ofnondeterministic automata is 2 n . In [11], it was shown that the complement of the languageaccepted by an n -state unambiguous automaton is accepted by an unambiguous automatonwith 2 . n +log n states.Many other notions of ambiguity were suggested and investigated. A recent paper [10]surveys works on the degree of ambiguity and on various nondeterminism measures for finiteautomata on words.An automaton is k -ambiguous if on every input it has at most k accepting runs; it is boundedly ambiguous if it is k -ambiguous for some k ; it is finitely ambiguous if on everyinput it has finitely many accepting runs.It is clear that an unambiguous automaton is k -ambiguous for every k >
0, and a k -ambiguous automaton is finitely ambiguous. The reverse implications fail. For ǫ -freeautomata over words (and over finite trees), on every input there are at most finitely manyaccepting runs. Hence, every ǫ -free automaton on finite words and on finite trees is finitelyambiguous. However, over ω -words there are nondeterministic automata with uncountablymany accepting runs. Over ω -words and over infinite trees, finitely ambiguous automata area proper subclass of the class of countably ambiguous automata, which is a proper subclassof nondeterministic automata.The cardinality of the set of accepting computations of an automaton over an infinitetree t is bounded by the cardinality of the set of functions from the nodes of t to thestate of the automaton, and therefore, it is at most continuum 2 ℵ . The set of acceptingcomputations on t is definable in Monadic Second-Order Logic (MSO). In B´ar´any et al. in[2] it was shown that the continuum hypothesis holds for MSO-definable families of sets.Therefore, if the set of accepting computations of an automaton on a tree t is uncountable,then its cardinality is 2 ℵ . Hence, there are exactly two infinite degrees of ambiguity.The degree of ambiguity of a regular language is defined in a natural way. A languageis k -ambiguous if it is accepted by a k -ambiguous automaton. A language is boundedlyambiguous if it is k -ambiguous for some k ; it is finitely (respectively, countably) ambiguousif it is accepted by a finitely (respectively, countably) ambiguous automaton.Over finite words, every regular language is accepted by a deterministic automaton.Over finite trees, every regular language is accepted by a deterministic bottom-up treeautomaton and by an unambiguous top-down tree automaton. Over ω -words every regularlanguage is accepted by an unambiguous B¨uchi automaton [1] and by a deterministic parityautomaton. MBIGUITY HIERARCHY OF REGULAR INFINITE TREE LANGUAGES 3
Hence, the regular languages over finite words, over finite trees and over ω -words areunambiguous.In [5] it was shown that the aforementioned situation is different for infinite trees.Carayol et al. [5] proved that the language L ∃ a of infinite full-binary trees over the alphabet { a, c } , defined as L ∃ a := { t | t has at least one node labeled by a } is ambiguous. The proofis based on the undefinability of a choice function in Monadic Second-Order logic (MSO)[9, 4].Our results imply that the complement of every countable regular language is notfinitely ambiguous. Since L ∃ a is the complement (with respect to the alphabet { a, c } ) ofthe language which consists of a single tree (i.e. the tree with all nodes labeled by c ), weconclude that L ∃ a is not finitely ambiguous (this strengthens the above mentioned resultof [5]). Our main result states that over infinite trees there is a hierarchy of degrees ofambiguity: Theorem 1.1 (Hierarchy) . (1) For every k > there are k -ambiguous languages whichare not ( k − -ambiguous. (2) There are finitely ambiguous languages which are not boundedly ambiguous. (3)
There are countably ambiguous languages which are not finitely ambiguous. (4)
There are uncountably ambiguous languages which are not countably ambiguous.
Some natural tree languages which witness items (1), (3) and (4) of Theorem 1.1 aredescribed in the examples below. We have not found a “natural” finitely ambiguous languagewhich is not boundedly ambiguous (Theorem 1.1(2)).
Examples 1.2.
Let T ω Σ be the set of all infinite full-binary trees over an alphabet Σ. LetΣ k = { c, a , a , ..., a k } , and let L ¬ a i := { t ∈ T ω Σ k | no node in t is labeled by a i } for 1 ≤ i ≤ n .Define:(1) L ¬ a ∨···∨¬ a k := L ¬ a ∪ · · · ∪ L ¬ a k . We show that this language is k -ambiguous, but is not( k − L ¬ a ∨¬ a is two ambiguous.(2) L ∃ a := { t ∈ T ω Σ | there exists an a -labeled node in t } . This is a countably ambiguouslanguage which is not finitely ambiguous (see Sect. 4).(3) L no − max − a := { t ∈ T ω Σ | above every a -labeled node in t there is an a -labeled node } .This is an uncountably ambiguous language which is not countably ambiguous (see Sect.7). Organization of the paper:
In Sect. 2 we recall notations and basic results about au-tomata and monadic second-order logic. In Sect. 3 simple properties of languages areproved. Sect. 4 gives a sufficient condition for a language to be not finitely ambiguous. Theproof techniques used in Sect. 4 refine the proof techniques of [5], and rely on the fact thata choice function is not MSO-definable. Sect. 5 deals with k -ambiguous languages - forevery k ∈ N , we describe a k -ambiguous language which is not ( k − A. RABINOVICH AND D. TIFERET
An extended abstract of this paper was published in [18]. In this paper we addedmissing proofs, presented natural examples of uncountably ambiguous languages (in Sect.7) and added Sect. 8 in which we prove that countable languages are unambiguous.2.
Preliminary
We recall here standard terminology and notations about trees, automata and logic [16, 17].2.1.
Trees.
We view the set { l, r } ∗ of finite words over alphabet { l, r } as the domain of afull-binary tree, where the empty word ǫ is the root of the tree, and for each node v ∈ { l, r } ∗ ,we call v · l the left child of v , and v · r the right child of v .We define a tree order “ ≤ ” as a partial order such that ∀ u, v ∈ { l, r } ∗ : u ≤ v iff u isa prefix of v . Nodes u and v are incomparable - denoted by u ⊥ v - if neither u ≤ v nor v ≤ u ; a set U of nodes is an antichain , if its elements are incomparable with each other.We say that an infinite sequence π = v , v , . . . is a tree branch if v = ǫ and ∀ i ∈ N : v i +1 = v i · l or v i +1 = v i · r .If Σ is a finite alphabet, then a Σ-labeled full-binary tree t is a labeling function t : { l, r } ∗ → Σ. We denote by T ω Σ the set of all Σ-labeled full-binary trees. We often use “tree”for “labeled full-binary tree.”Given a Σ-labeled tree t and a node v ∈ { l, r } ∗ , the tree t ≥ v (called the subtree of t ,rooted at v ) is defined by t ≥ v ( u ) := t ( v · u ) for each u ∈ { l, r } ∗ . Grafting . Given two labeled trees t and t and a node v ∈ { l, r } ∗ , the grafting of t on v in t , denoted by t ◦ v t , is the tree t which is obtained from t by replacing the subtree of t rooted at v by t . Formally, t ( u ) := ( t ( w ) ∃ w ∈ { l, r } ∗ : u = v · wt ( u ) otherwiseMore generally, given a tree t , an antichain Y ⊆ { l, r } ∗ and a tree t , the grafting of t on Y in t , denoted by t ◦ Y t , is obtained by replacing each subtree of t rooted at anode y ∈ Y by the tree t . Tree Language . A language L over an alphabet Σ is a set of Σ-labeled trees. We denoteby L := T ω Σ \ L the complement of L .2.2. Automata. ω -word Automata. Parity ω -word Automata (PWA) . A PWA is a tuple ( Q A , Σ , Q I , δ, C ) where Σ is afinite alphabet, Q is a finite set of states, Q I ⊆ Q is a set of initial states, δ ⊆ Q × Σ × Q is a transition relation, and C : Q → N is a coloring function. A run of A on an ω -word y = a a . . . is an infinite sequence ρ = q q . . . such that q ∈ Q I , and ( q i , a i , q i +1 ) ∈ δ forall i ∈ N . We say that ρ is accepting if the maximal number which occurs infinitely oftenin C ( q ) C ( q ) . . . is even. Language . We denote the set of all accepting runs of A on y by ACC ( A , y ). The languageof A is defined as L ( A ) := { y ∈ Σ ω | ACC ( A , y ) = ∅} . MBIGUITY HIERARCHY OF REGULAR INFINITE TREE LANGUAGES 5
Infinite Tree Automata.
Parity Tree Automata (PTA) . A PTA is a tuple ( Q A , Σ , Q I , δ, C ) where δ ⊆ Q × Σ × Q × Q , and Σ, Q , Q I , F are defined as in PWA. A computation of A on a tree t is a function φ : { l, r } ∗ → Q such that φ ( ǫ ) ∈ Q I , and ∀ v ∈ { l, r } ∗ : ( φ ( v ) , t ( v ) , φ ( v · l ) , φ ( v · r )) ∈ δ . Wesay that φ is accepting if for each tree branch π = v v . . . , the maximal number whichoccurs infinitely often in C ( φ ( v )) C ( φ ( v )) . . . is even.Given a PTA A = ( Q A , Σ , Q I , δ A , C A ) and a set Q ′ ⊆ Q A , we define A Q ′ := ( Q A , Σ , Q ′ , δ A , C A )as the automaton obtained from A by replacing the set of initial states Q I with Q ′ . For asingleton Q ′ = { q } , we simplify this notation by A q := A Q ′ . Language . We denote the set of all accepting computations of A on t by ACC ( A , t ). Thelanguage of A is defined as L ( A ) := { t ∈ T ω Σ | ACC ( A , t ) = ∅} . A tree language is said tobe regular if it is accepted by a PTA.A state q ∈ Q of a PTA A is called useful if there is a tree t ∈ L ( A ), a computation φ ∈ ACC ( A , t ) and a node v ∈ { l, r } ∗ such that φ ( v ) = q . Throughout the paper we willassume all states of PTA are useful. Degree of Ambiguity of an Automaton . We denote by | X | the cardinality of a set X .An automaton A is k -ambiguous if | ACC ( A , t ) | ≤ k for all t ∈ L ( A ); A is unambiguous if itis 1-ambiguous; A is boundedly ambiguous if there is k ∈ N such that A is k -ambiguous; A is finitely ambiguous if ACC ( A , t ) is finite for all t ; A is countably ambiguous if ACC ( A , t )is countable for all t .The degree of ambiguity of A (notation da ( A )) is defined by da ( A ) := k if A is k -ambiguous and either k = 1 or A is not k − da ( A ) := f inite if A is finitelyambiguous and not boundedly ambiguous, da ( A ) := ℵ if A is countably ambiguous andnot finitely ambiguous, and da ( A ) := 2 ℵ if A is not countably ambiguous.We order the degrees of ambiguity in a natural way: i < j < f inite < ℵ < ℵ , for i < j ∈ N . Degree of Ambiguity of a Language . We say that a regular tree language L is unambigu-ous (respectively, k -ambiguous, finitely ambiguous, countably ambiguous) if it is acceptedby an unambiguous (respectively, k -ambiguous, finitely ambiguous, countably ambiguous)automaton. We define da ( L ) := min A { da ( A ) | L ( A ) = L } .2.3. Monadic Second-Order Logic.
We use standard notations and terminology aboutmonadic second-order logic (MSO) [17, 21, 20].Let τ be a relational signature. A structure (for τ ) is a tuple M = ( D, { R M | R ∈ τ } )where D is a domain, and each symbol R ∈ τ is interpreted as a relation R M on D .MSO-formulas use first-order variables, which are interpreted by elements of the struc-ture, and monadic second-order variables, which are interpreted as sets of elements. AtomicMSO-formulas are of the following form: • R ( x , . . . , x n ) for an n -ary relational symbol R and first order variables x , . . . , x n • x = y for two first-order variables x and y • x ∈ X for a first-order variable x and a second-order variable X A. RABINOVICH AND D. TIFERET
MSO-formulas are constructed from the atomic formulas, using boolean connectives, thefirst-order quantifiers, and the second-order quantifiers.We write ψ ( X , . . . , X n , x , . . . , x m ) to indicate that the free variables of the formula ψ are X , . . . , X n (second order variables) and x , . . . , x m (first order variables). We write M | = ψ ( A , . . . , A n , a , . . . a m ) if ψ holds in M when subsets A i are assigned to X i for i = 1 , . . . , n and elements a i are assigned to variables x , . . . , x m for i = 1 , . . . , m . Coding . Let ∆ be a finite set. We can code a function from a set D to ∆ by a tuple ofunary predicates on D . This type of coding is standard, and we shall use explicit variableswhich range over such mappings and expressions of the form “ F ( u ) = d ” (for d ∈ ∆) inMSO-formulas, rather than their codings.Formally, for each finite set ∆ we have second-order variables X ∆1 , X ∆2 , . . . which rangeover the functions from D to ∆, and atomic formulas X ∆ i ( u ) = d for d ∈ ∆ and u a firstorder variables [21]. Often the type of the second order variables will be clear from thecontext and we drop the superscript ∆. Definable Relations . The powerset of D is denoted by P ( D ). We say that a relation R ⊆ P ( D ) n × D m is MSO-definable in a structure S with universe D if there is an MSO-formula ψ ( X , . . . , X n , x , . . . , x m ) such that R = { ( D , . . . , D n , u , . . . , u m ) ∈ P ( D ) n × D m | S | = ψ ( D . . . , D n , u . . . , u n ) } .An element d ∈ D is MSO-definable in a structure S if there is a formula ψ ( x ) suchthat S | = φ ( u ) iff u = d . A set U ⊆ D is MSO-definable if there is a formula φ ( X ) suchthat S | = φ ( V ) iff V = U . A function is MSO-definable if its graph is.The unlabeled binary tree is the structure ( { l, r } ∗ , { E l , E r } ) where E l and E r are binarysymbols, respectively interpreted as { ( v, v · l ) | v ∈ { l, r } ∗ ) } and { ( v, v · r ) | v ∈ { l, r } ∗ ) } .It is easy to verify the correctness of the following lemma: Lemma 2.1.
The following relations are MSO-definable in the unlabeled full-binary tree. • The ancestor relation ≤ . • “A set of nodes is a branch,” “A set of nodes is an antichain.” • Let A = ( Q, Σ , Q I , δ, C ) be a PTA. We use φ for a function { l, r } ∗ → Q and σ for afunction { l, r } ∗ → Σ . – “ φ is a computation of A on the tree σ .” – “ φ is an accepting computation of A on the tree σ .” Theorem 2.2 (Rabin [17]) . A tree language is regular iff it is MSO-definable in the unla-beled binary tree structure.
A labeled tree is regular iff it has finitely many different subtrees. An equivalent def-inition is: a tree is regular iff its labeling is MSO-definable [17]. Hence, for every regularΣ-labeled tree t , there is an MSO-formula ψ t ( σ Σ ) which is satisfied by t iff t = t . Theorem 2.3 (Rabin’s basis theorem [17]) . Any non-empty regular tree language containsa regular tree.
MBIGUITY HIERARCHY OF REGULAR INFINITE TREE LANGUAGES 7
Choice Function . A choice function is a mapping which assigns to each non-empty set ofnodes one element from the set.
Theorem 2.4 (Gurevich and Shelah [9]) . There is no MSO-definable choice function onthe full-binary tree.
The following lemma follows from Theorem 2.4.
Lemma 2.5.
There is no MSO-definable function which assigns to every non-empty an-tichain Y a finite non-empty subset X ⊆ Y .Proof. Assume, for the sake of contradiction, that a function which returns a non-emptysubset for each non-empty antichain is MSO-definable in the unlabeled full-binary tree, byan MSO-formula
F initeAntichainSubset ( X, Y ). Claim 2.5.1 (Choice function over finite sets) . There is an MSO-definable function whichassigns to each non-empty finite set X ⊆ { l, r } ∗ an element x ∈ X .Proof. We first define a lexicographic order “ ≤ lex ” on { l, r } ∗ , by u ≤ lex v iff u is a prefix of v or u = w · l · u ′ and v = w · r · v ′ for some w, u ′ , v ′ ∈ { l, r } ∗ .It is easy to verify that ≤ lex is MSO-definable in the unlabeled full-binary tree. ≤ lex is alinear order, and therefore each non-empty finite set has a exactly one ≤ lex -minimal element.We conclude that a finite set choice function is definable by F initeChoice ( X, x ) :=“ x is the ≤ lex -minimal element in X ”. (cid:4) Let
F initeChoice ( X, x ) be an MSO-formula which defines a function as in Claim 2.5.1.We will use formulas
F initeAntichainSubset ( X, Y ) and
F initeChoice ( X, x ) to define achoice function by an MSO-formula
Choice ( X, x ) which is the conjunction of the followingconditions:(1) ∃ Z : “ Z is the set of ≤ -minimal elements in X ”(2) ∃ Y : F initeAntichainSubset ( Z, Y )(3)
F initeChoice ( Y, x )For each non-empty set X there is a unique subset Z ⊆ X of the ≤ -minimal elements in X . This set is a non-empty antichain, and therefore F initeAntichainSubset ( Z, Y ) returnsa finite subset Y ⊆ Z . Therefore, F initeChoice ( Y, x ) returns an element in Y . We concludethat Choice ( X, x ) returns an element x ∈ X and therefore defines a choice function in theunlabeled full-binary tree, in contradiction to Theorem 2.4.3. Simple Properties of Automata and Languages
In this section some simple lemmas are collected.
Lemma 3.1.
Let A = ( Q , Σ , Q I , δ , C ) and A = ( Q , Σ , Q I , δ , C ) be two PTA.Then: (1) There exists an automaton B such that L ( B ) = L ( A ) ∪ L ( A ) and for each t ∈ L ( A ) ∪ L ( A ) , | ACC ( B , t ) | ≤ | ACC ( A , t ) | + | ACC ( A , t ) | (2) There exists an automaton B such that L ( B ) = L ( A ) ∩ L ( A ) and for each t ∈ L ( A ) ∩ L ( A ) , | ACC ( B , t ) | ≤ | ACC ( A , t ) | · | ACC ( A , t ) | A. RABINOVICH AND D. TIFERET
Proof. (1) Assume that Q and Q are disjoint, and let B := ( Q ∪ Q , Σ ∪ Σ , Q I ∪ Q I , δ ∪ δ , C ∪ C ). It is clear that L ( B ) = L ( A ) ∪ L ( A ).Let t ∈ L ( B ). By definition of B , for each φ ∈ ACC ( B , t ) we either have φ ∈ ACC ( A , t )or φ ∈ ACC ( A , t ). Therefore, we obtain | ACC ( B , t ) | = | ACC ( A , t ) | + | ACC ( A , t ) | .(2) It is easy to verify that there is an MSO-formula over ω -words which holds for w = ( c , c ′ ) , . . . , ( c i , c ′ i ) , · · · ∈ ( Image ( C ) × Image ( C )) ω iff the maximal color whichappears infinitely often in the first coordinate of w and the maximal color which ap-pears infinitely often in the second coordinate of w are both even. Therefore (by Mc-Naughton’s Theorem [14]) there is a deterministic PWA D = ( Q D , Σ D , q D I , δ D , C D ) overalphabet Σ D = Image ( C ) × Image ( C ) such that w ∈ L ( D ) iff the maximal color whichappears infinitely often in the first coordinate of w and the maximal color which appearsinfinitely often in the second coordinate of w are both even.We will use the automata A , A and D to define a PTA B = ( Q B , Σ B , Q B I , δ B , C B )which accepts L ( A ) ∩ L ( A ). • Q B = Q × Q × Q D • Σ B := Σ ∩ Σ • Q B I := Q I × Q I × { q D I }• (( q, p, s ) , a, ( q , p , s ) , ( q , p , s )) ∈ δ B iff ( q, a, q , q ) ∈ δ , ( p, a, p , p ) ∈ δ , and s = s = δ D ( s, ( C ( q ) , C ( p ))). • C B ( q , q , p ) := C D ( p )It is easy to verify that L ( B ) = L ( A ) ∩ L ( A ).Assume, for the sake of contradiction, that there exists t such that | ACC ( B , t ) | > | ACC ( A , t ) | · | ACC ( A , t ) | . Since D is deterministic, it follows that there is a computationin ACC ( B , t ) such that either the projection of the first coordinate of φ on Q , denoted φ , isnot in ACC ( A , t ) or the projection of the second coordinate of φ on Q , denoted φ , is notin ACC ( A , t ). Assume w.l.o.g. that φ / ∈ ACC ( A , t ). Therefore, there is a tree branch π = v , v , . . . such that the maximal color which C assigns to the states which occurs infinitelyoften in φ ( π ) is odd. By definition of D we conclude that w := ( c , c ′ ) , ( c , c ′ ) , . . . / ∈ L ( D ),where c i := C ( φ ( v i )) and c ′ i := C ( φ ( v i )). Hence, by definition of B we conclude thatthe sequence of colors which C B assigns to the states φ ( π ) is exactly w , and therefore φ / ∈ ACC ( B , t ) - a contradiction.From Lemma 3.1, we obtain: Corollary 3.2.
Boundedly, finitely and countably ambiguous tree languages are closed un-der finite union and intersection.
We often use implicitly the following simple Lemma.
Lemma 3.3 (Grafting) . Let A be an automaton, t , t trees, v ∈ { l, r } ∗ and φ ∈ ACC ( A , t ) ,and φ ∈ ACC ( A q , t ) . If φ ( v ) = q , then φ ◦ v φ is an accepting computation of A on t ◦ v t . A similar lemma holds for general grafting. As an immediate consequence, we obtainthe following lemma:
Lemma 3.4. da ( A ) ≥ da ( A q ) for every useful state q of A . Corollary 3.5.
Let A be a boundedly (respectively, finitely, countably) ambiguous PTAwith a set Q of useful states, and let Q ′ ⊆ Q . Then A Q ′ is boundedly (respectively, finitely,countably) ambiguous. MBIGUITY HIERARCHY OF REGULAR INFINITE TREE LANGUAGES 9
Lemma 3.6.
Let L and L be two tree languages such that da ( L ) = da ( L ) and L ⊆ L .Then, there exists a tree t ∈ L \ L .Proof. The lemma follows immediately, since otherwise we have L = L and therefore da ( L ) = da ( L ), in contradiction to da ( L ) = da ( L ). Lemma 3.7.
Let A = ( Q, Σ , Q I , δ, C ) be a PTA. Then, there is a PTA B = ( Q B , Σ , { q B I } , δ B , C ) with single initial state such that L ( B ) = L ( A ) , and da ( B ) ≤ da ( A ) .Proof. Let Q B := Q ∪ { q B I } and δ B := δ A ∪ { ( q B I , a, q l , q r ) | q I ∈ Q I and ( q I , a, q l , q r ) ∈ δ } . Itis easy to see that L ( B ) = L ( A ).Let t ∈ L ( A ), and let g t be a function from ACC ( A , t ) to ACC ( B , t ) which maps eachcomputation φ ∈ ACC ( A , t ) to a computation φ ′ which assigns q B I to node ǫ , and φ ( v ) toother nodes. It is easy to see that φ ′ ∈ ACC ( B , t ), and that g t is surjective, and therefore ∀ t : | ACC ( A , t ) | ≥ | ACC ( B , t ) | , as requested. Definition 3.8 (Moore machine) . A Moore machine is a tuple M = (Σ , Γ , Q, q I , δ, out ),where Σ is a finite input alphabet, Q is a finite set of states, q I ∈ Q is an initial state, δ : Q × Σ → Q is a transition function, Γ is an output alphabet, and out : Q → Γ is anoutput function.Define b δ : Σ ∗ → Q by b δ ( ǫ ) := q I and b δ ( w ) := δ ( b δ ( w ′ ) , a ) for w = w ′ · a where w ′ ∈ Σ ∗ and a ∈ Σ. We say that a function F : Σ ∗ → Γ is definable by a Moore machine if there isa Moore machine M such that F ( w ) = out ( b δ ( w )) for all w ∈ Σ ∗ . Definition 3.9.
Let F : Σ ∗ → Σ be a function definable by a Moore machine, and let t ∈ T ω Σ . We define t := b F ( t ) as a tree in T ω Σ such that t ( v ) := F ( t ( v ) · · · · · t ( v k ))where v , v , . . . , v k is the path from the root to v .For a tree language L ⊆ T ω Σ , we define b F ( L ) := { b F ( t ) | t ∈ L } ⊆ T ω Σ . Lemma 3.10 (Reduction) . Let L and L be regular tree languages over alphabets Σ and Σ , respectively. Let F : Σ ∗ → Σ be a function definable by a Moore machine. Assumethat for each t ∈ T ω Σ , t ∈ L iff b F ( t ) ∈ L . Then da ( L ) ≤ da ( L ) .Proof. Let A = ( Q , Σ , Q I , δ , C ) such that A accepts L and da ( A ) = da ( L ).Let M = (Σ , Σ , Q M , q MI , δ M , out M ) be a Moore machine defining F . We will use A and M to define an automaton A = ( Q , Σ , Q I , δ , C ) such that t ∈ L ( A ) iff b F ( t ) ∈ L ( A ), by: • Q := Q × Q M • Q I := Q I × { q MI }• (( q, p ) , a, ( q , p ) , ( q , p )) ∈ δ iff p = p = δ M ( p, a ) and ( q, out M ( p ) , q , q ) ∈ δ • C ( q, p ) := C ( q )First notice that ∀ t ∈ T ω Σ : t ∈ L ( A ) ⇔ b F ( t ) ∈ L ( A ) ⇔ b F ( t ) ∈ L ⇔ t ∈ L , andtherefore L ( A ) = L as needed.Let φ ∈ ACC ( A , t ), and define a computation φ ′ by φ ′ ( v ) = q for φ ( v ) = ( q , q ) ∈ Q × Q M . It is easy to see that φ ′ ∈ ACC ( A , b F ( t )) and since M is deterministic, weconclude that | ACC ( A , t ) | ≤ | ACC ( A , b F ( t ) | ) | , and therefore da ( A ) ≤ da ( A ).We conclude that da ( L ) ≤ da ( A ) ≤ da ( A ) = da ( L ), as requested.Let us state another well-known characterization of regular trees. Fact 3.11.
A tree t is regular iff its labelling t : { l, r } ∗ → Σ is definable by a Mooremachine. 4.
Not-Finitely Ambiguous Languages
We provide here sufficient conditions for a language to be not finitely ambiguous. First,we state our main technical result - Proposition 4.1. Then, we derive some consequences.Finally, a proof of Proposition 4.1 is given. Our proof relies on the fact that there is noMSO-definable function which assigns to every non-empty antichain Y a finite non-emptysubset X ⊆ Y (Lemma 2.5), and our proof techniques refine the proof techniques of [5]. Notations . For trees t and t ′ and an antichain Y , we denote by t [ t ′ /Y ] the tree obtainedfrom t by grafting t ′ at every node in Y . Proposition 4.1.
Let t and t be regular trees and L be a regular language such that t L and t [ t /Y ] ∈ L for every non-empty antichain Y . Then L is not finitely ambiguous. Definition 4.2.
For a tree language L over alphabet Σ, we denote by Subtree ( L ) the treelanguage { t ∈ T ω Σ | ∃ t ′ ∈ L ∃ v : t ′≥ v = t } . Corollary 4.3.
Let L be a non-empty regular language over an alphabet Σ such that Subtree ( L ) = T ω Σ . Then, the complement of L is not finitely ambiguous.Proof. Let L be as in Corollary 4.3. We claim that there are regular Σ-labeled trees t ∈ L and t Subtree ( L ). Indeed, by Rabin’s basis theorem there is a regular t ∈ L . Since L is regular, there is an automaton B = ( Q, Σ , { q I } , δ, C ) (with only useful states) whichaccepts L . It is clear that B Q accepts Subtree ( L ), and therefore Subtree ( L ) is regular.The complement of Subtree ( L ) is regular (as the complement of a regular language) andnon-empty (since Subtree ( L ) = T ω Σ ), and therefore contains a regular tree t (by Rabin’sbasis theorem). Note that t [ t /Y ] L for every non-empty antichain Y .The complement of L satisfies the assumption of Proposition 4.1. Therefore, it is notfinitely ambiguous. Corollary 4.4 (not finitely ambiguous languages) . The following languages are not finitelyambiguous: (1)
The complement of a non-empty regular countable tree language. (2)
The complement of a regular language which contains a single tree. (3)
The language L ∃ a := { t ∈ T ω Σ | t has at least one node labeled by a } over alphabet Σ = { a , . . . , a m , c } .Proof. (1) Every tree has countably many subtrees. Since L is countable, we conclude that Subtree ( L ) is countable. Therefore, Subtree ( L ) does not contain all trees. By Proposition4.3, we conclude that L is not finitely ambiguous.(2) Follows immediately from (1).(3) By the definition of L ∃ a we have L ∃ a ∩ T ω { c,a } = T ω { c,a } \ { t c } , and therefore by(2), L ∃ a ∩ T ω { c,a } is not finitely ambiguous. It is easy to see that T ω { c,a } is unambiguous(since there is a deterministic automaton which accepts it). Therefore, by Corollary 3.2 weconclude that L ∃ a is not finitely ambiguous. MBIGUITY HIERARCHY OF REGULAR INFINITE TREE LANGUAGES 11
It is easy to prove that the complement of every finite language is countably ambiguous.Therefore, we obtain:
Corollary 4.5. If L is regular and its complement is finite and non-empty, then da ( L ) = ℵ .Proof of Corollary 4.5. We first prove the following claim:
Claim 4.5.1.
Let L be a regular tree language containing a single tree. Then L is countablyambiguous.Proof. Assume that L = { t } . L is a regular language, and therefore t is regular. Weconclude that there is a Moore machine M = ( { l, r } , Σ , Q M , q MI , δ M , out M ) such that foreach v ∈ { l, r } ∗ , out ( b δ ( v )) = σ iff t ( v ) = σ (that is, M defines the function t : { l, r } ∗ → Σ).We will use M to construct a countably ambiguous automaton A which accepts L byguessing a node v ∈ { l, r } ∗ such that t ( v ) = t ′ ( v ) for each tree t ′ ∈ L .Let A := ( Q A , Σ , Q I , δ, C ) such that: • Q A := { q, q ′ } × Q M • Q I := { ( q ′ , q MI ) }• δ is defined by: – (( q, p ) , a, ( q, p ′ ) , ( q, p ′′ )) ∈ δ iff δ M ( p, l ) = p ′ , δ M ( p, r ) = p ′′ – (( q ′ , p ) , a, ( q, p ′ ) , ( q, p ′′ )) ∈ δ iff δ M ( p, l ) = p ′ , δ M ( p, r ) = p ′′ and out ( p ) = a – (( q ′ , p ) , a, ( q ′ , p ′ ) , ( q, p ′′ )) , (( q ′ , p ) , a, ( q, p ′ ) , ( q ′ , p ′′ )) ∈ δ iff δ M ( p, l ) = p ′ , δ M ( p, r ) = p ′′ and out ( p ) = a . • ∀ p ∈ Q M : C ( q, p ) := 0 and C ( q ′ , p ) := 1By definition of A , it is clear that t ′ ∈ L ( A ) iff there is a node v such that t ′ ( v ) = t ( v ),and therefore t ′ ∈ L ( A ) iff t ′ = t .For each computation φ of A on t ′ , the Q M component is determined deterministicallyby M and t . If φ is accepting, there are finitely many nodes v such that the first componentof φ ( v ) is q ′ - otherwise, there would be a branch where the maximal color assigned infinitelyoften by C is odd, in contradiction to φ being an accepting computation. Therefore, thereare countably many accepting computations on each tree t ′ ∈ L ( A ), and A is countablyambiguous. (cid:4) L is finite and therefore there are t , . . . , t k ∈ T ω Σ such that L = { t , . . . , t k } . A finitetree language does not contain a non-regular tree, and therefore t , . . . , t k are regular. ByClaim 4.5.1, for each tree t i ∈ L , there is a countably ambiguous automaton A i such that t ∈ L ( A i ) iff t = t i . Notice that L = L ( A ) ∩ . . . , ∩ L ( A k ), and therefore by Lemma 3.1 weconclude that L is countably ambiguous. On the proof of Proposition 4.1 . In the rest of this section, Proposition 4.1 is proved.Let us sketch some ideas of the proof. For a language L , as in Proposition 4.1, and anynon-empty antichain Y we show that if A does not accept t and accepts t := t [ t /Y ], thenevery φ ∈ ACC ( A , t ) chooses (in an MSO-definable way) an element from Y . Hence, thecomputations in ACC ( A , t ) choose together a subset X of Y of cardinality ≤ | ACC ( A , t ) | (each computation chooses a single element). Therefore, if A accepts L and is finitelyambiguous, then X is finite - a contradiction to Lemma 2.5. To implement this plan,in Subsect. 4.1 we recall a game theoretical interpretation of “a tree is accepted by anautomaton.” Then, in Subsect. 4.2 we analyze which concepts related to these games areMSO-definable. Finally, in Subsect. 4.3, the proof is completed. Membership Game.
Let A = ( Q, Σ , { q I } , δ, C ) be a PTA, and let t be a Σ-labeledtree. A two-player game G t, A (called a “membership game”) between Automaton andPathfinder is defined as follows. The positions of Automaton are { l, r } ∗ × Q , and thepositions of Pathfinder are { l, r } ∗ × Q × Q . The initial position is ( ǫ, q I ).From a position ( v, q ) ∈ { l, r } ∗ × Q Automaton chooses a tuple ( q l , q r ) ∈ Q × Q suchthat ∃ a ∈ Σ : ( q, a, q l , q r ) ∈ δ , and moves to the position ( v, q l , q r ). From a position( v, q l , q r ) ∈ { l, r } ∗ × Q × Q Pathfinder chooses a direction d ∈ { l, r } , and moves to theposition ( v · d, q d ).We define a play s := e , d , e , d , . . . , e i , d i , · · · ∈ ( Q × Q × { l, r } ) ω as an infinitesequence of moves, corresponding to the choices of Automaton and Pathfinder from theinitial position. We say that the move e i = ( q l , q r ) from position ( q, v ) is invalid forAutomaton if ( q, t ( v ) , q l , q r ) / ∈ δ .A strategy for a player in G t, A is a function which determines the next move of theplayer based on previous moves of both players.A positional strategy for a player in G t, A is a strategy which determines the nextmove of the player based only on the current position. A positional strategy for Automatonis a function str : { l, r } ∗ × Q → Q × Q , and a positional strategy for Pathfinder is a function ST R : { l, r } ∗ × Q × Q → { l, r } .Let C G be a coloring function which maps each position in G t, A to a color in N . Wedefine C G ( v, q ) := C ( q ) for Automaton’s positions, and C G ( v, q l , q r ) := 0 for Pathfinder’spositions.For each play s define π s as the infinite sequence of positions corresponding to themoves in s . A play s is winning for Automaton iff s does not contain an invalid move forAutomaton, and the maximal color which C G assigns infinitely often to the positions in π s is even. Since all Pathfinder’s positions are colored by 0, it is sufficient to consider thecoloring of Automaton’s positions in π s .We say that a play is consistent with a strategy of a player if all moves of the player areaccording to the strategy. A winning strategy for a player is a strategy such that eachplay which is consistent with the strategy is winning for the player.Parity games are positionally determined [7], i.e., for each parity game, one of theplayers has a positional winning strategy. Therefore, if a player has a winning strategy,then he has a positional winning strategy. Additionally, if a positional strategy of a playerwins against all positional strategies of the other player, then it is a winning strategy.We recall standard definitions and facts about the connections between games and treeautomata [8, 16].Let φ : { l, r } ∗ → Q be a function such that φ ( ǫ ) = q I and ∀ v ∈ { l, r } : ∃ a ∈ Σ :( φ ( v ) , a, φ ( v · l ) , φ ( v · r )) ∈ δ . We define a positional strategy str φ : { l, r } ∗ × Q → Q × Q forAutomaton, by str φ ( v, q ) := ( φ ( v · l ) , φ ( v · r )). Conversely, for each positional strategy str : { l, r } ∗ × Q → Q × Q of Automaton we construct a function φ str : { l, r } ∗ → Q by φ ( ǫ ) := q I and for all v ∈ { l, r } ∗ we set φ ( v · l ) := q l , and φ ( v · r ) := q r where str ( v, φ ( v )) = ( q l , q r ). Claim 4.1.1. (1)
Let s be a play which is consistent with str φ , and let ( v i , q i ) be the i -thposition of Automaton in π s . Then, φ ( v i ) = q i . (2) If φ ∈ ACC ( A , t ) , then str φ is a positional winning strategy for Automaton. (3) If str is a positional winning strategy for Automaton, then φ str ∈ ACC ( A , t ) . MBIGUITY HIERARCHY OF REGULAR INFINITE TREE LANGUAGES 13
Proof. (1) We will prove by induction on i . For i = 0 we have ( v , q ) = ( ǫ, q I ) (by definitionof G t, A ), and indeed φ ( v ) = φ ( ǫ ) = q I . Assume the claim holds for i = k and we prove for i = k + 1.Let d ∈ { l, r } be the i -th move of Pathfinder in s . By definition of G t, A we have v i +1 = v i · d , and q i +1 = q d , where str φ ( v i , q i ) = ( q l , q r ).By definition str φ we have ( q l , q r ) = ( φ ( v i · l ) , φ ( v i · l )), and therefore q i +1 = φ ( v i · d ) = φ ( v i +1 ), as requested.(2) and (3) are well known results about membership games [16]. (cid:4) The next claim describes what happens when Pathfinder plays his winning strategy in G t, A against an Automaton’s winning strategy in G t ′ , A (for t ′ = t ). Claim 4.1.2.
Assume t / ∈ L ( A ) and let φ be an accepting computation of A on a tree t ′ ,and ST R be a winning strategy of Pathfinder in G t, A . Let s := e , d , e , d , . . . , e i , d i , . . . be the play which is consistent with str φ and ST R . Then, there is i ∈ N such that e i is aninvalid move for Automaton in G t, A . Moreover, if e i is the first invalid move for Automatonin s , then t ( v ) = t ′ ( v ) for v := d . . . d i − .Proof. Assume, for the sake of contradiction, that s does not contain an invalid move forAutomaton, and let ( v i , q i ) be the i -th position of Automaton in π s . By definition of G t, A it is easy to see that π = v , . . . , v i , . . . is a branch in the full-binary tree. Since φ isan accepting computation of A on t ′ , we conclude that the maximal color which C assignsinfinitely often to states in φ ( π ) is even. By Claim 4.1.1(1) we have φ ( v i ) = q i , and therefore φ ( π ) = q . . . q i . . . . By the definition of C G we have C G ( v i , q i ) = C ( q i ) and we concludethat the maximal color which C assigns infinitely often in π s is even, and therefore the playis winning for Automaton - a contradiction to ST R being a winning strategy of Pathfinder.Therefore, Automaton makes an invalid move in s . Let e i = ( q l , q r ) be the first invalidmove of Automaton in s . Since e i is invalid we have ( q i , t ( v i ) , q l , q r ) / ∈ δ , and by definition of str φ we obtain ( q l , q r ) = ( φ ( v i · l ) , φ ( v i · r )). Since φ ( v i ) = q i we have ( φ ( v i ) , t ( v i ) , φ ( v i · l ) , φ ( v i · r )) / ∈ δ . φ is a computation of A on t ′ and therefore ( φ ( v i ) , t ′ ( v i ) , φ ( v i · l ) , φ ( v i · r )) ∈ δ , andwe conclude that t ( v i ) = t ′ ( v i ). Notice that by the definition of G t, A we have v i = d . . . d i − ,and the claim follows. (cid:4) MSO-definability.
Throughout this section we will use the following conventions andterminology.
Positional Pathfinder strategies as labeled trees:
A positional strategy
ST R for Pathfinderis a function in { l, r } ∗ × Q × Q → { l, r } . Hence, it can be considered as a Q × Q → { l, r } labeled tree. Below we will not distinguish between a positional Pathfinder’s strategyand the corresponding Q × Q → { l, r } labeled full-binary tree. In particular, we callsuch a strategy regular, if the corresponding tree is regular. MSO-definability:
We will use “MSO-definable” for “MSO-definable in the unlabeledfull-binary tree.”The rest of the proof deals with MSO-definability. By Claim 4.1.2, there is a function
Invalid A ( φ, ST R, t, v ) which, for every accepting computation φ of A on t ′ , returns a node v such that t ′ ( v ) = t ( v ). This function depends on the strategy ST R of Pathfinder. Therestriction of
Invalid A to the Pathfinder positional winning strategies in G t, A is MSO-definable (with parameters t and ST R ) by the following formula
Leads A ( φ, ST R, t, v ), which describes in MSO the play of φ against ST R up to the first invalid move of Au-tomaton (at the position ( v, φ ( v )).Define Leads A ( φ, ST R, t, v ) as the conjunction of:(1) φ ( ǫ ) = q I -the play starts from the initial position.(2) ∀ u < v : (( φ ( u ) , t ( u ) , φ ( u · l ) , φ ( u · r )) ∈ δ - all Automaton’s moves at the positions ( u, q ),where u is an ancestor of v respect δ . (By Claim 4.1.1(1), in any play consistent with φ , Automaton can reach only the positions of the form ( u, φ ( u ))).(3) ( φ ( v ) , t ( v ) , φ ( v · l ) , φ ( v · r )) / ∈ δ - the Automaton move at ( v, φ ( v )) is invalid.(4) ∀ u < v : ( ST R ( u, φ ( u · l ) , φ ( u · r )) = l ) ↔ u · l ≤ v )) - the Pathfinder moves d . . . d j . . . are consistent with ST R and are along the path from the root to v , i.e., d d . . . d j ≤ v .To sum up, we have the following claim: Claim 4.1.3.
Leads A ( φ, ST R, t, v ) defines a function which, for every tree t L ( A ) , everyPathfinder’s positional (in G t, A ) winning strategy ST R , and every φ ∈ ACC ( A , t ′ ) , returnsa node v such that t ( v ) = t ′ ( v ) . Claim 4.1.3 plays a crucial role in our proof. It is instructive to compare it with Theorem2.4 which implies that there is no MSO-definable function F ( t, D, v ) which for a tree t = t ′ and D := { u | t ( u ) = t ′ ( u ) } returns a node v such that t ( v ) = t ′ ( v ).The following claim is folklore. Due to the lack of references, it is proved in the Appen-dix. Claim 4.1.4.
Let t be a regular tree such that t / ∈ L ( A ) . Then, Pathfinder has a regularpositional winning strategy in G t , A . Let t be a regular tree such that t / ∈ L ( A ). By Claim 4.1.4 there is a regular positionalwinning strategy [ ST R of Pathfinder in G t , A . Now, we can substitute [ ST R and t forarguments ST R and t of Leads A and obtain the following Proposition: Proposition 4.6.
For every regular tree t / ∈ L ( A ) and a regular positional winning strategy [ ST R for Pathfinder in G t , A , there is an MSO-definable function which, for each acceptingcomputation φ of A on t ′ , returns a node v such that t ( v ) = t ′ ( v ) .Proof. Let ψ t ( σ ) and ψ [ ST R ( ST R ) be MSO-formulas that define t and [ ST R . Then, byClaim 4.1.3, ∃ σ ∃ ST R : ψ t ( σ ) ∧ ψ [ ST R ( ST R ) ∧ Leads A ( φ, ST R, σ, v ) defines such a function. (cid:4) Let us continue with the proof of Proposition 4.1. Recall that for trees t and t ′ and anantichain Y , we denote by t [ t ′ /Y ] the tree obtained from t by grafting t ′ at every node in Y . Claim 4.1.5.
Let t and t be regular trees. Then, there is an MSO-formula graft t ,t ( Y, σ ) defining a function which for every antichain Y returns the tree t [ t /Y ] .Proof of Claim 4.1.5. t and t are regular, and therefore there are MSO-formulas ψ t ( σ )and ψ t ( σ ) which defines t and t .Let ψ ≥ yt ( y, σ ) be a formula which is obtained from ψ t ( σ ) by relativizing the first-order quantifiers to ≥ y , i.e., by replacing subformulas of the form ∃ x ( . . . ) and ∀ x ( . . . ) by ∃ x ( x ≥ y ) ∧ ( . . . ) and ∀ x ( x ≥ y ) → ( . . . ). Then, v, t | = ψ ≥ yt ( y, σ ) iff t ≥ v = t . Hence, graft t ,t ( Y, σ ) can be defined as the conjunction of:
MBIGUITY HIERARCHY OF REGULAR INFINITE TREE LANGUAGES 15 (1) ∃ σ ψ t ( σ ) ∧ ∀ v - “if no Y node is an ancestor of v then σ ( v ) = σ ( v ),” and(2) ∀ y ( y ∈ Y ) → ψ ≥ yt ( y, σ ) - “at every node in Y a tree t is grafted.” (cid:4) Finishing Proof of Proposition 4.1.
Now, we have all the ingredients ready for theproof of Proposition 4.1.Let A be such that L ( A ) = L , and let α t , A , [ ST R ( φ, v ) be a formula which defines thefunction from Proposition 4.6 ( t [ t /Y ] now takes the role of t ′ ).Define a formula: Choice A ,t ,t , [ ST R ( Y, φ, y ) := y ∈ Y ∧ ∃ v ( α t , A , [ ST R ( φ, v ) ∧ v ≥ y ). Claim 4.1.6.
Choice A ,t ,t , [ ST R ( Y, φ, y ) defines a function which for every non-empty an-tichain Y and an accepting computation φ of A on t [ t /Y ] , returns a node y ∈ Y .Proof. By Proposition 4.6, α t , A , [ ST R ( φ, v ) returns a node v such that t ( v ) = ( t [ t /Y ])( v ).By definition of t [ t /Y ], there is a unique node y ∈ Y such that v ≥ y . (cid:4) Define
ChooseSubset A ,t ,t , [ ST R ( Y, X ) := ∀ x : x ∈ X iff the following conditions hold:(1) x ∈ Y and(2) ∃ σ such that(a) graft t ,t ( Y, σ ) - “ σ = t [ t /Y ]” and(b) ∃ φAcceptingRun A ( σ, φ ) ∧ Choice A ,t ,t , [ ST R ( Y, φ, x ), where
AcceptingRun A ( σ, φ )defines “ φ is an accepting computation of A on the tree σ .” Claim 4.1.7.
ChooseSubset A ,t ,t , [ ST R ( Y, X ) defines a function which maps every non-empty antichain Y to a non-empty subset X ⊆ Y . Moreover, | X | ≤ | ACC ( A, t [ t /Y ]) | .Proof. If Y is non-empty, then t [ t /Y ] ∈ L . Hence, A has at least one accepting com-putation on t [ t /Y ]. Therefore, X is non-empty, by Claim 4.1.6. The “Moreover” partimmediately follows from Claim 4.1.6. (cid:4) Let A be such that L ( A ) = L and assume, for the sake of contradiction, that A is finitelyambiguous. In particular, there are finitely many accepting computations of A on t [ t /Y ],and therefore by Claim 4.1.7, we conclude that ChooseSubset A ,t ,t , [ ST R ( Y, X ) assigns toevery non-empty antichain Y a finite non-empty X ⊆ Y - a contradiction to Lemma 2.5.5. k -Ambiguous Languages In this section we prove that for every 0 < k ∈ N , there is a tree language with the degreeof ambiguity equal to k . First, we introduce some notations. For a letter σ , we denote by t σ , the full-binary tree with all nodes labeled by σ . Let L ¬ a ∨···∨¬ a k := L ¬ a ∪ · · · ∪ L ¬ a k bea tree language over alphabet Σ n = { c, a , a , ..., a n } , where L ¬ a i := { t ∈ T ω Σ n | no node in t is labeled by a i } . Proposition 5.1.
The degree of ambiguity of L ¬ a ∨···∨¬ a k for k ≤ n is k . It is easy to see that L ¬ a i are accepted by deterministic PTA. Therefore, by Lemma3.1, we obtain that L ¬ a ∨···∨¬ a k is k -ambiguous. In the rest of this section we will show that L ¬ a ∨···∨¬ a k is not ( k − L ¬ a ∨¬ a is ambiguous. Lemma 5.2.
Let L ∃ a ∧···∧∃ a m := { t ∈ T ω Σ n | for every i ≤ m there is a node in t labeled by a i } , and let L be a tree language such that t c / ∈ L and L ∃ a ∧···∧∃ a m ∩ T ω { c,a ,...,a m } ⊆ L . Then, L is not finitely ambiguous.Proof. Define a function F : Σ ∗ → Σ such that F ( σ . . . σ k ) := a k − i +1 if there is i such that σ i = a , for all j < i : σ j = a and k − i + 1 ≤ m . Otherwise, F ( σ . . . σ k ) := c .It is easy to see that F is definable by a Moore machine, and ∀ t ∈ T ω Σ : t ∈ L ∃ a iff b F ( t ) ∈ L . Therefore, by Lemma 3.10 we conclude that da ( L ) ≥ da ( L ∃ a ). Since L ∃ a is notfinitely ambiguous (by Corollary 4.4 (3)), we conclude that L is not finitely ambiguous. Notations.
Let a ∈ Σ, t ∈ T ω Σ and t ∈ T ω Σ . We define T ree ( a, t , t ) ∈ T ω Σ as a tree t where t ( ǫ ) = a , t ≥ l = t and t ≥ r = t . Lemma 5.3.
Let A be a finitely ambiguous automaton over alphabet Σ n such that L ( A ) = L ¬ a ∨···∨¬ a k for k ≤ n . Then | ACC ( A , t c ) | ≥ k .Proof. We will prove by induction on k . For k = 1 the claim holds trivially, since t c ∈ L ( A )implies that | ACC ( A , t c ) | ≥ k < m ≤ n and prove for k = m .Let A = ( Q, Σ , Q I , δ, C ) be a finitely ambiguous automaton which accepts L ¬ a ∨···∨¬ a m .Define R := { ( q , q ) ∈ Q × Q | ∃ q i ∈ Q I : ( q i , c, q , q ) ∈ δ ) } , and let R [1] and R [2] be theprojections of the first and second coordinate of R on Q , respectively.Define Q ∃ a m := { q ∈ R [1] | L ( A q ) ∩ L ∃ a m = ∅} , and let Q ∃ a m ∧ t c := { q ∈ Q ∃ a m | t c ∈ L ( A q ) } and Q ∃ a m ∧¬ t c := Q ∃ a m \ Q ∃ a m ∧ t c .By definition of Q ∃ a m ∧¬ t c we have t c / ∈ L ( A Q ∃ am ∧¬ tc ) and therefore L ( A Q ∃ am ∧¬ tc ) ∩ T ω { c,a m } ⊆ T ω { c,a m } \{ t c } . The language T ω { c,a m } \{ t c } is not finitely ambiguous by Corollary 4.4(2). L ( A Q ∃ am ∧¬ tc ) is finitely ambiguous (by Corollary 3.5) and since T ω { c,a m } is unambiguouswe conclude that L ( A Q ∃ am ∧¬ tc ) ∩ T ω { c,a m } is finitely ambiguous, by Corollary 3.2. Therefore,by Lemma 3.6, there is a tree t ′ ∈ T ω { c,a m } \{ t c } = L ∃ a m ∩ T ω { c,a m } such that t ′ / ∈ L ( A Q ∃ am ∧¬ tc ),and since L ∃ a m ∩ T ω { c,a m } ⊆ L ( A Q ∃ am ) = L ( A Q ∃ am ∧ tc ) ∪ L ( A Q ∃ am ∧¬ tc ) we conclude that t ′ ∈ L ( A Q ∃ am ∧ tc ).Define Q ′ := { q ∈ R [1] | t ′ ∈ L ( A q ) } and R ′ := { ( q , q ) ∈ R | q ∈ Q ′ } . Since t ′ ∈ L ∃ a m ∩ T ω { c,a m } , we conclude that { t ∈ T ω Σ | T ree ( c, t ′ , t ) ∈ L ¬ a ∨···∨¬ a m } = L ¬ a ∨···∨¬ a m − .Therefore, L ( A R ′ [2] ) = L ¬ a ∨···∨¬ a m − , and by induction assumption we obtain | ACC ( A R ′ [2] , t c ) | ≥ m − φ ∈ ACC ( A R ′ [2] , t c ) we will construct a computation g ( φ ) ∈ ACC ( A , t c ), as following. Let q := φ ( ǫ ). By the definition of R ′ , there is ( q , q ) ∈ R ′ such that t ′ ∈ L ( A q ). Since t ′ ∈ L ( A Q ∃ am ∧ tc ) we have t c ∈ L ( A q ), and therefore thereis a computation φ c ∈ ACC ( A q , t c ). Let q i ∈ Q I such that ( q i , c, q , q ) ∈ δ . By defining g ( φ ) := T ree ( q i , φ c , φ ) we obtain that g ( φ ) ∈ ACC ( A , t c ), as requested.Let Φ := { g ( φ ) | φ ∈ ACC ( A R ′ [2] , t c ) } . g ( φ ) ≥ r = φ and therefore g is injective, and weconclude that | Φ | = | ACC ( A R ′ [2] , t c ) | ≥ m − φ ∈ ACC ( A , t c ) such that φ / ∈ Φ,resulting | ACC ( A , t c ) | ≥ m .Let Q ∃ a ∧···∧∃ a m − := { q ∈ R [2] | L ( A q ) ∩ L ∃ a ∧···∧∃ a m − = ∅} and let Q t c ∧∃ a ∧···∧∃ a m − := { q ∈ Q ∃ a ∧···∧∃ a m − | t c ∈ L ( A q ) } and Q ¬ t c ∧∃ a ∧···∧∃ a m − := Q ∃ a ∧···∧∃ a m − \ Q t c ∧∃ a ∧···∧∃ a m − . MBIGUITY HIERARCHY OF REGULAR INFINITE TREE LANGUAGES 17
Claim 5.3.1.
There is a tree t ′′ ∈ L ∃ a ∧···∧∃ a m − ∩ T ω { c,a ,...,a m − } such that t ′′ ∈ L ( A Q tc ∧∃ a ∧···∧∃ am − ) and t ′′ / ∈ L ( A Q ¬ tc ∧∃ a ∧···∧∃ am − ) .Proof. By the definition of R [2] we have L ∃ a ∧···∧∃ a m − ∩ T ω { c,a ,...,a m − } ⊆ L ( A R [2] ) and there-fore by the definition of Q t c ∧∃ a ∧···∧∃ a m − and Q ¬ t c ∧∃ a ∧···∧∃ a m − , we have L ∃ a ∧···∧∃ a m − ∩ T ω { c,a ,...,a m − } ⊆ L ( A Q tc ∧∃ a ∧···∧∃ am − ) ∪ L ( A Q ¬ tc ∧∃ a ∧···∧∃ am − ).Assume, for the sake of contradiction, that the claim does not hold. Then, we obtain L ∃ a ∧···∧∃ a m − ∩ T ω { c,a ,...,a m − } ⊆ L ( A Q ¬ tc ∧∃ a ∧···∧∃ am − ). We have t c / ∈ L ( A Q ¬ tc ∧∃ a ∧···∧∃ am − ),and therefore by Lemma 5.2 we conclude that L ( A Q ¬ tc ∧∃ a ∧···∧∃ am − ) is not finitely ambigu-ous - a contradiction to A being finitely ambiguous. (cid:4) Let t ′′ be a tree as in Claim 5.3.1. We have t ′′ ∈ L ∃ a ∧···∧∃ a m − ∩ T ω { c,a ,...,a m − } ,and therefore T ree ( c, t c , t ′′ ) ∈ L ¬ a ∨···∨¬ a m = L ( A ), and there is a computation φ ∈ ACC ( A , T ree ( c, t c , t ′′ )). Let q := φ ( r ). By definition of t ′′ , we have q ∈ Q t c ∧∃ a ∧···∧∃ a m − and therefore t c ∈ L ( A q ). Let φ c ∈ ACC ( A q , t c ), and let φ ′ be the computation obtainedfrom φ by grafting φ c on r . We conclude that φ ′ ∈ ACC ( A , t c ).Assume, for the sake of contradiction, that φ ′ ∈ Φ, and let q := φ ′ ( l ) and q := φ ′ ( r ). We have t ′ ∈ L ( A q ) (by definition of | Φ | ) and t ′′ ∈ L ( A q ) (by definition of φ ′ ).Therefore, by grafting computations φ t ′ ∈ ACC ( A q , t ′ ) and φ t ′′ ∈ ACC ( A q , t ′′ ) to the leftand right children of the root of t c , respectively, we obtain T ree ( c, t ′ , t ′′ ) ∈ L ( A ). That isa contradiction, since t ′ contains an a m labeled node, and t ′′ contains a , . . . , a m − labelednodes, and therefore T ree ( c, t ′ , t ′′ ) / ∈ L ¬ a ∨···∨¬ a m .We conclude that φ ′ / ∈ Φ, and therefore | ACC ( A , t c ) | ≥ | Φ | = 1 + ( m −
1) = m .6. Finitely Ambiguous Languages
Definition 6.1.
Let Σ = { a , a , c } . We define the following languages over Σ: • For k, m ∈ N such that k < m , we define L k,m as the set of trees t which are obtainedfrom t c by grafting a tree t ′ ∈ L ¬ a ∨¬ a on node l k r , and grafting t a on node l m . • For m ∈ N we define L m := ∪ k The degree of ambiguity of L fa is finite. The proposition follows from Lemma 6.3 and Lemma 6.6 proved below. Lemma 6.3. There is a finitely ambiguous automaton which accepts L fa Proof. On a tree t ∈ L m the automaton “guesses” a position i < m , checks that t ≥ l i r ∈ L ¬ a ∨¬ a (using a 2-ambiguous automaton), checks that t ≥ l j r = t c for all j = i ∧ j < m ,and checks that t ≥ l m = t a (using deterministic automata). Below, a more detailed proof isgiven.First, notice that there are deterministic PTA A c , A a , A ¬ a and A ¬ a which acceptslanguages { t c } , { t a } , L ¬ a and L ¬ a , respectively.By Lemma 3.1, there is a 2-ambiguous automaton A ¬ a ∨¬ a which accepts the language L ¬ a ∨¬ a := L ¬ a ∪ L ¬ a .We will construct an automaton B := ( Q B , Σ B , Q I B , δ B , C B ) which accepts L fa . • Q B is defined as the union of states of A a , A c and A ¬ a ∨¬ a , along with additional states q , q . • Σ B := { a , a , c }• Q I B := { q }• δ B will consists of the transitions of A a , A c and A ¬ a ∨¬ a , along with additional transi-tions: – ( q , c, q , p ) ∈ δ B for p an initial state in A c – ( q , c, q , p ) ∈ δ B for p an initial state in A ¬ a ∨¬ a – ( q , c, q , p ) ∈ δ B for p an initial state in A c – ( q , a , p, p ) ∈ δ B for p an initial state in A a • C B ( q ) := 1, C B ( q ) := 1, and for other states, the assigned color would be the same asin the automaton the state has originated from ( A a , A c or A ¬ a ∨¬ a )It is easy to see that L ( B ) = L fa .Let t ∈ L ( B ). By definition of L fa , there is m ∈ N such that t ∈ L m . If φ is anaccepting computation on t , then φ assigns to the first m + 2 nodes on the leftmost branchthe sequence q , . . . , q | {z } i times · q , . . . , q | {z } m − i + 1 times · q a for some i ∈ { , . . . , m } , where q a is the initialstate of A a (total m possibilities). φ assigns to l j · r the initial state of A c if j < i − i − < j < m ; the initial state of A ¬ a ∨¬ a if j = i − 1; and the initial state of A a if j ≥ m .Since A c and A a are deterministic and A ¬ a ∨¬ a is 2-ambiguous, the number of acceptingcomputations on t is at most 2 m , hence, finite. Lemma 6.4. Let L be a tree language such that L m ⊆ L ⊆ L fa . Then, L is not m − ambiguous.Proof. Let A be an automaton with states Q which accepts L , and assume A is finitelyambiguous. Define a set Q ′ ⊆ Q by Q ′ := { φ ( l i r ) | i < m ∧ ∃ t ∈ L : φ ∈ ACC ( A , t ) } and Q ∃ a := { q ∈ Q ′ | L ∃ a ∩ L ( A q ) = ∅} , and let Q t c ∧∃ a := { q ∈ Q ∃ a | t c ∈ L ( A q ) } and Q ¬ t c ∧∃ a := Q ∃ a \ Q t c ∧∃ a .Relying on the fact that T ω { c,a } \ { t c } is not finitely ambiguous (by Corollary 4.4 (2)),we derive the following claim: Claim 6.4.1. There is a tree t ∃ a ∈ (cid:0) T ω { c,a } \ { t c } (cid:1) ∩ (cid:0) L ( A Q tc ∧∃ a ) \ L ( A Q ¬ tc ∧∃ a ) (cid:1) . (cid:4) Recall that t m is the tree which is obtained from t c by grafting t a on node l m . For each i < m , define t mi as the tree which is obtained from t m by grafting t ∃ a on node l i r . It isclear that t mi ∈ L ( A ), and therefore there is an accepting computation φ i of A on t mi . t ∃ a ∈ L ( A Q tc ∧∃ a ) \ L ( A Q ¬ tc ∧∃ a ) and since t ∃ a ∈ A φ i ( l i r ) we conclude that φ i ( l i r ) ∈ Q t c ∧∃ a and therefore t c ∈ L ( A φ i ( l i r ) ). Let φ ci ∈ ACC ( A φ i ( l i r ) , t c ), and construct a compu-tation φ ′ i from φ i by grafting φ ci on l i r . This tree which is obtained from t mi by grafting t c on l i r is the tree t m and therefore φ ′ i ∈ ACC ( A , t m ).We are going to show that for all i < j < m , the computations φ ′ i , φ ′ j ∈ ACC ( A , t m )are different. Assume towards a contradiction φ ′ i = φ ′ j and let b φ := φ ′ i . Define p i := b φ ( l i r ), p j := b φ ( l j r ), and let φ p i ∈ ACC ( A p i , t ∃ a ) and φ p j ∈ ACC ( A p , t ∃ a ). Construct t ′ from t m by grafting t ∃ a on nodes l i r and l j r , and construct φ ′ from b φ by grafting φ p i on l i r and φ p on l j r . It follows that φ ′ is an accepting computation of A on t ′ , which is a contradiction,since t ′ / ∈ L fa (since t ′≥ l j r = t ′≥ l i r = t ∃ a = t c ) and therefore t ′ / ∈ L (since L ⊆ L fa ). Weconclude that there are at least m different accepting computations of A on t m . MBIGUITY HIERARCHY OF REGULAR INFINITE TREE LANGUAGES 19 Remark 6.5. The language L m is 2 m ambiguous but not m − Lemma 6.6. L fa is not boundedly ambiguousProof. ∀ m ∈ N : L m ⊆ L fa , and therefore from Lemma 6.4 it follows that L fa is not( m − L fa is not boundedly ambiguous.7. Uncountably Ambiguous Languages In this section we introduce a scheme for obtaining uncountably ambiguous languages fromlanguages which are not boundedly ambiguous. We then use this scheme to obtain naturalexamples of tree languages which are uncountably ambiguous. Definition 7.1. Let L ¬ ba be an arbitrary regular tree language over alphabet Σ which isnot boundedly ambiguous, and let L be an arbitrary regular tree language over alphabetΣ such that L ∩ L ¬ ba = ∅ . Let c ∈ Σ and define a language L [ L , L ¬ ba ] over alphabet Σ: t ∈ L [ L , L ¬ ba ] iff the following conditions hold: • ∀ v ∈ l ∗ : t ( v ) = c • There is an infinite set I ⊆ N such that ∀ i ∈ I : t ≥ l i · r ∈ L ¬ ba and ∀ i I : t ≥ l i · r ∈ L . Proposition 7.2. The degree of ambiguity of L [ L , L ¬ ba ] is ℵ .Proof. Let A = ( Q, Σ , Q I , δ, C ) be a PTA which accepts L [ L , L ¬ ba ]. We will show that da ( A ) = 2 ℵ .Let Q ′ := { φ ( u ) | u ∈ l ∗ · r and ∃ t : φ ∈ ACC ( A , t ) } , and define Q unamb ∧¬ L := { q ∈ Q ′ | A q is unambiguous and L ( A q ) ∩ L = ∅} . Claim 7.2.1. L ( A Q unamb ∧¬ L ) ⊆ L ¬ ba .Proof. Assume, for the sake of contradiction, that there is a tree t ∈ L ( A Q unamb ∧¬ L ) suchthat t / ∈ L ¬ ba . By definition of Q unamb ∧¬ L we conclude that t / ∈ L .Let q ∈ Q unamb ∧¬ L such that t ∈ L ( A q ) and let φ ∈ ACC ( A q , t ). Recall that q ∈ Q ′ (since Q unamb ∧¬ L ⊆ Q ′ ) and therefore there is a tree t ′ ∈ L ( A ), a computation φ ′ ∈ ACC ( A , t ) and a node u ∈ l ∗ · r such that φ ′ ( u ) = q . By the grafting lemma we concludethat φ ′ ◦ u φ is an accepting computation of A on t ′ ◦ u t . Therefore, t ′ ◦ u t ∈ L ( A ) for t / ∈ L ¬ ba ∪ L - a contradiction to definition of A . (cid:4) Notice that L ( A Q unamb ∧¬ L ) is boundedly ambiguous by Corollary 3.2 (as a finite unionof unambiguous languages), and since L ¬ ba is not boundedly ambiguous we conclude that da ( L ( A Q unamb ∧¬ L )) = da ( L ¬ ba ). By Claim 7.2.1 we obtain L ( A Q unamb ∧¬ L ) ⊆ L ¬ ba , and ap-plying Lemma 3.6 we conclude that there is a tree t ¬ ba ∈ L ¬ ba such that t ¬ ba / ∈ L ( A Q unamb ∧¬ L ).Let c ∈ Σ be as in the definition of L [ L , L ¬ ba ], and let t c be a tree where all nodes arelabeled by c . Let A := l ∗ · r be an antichain, and define t ′′ := t c ◦ A t ¬ ba . By the definition of A it is clear that t ′′ ∈ L ( A ). Let φ ′′ ∈ ACC ( A , t ′′ ), and let B := { u ∈ A | L ( A φ ′′ ( u ) ) ∩ L = ∅} .For each u ∈ B there is a tree t u ∈ L and a computation φ u ∈ ACC ( A φ ′′ ( u ) , t u ).Therefore, by the grafting lemma, we conclude that the tree t ′′′ which is obtained from t ′′ by grafting t u on each node u ∈ B is in L ( A ). Assume, for the sake of contradiction, that A \ B is finite. By definition of t ′′′ , for each i ∈ N such that u := l i · r ∈ B we have t ′′′≥ l i · r = t u ∈ L . Therefore, |{ i ∈ N | t ′′′≥ l i · r ∈ L ¬ ba }| = |{ u ∈ A | t ′′′≥ u ∈ L ¬ ba }| = |{ u ∈ A \ B | t ′′′≥ u ∈ L ¬ ba }| = | A \ B | < ℵ , and by definition of L [ L , L ¬ ba ] we conclude that t ′′′ / ∈ L [ L , L ¬ ba ] - a contradiction to the definition of A . A \ B is infinite, and therefore there is a state q and an infinite set b A ⊆ A \ B suchthat φ ′′ ( u ) = q for all u ∈ b A . Recall that ∀ u ∈ b A : t ′′≥ u = t ¬ ba . Notice that for each u ∈ b A we have u / ∈ B , and by definition of B we obtain L ( A φ ′′ ( u ) ) ∩ L = L ( A q ) ∩ L = ∅ . Since t ¬ ba / ∈ L ( A Q unamb ∧¬ L ) we conclude that q / ∈ Q unamb ∧¬ L - hence, A q is ambiguous.Let t amb ∈ L ( A q ) be a tree with at least two accepting computations φ , φ ∈ ACC ( A q , t amb ).Let b t := t ′′ ◦ b A t amb , and b φ := φ ◦ b A φ . By the grafting lemma we obtain b φ ∈ ACC ( A , b t ). Foreach A ′ ⊆ b A , define a computation φ A ′ := b φ ◦ A ′ φ . Notice that φ A ′ ∈ ACC ( A , b t ) (by thegrafting lemma) and that ∀ A , A ⊆ b A : A = A → φ A = φ A (since φ = φ ). Therefore, | ACC ( A , b t ) | ≥ |{ A ′ | A ′ ⊆ b A }| = 2 ℵ , and da ( A ) = 2 ℵ , as requested.We will now introduce a couple of definitions, and present three natural examples ofinfinite tree languages which are not countable ambiguous. Definition 7.3 (characteristic tree) . The characteristic tree of U , . . . , U n ⊆ { l, r } ∗ is a { , } n -labeled tree t [ U , . . . , U n ] such that t [ U , . . . , U n ]( u ) := ( b , . . . , b n ) where b i = 1 iff u ∈ U i for each 1 ≤ i ≤ n . Definition 7.4. For a set U ⊆ { l, r } ∗ we define U ↓ as the downward closure of U . Definition 7.5. A set X ⊆ { l, r } ∗ is called perfect if X = ∅ and ∀ u ∈ X : ∃ v , v ∈ X such that v , v > u and v ⊥ v . Proposition 7.6. The following regular languages are not countably ambiguous: (1) L X ⊆ Y ↓ := { t [ X, Y ] | X ⊆ Y ↓} - “for each node in X there is a greater or equal nodein Y .” (2) L no − max := { t [ X ] | X has no maximal element } - “for each node in X there is a greaternode in X .” (3) L perf := { t [ X ] | X is perfect } - “for each node in X there are at least two greaterincomparable nodes in X .” In the rest of this section we will prove Proposition 7.6. Proof of Proposition 7.6(1). Let L left := { t [ X, Y ] | X = l ∗ and Y ∩ l ∗ = ∅} . It is easy tosee that L left can be accepted by a deterministic PTA, and therefore da ( L left ) = 1.By Lemma 3.1 we conclude that da ( L X ⊆ Y ↓ ∩ L left ) ≤ da ( L X ⊆ Y ↓ ) · da ( L left ) = da ( L X ⊆ Y ↓ ).We will show that L X ⊆ Y ↓ ∩ L left is not countably ambiguous. By the above inequality, thisimplies that L X ⊆ Y ↓ is not countably ambiguous. Claim 7.6.1. Let L X = ∅ ,Y = ∅ := { t [ X, Y ] | X = ∅ and Y = ∅} . Then t ′ ∈ L X ⊆ Y ↓ ∩ L left iffthe following conditions hold: (1) ∀ u ∈ l ∗ : t ′ ( u ) = (1 , There is an infinite set I ⊆ N such that: (a) If i ∈ I then t ′≥ l i · r ∈ L X = ∅ ,Y = ∅ (b) If i / ∈ I then t ′≥ l i · r ∈ { t [ ∅ , ∅ ] } MBIGUITY HIERARCHY OF REGULAR INFINITE TREE LANGUAGES 21 Proof. ⇒ : Let t ′ ∈ L X ⊆ Y ↓ ∩ L left . By definition of L left it is clear that the condition (1)holds, and that for each i ∈ N : t ′≥ v i · l ∈ L X = ∅ ,Y = ∅ or t ′≥ v i · l = t [ ∅ , ∅ ]. Assume, for the sakeof contradiction, that the set { i ∈ N | t ′≥ v i · l ∈ L X = ∅ ,Y = ∅ } is finite. Therefore, by the secondcondition, there is an index k ∈ N such that ∀ i ≥ k : t ′≥ v i · l = t [ ∅ , ∅ ]. Let u := l k . By thedefinition of L left we have u ∈ X , and for each v ≥ u we have either t ′ ( v ) = (1 , 0) if v ∈ l ∗ ,or t ′ ( v ) = (0 , 0) otherwise. Hence, ∀ v ≥ u : v / ∈ Y , in contradiction to t ′ ∈ L X ⊆ Y ↓ . ⇐ : Assume that the conditions hold for t ′ . It is easy to see that t ′ ∈ L left . We willshow that t ′ ∈ L X ⊆ Y ↓ . Assume, for the sake of contradiction, that there is a node u ∈ X such that v / ∈ Y for each node v ≥ u . Since all nodes in X are in l ∗ we conclude that thereis i ∈ N such that u = l i . Notice that the set I ⊆ N is infinite, and therefore there is j > i such that t ′≥ l j · r ∈ L X = ∅ ,Y = ∅ . Therefore, there is a node v ≥ l j · r > l i = u such that v ∈ Y -a contradiction. (cid:4) Observe that the language L X = ∅ ,Y = ∅ := { t [ X, Y ] | X = ∅ and Y = ∅} can be consideredas a tree language over alphabet { } × { , } , and that L X = ∅ ,Y = ∅ = T ω { }×{ , } \ { t [ ∅ , ∅ ] } .Therefore, by Corollary 4.4(2) we conclude that L X = ∅ ,Y = ∅ is not finitely ambiguous.Notice that by Claim 7.6.1 we obtain L X ⊆ Y ↓ ∩ L left = L [ L , L ¬ ba ], for L = { t [ ∅ , ∅ ] } and L ¬ ba = L X = ∅ ,Y = ∅ . Therefore, applying Proposition 7.2 we conclude that L X ⊆ Y ↓ ∩ L left is not countably ambiguous.To prove Proposition 7.6(2), we will first prove the following lemma: Lemma 7.7. L no − max is not finitely ambiguous.Proof. Let A = ( Q, Σ , Q I , δ, C ) be a PTA which accepts L no − max . Let Q ′ := { q ∈ Q | ∃ q i ∈ Q I ∃ q ′ ∈ Q : ( q i , , q, q ′ ) ∈ δ and t [ ∅ ] ∈ L ( A q ′ ) } . Claim 7.7.1. Define L ¬∅ := T ω Σ \ { t [ ∅ ] } . Then: (1) L no − max \ { t [ ∅ ] } ⊆ L ( A Q ′ )(2) L ( A Q ′ ) ⊆ L ¬∅ Proof. (1) Let t ′ ∈ L no − max \ { t [ ∅ ] } , and let t ǫ := t [ { ǫ } ] (that is, t ǫ ( ǫ ) := 1, and ∀ u = ǫ : t ǫ ( u ) := 0). Let t ′′ := t ǫ ◦ l t ′ ◦ r t [ ∅ ]. By the definition of L no − max we obtain t ′′ ∈ L no − max .Therefore, there is a computation φ ∈ ACC ( A , t ′′ ) such that φ ( l ) ∈ Q ′ and t ′ ∈ L ( A φ ( l ) ), asrequested.(2) Assume, for the sake of contradiction, that t [ ∅ ] ∈ L ( A Q ′ ). Then there is a transition( q i , , q , q ) ∈ δ from an initial state q i such that t [ ∅ ] ∈ L ( A q ) and t [ ∅ ] ∈ L ( A q ). Therefore,we conclude that t ǫ := t [ { ǫ } ] is accepted by A - a contradiction to the definition of L no − max . (cid:4) Let Σ := { , } . Define a function F : Σ ∗ → Σ such that F ( σ , . . . , σ m ) := ( ∃ ≤ i ≤ m : σ i = 10 otherwiseIt is easy to see that F is definable by a Moore machine. We show that F reduces L ¬∅ to L ( A Q ′ ).Notice that ∀ t ′ ∈ T ω Σ : t ′ ∈ L ¬∅ → b F ( t ′ ) ∈ L no − max \ { t [ ∅ ] } . Since L no − max \ { t [ ∅ ] } ⊆ L ( A Q ′ ) (by Claim 7.7.1(1)) we conclude that ∀ t ′ ∈ T ω Σ : t ′ ∈ L ¬∅ → b F ( t ′ ) ∈ L ( A Q ′ ). Conversely, ∀ t ′ ∈ T ω Σ : b F ( t ′ ) ∈ L ¬∅ → t ′ ∈ L ¬∅ , and since L ( A Q ′ ) ⊆ L ¬∅ (by Claim 7.7.1(2))we obtain ∀ t ′ ∈ T ω Σ : b F ( t ′ ) ∈ L ( A Q ′ ) → t ′ ∈ L ¬∅ .Therefore, by Lemma 3.10, we conclude that da ( L ( A Q ′ )) ≥ da ( L ¬∅ ). Notice that L ¬∅ = T ω Σ \ { t [ ∅ ] } and by Corollary 4.4(2) we obtain da ( L ¬∅ ) ≥ ℵ . Hence, A Q ′ is notfinitely ambiguous, and by Corollary 3.5 we conclude that da ( A ) ≥ ℵ . Proof of Proposition 7.6(2). Let L l ∗ ∩ X = ∅ := { t [ X ] | X ∩ l ∗ = ∅} . It is easy to construct adeterministic PTA which accepts L l ∗ ∩ X = ∅ , and therefore da ( L l ∗ ∩ X = ∅ ) = 1.By Lemma 3.1 we conclude that da ( L no − max ∩ L l ∗ ∩ X = ∅ ) ≤ da ( L no − max ) · da ( L l ∗ ∩ X = ∅ ) = da ( L no − max ). We will show that da ( L no − max ∩ L l ∗ ∩ X = ∅ ) = 2 ℵ , and the lemma will follow.Notice that t ′ ∈ L no − max ∩ L l ∗ ∩ X = ∅ iff the following hold: • ∀ u ∈ l ∗ : t ( u ) = 0 • ∀ u ∈ l ∗ · r : t ′≥ u ∈ L no − max It is easy to see that L no − max ∩ L l ∗ ∩ X = ∅ = L [ L , L ¬ ba ] for L ¬ ba := L no − max (which isnot boundedly ambiguous, by Lemma 7.7) and L := ∅ . Therefore, by Proposition 7.2 weconclude that da ( L no − max ∩ L l ∗ ∩ X = ∅ ) = 2 ℵ , as requested. Proof of Proposition 7.6(3). Let L contains − l ∗ := { t [ X ] | l ∗ ⊆ X } . It is easy to see that L contains − l ∗ can be accepted by a deterministic PTA, and therefore da ( L contains − l ∗ ) = 1.Look at the language L perf ∩ L contains − l ∗ . By Lemma 3.1 we obtain da ( L perf ∩ L contains − l ∗ ) ≤ da ( L perf ) · da ( L contains − l ∗ ) = da ( L perf ). We will show that L perf ∩ L contains − l ∗ is not count-ably ambiguous. By the above inequality, this implies that da ( L perf ) = 2 ℵ . Claim 7.7.2. L perf is not finitely ambiguous.Proof. Define a function F : Σ ∗ → Σ such that F ( σ , . . . , σ m ) := ( ∃ ≤ i ≤ m : σ i = 10 otherwise. .It is easy to see that F is definable by a Moore machine, and that ∀ t ′ ∈ T ω Σ : t ′ ∈ T ω Σ \{ t [ ∅ ] } ↔ b F ( t ) ∈ L perf . Notice that T ω Σ \ { t [ ∅ ] } is not finitely ambiguous (by Corollary 4.4(2)), andtherefore by Lemma 3.10 we conclude that L perf is not finitely ambiguous. (cid:4) Claim 7.7.3. t ′ ∈ L perf ∩ L contains − l ∗ iff the following conditions hold: (1) ∀ u ∈ l ∗ : t ′ ( u ) = 1(2) There is an infinite set I ⊆ N such that ∀ i ∈ I : t ′≥ l i · r ∈ L perf and ∀ i I : t ′≥ l i · r ∈ { t [ ∅ ] } .Proof. ⇒ : Let t ′ ∈ L perf ∩ L contains − l ∗ . By definition of L contains − l ∗ it is clear that condition(1) holds for t ′ . Notice that ∀ i ∈ N : t ′≥ l i · r ∈ L perf or t ′≥ l i · r = t [ ∅ ]. Assume, for the sakeof contradiction, that { i ∈ N | t ′≥ l i · r ∈ L perf } is finite. Therefore, there is k ∈ N such that ∀ i ≥ k : t ′≥ l i · r = t [ ∅ ]. Let u := l k , and notice that t ′ ( u ) = 1, and ∀ v > u : t ′ ( v ) = 1 ↔ v ∈ l ∗ .Hence, each pair of 1-labeled nodes which are greater than u are comparable - a contradictionto the definition of L perf . ⇐ : Let t ′ such that the conditions hold. By the first condition it is clear that t ′ ∈ L contains − l ∗ . We will prove that t ′ ∈ L perf , and the claim will follow. First, notice that t ′ ( ǫ ) = 1, and therefore t ′ = t [ ∅ ]. Let u be a node such that t ′ ( u ) = 1. If u ∈ l ∗ then bythe second condition, there is a node v ∈ l ∗ · r such that v > u and t ≥ v ∈ L perf . Therefore,there are two nodes w , w > v > u such that w ⊥ w and t ′ ( w ) = t ′ ( w ) = 1. Otherwise( u / ∈ l ∗ ), there is a node v ∈ l ∗ · r , such that u > v and t ≥ v ∈ L perf , and by definition of L perf MBIGUITY HIERARCHY OF REGULAR INFINITE TREE LANGUAGES 23 we conclude that there are two nodes w , w > u such that w ⊥ w and t ′ ( w ) = t ′ ( w ) = 1- hence, t ′ ∈ L perf . (cid:4) It is easy to see that L perf ∩ L contains − l ∗ = L [ L , L ¬ ba ] for L ¬ ba := L perf (which is notboundedly ambiguous, by Claim 7.7.2) and L := { t [ ∅ ] } . Therefore, by Proposition 7.2 weconclude that L perf ∩ L contains − l ∗ = 2 ℵ , as requested.Observe that our proof shows that L perf ∧ min := { t [ X ] | X is perfect and has the ≤ -minimal element } is also uncountable ambiguous. We conclude with an instructive exampleof an unambiguous language which is similar to L perf ∧ min . Let X ⊆ { l, r } ∗ be a set of nodes.We say that u ∈ X is a X -successor of v if u > v and there is no node w ∈ X such that v < w < u . We call X a full-binary subset-tree if X has a minimal node, and each node in X has two X -successors.Note that if X is a full-binary subset tree then X is perfect and has the ≤ -minimalelement. However the language L binary := { t [ X ] | X is a full-binary subset tree } is unam-biguous. 8. Countable Languages are Unambiguous In this section we prove the following Proposition: Proposition 8.1. Each regular countable tree language is unambiguous To prove Proposition 8.1 we first recall finite tree automata (Subsec. 8.1). Then, wepresent Niwi´nski’s Representation for Countable Languages (Subsec. 8.2). Finally, theproof of Proposition 8.1 is given (Subsec. 8.3).8.1. Finite Trees and Finite Tree Automata.Finite Trees . A finite tree is a finite set U ⊆ { l, r } ∗ which is closed under prefix relation. U is called a finite binary tree if ∀ u ∈ U : u · l ∈ U ↔ u · r ∈ U . Finite Σ -labeled Binary Trees . Let Σ be partitioned into two sets: Σ - labels of internalnodes, and Σ - labels of leaves. A finite Σ-labeled binary tree is a function t U : U → Σ,where U ⊆ { l, r } ∗ is a finite binary tree, t U ( v ) ∈ Σ if v is a leaf, and t U ( v ) ∈ Σ if v haschildren.When it is clear from the context, we will use “finite tree” or “labeled finite tree” for“Σ-labeled finite binary tree”. Finite Tree Automata (FTA) . An automaton over Σ-labeled finite trees is a tuple B =( Q, Σ , Q I , δ ), where Q is a finite set of states, Σ = Σ ∪ Σ is an alphabet, Q I is a set ofinitial states, and δ ⊆ ( Q × Σ ) ∪ ( Q × Σ × Q × Q ) is a set of transitions.An accepting computation of B on a finite tree t U is a function φ : U → Q , such that φ ( ǫ ) ∈ Q I , and for each node u ∈ U , if u is not a leaf then ( φ ( u ) , t U ( u ) , φ ( u · l ) , φ ( u · r )) ∈ δ ,and otherwise ( φ ( u ) , t U ( u )) ∈ δ .The language of a FTA B is the set of finite trees t such that B has an acceptingcomputation on t . A finite tree language is regular iff it is accepted by a FTA. It is well-known that every regular finite tree language is unambiguous (i.e., for every finite treelanguage, there is an unambiguous automaton which accepts it). Niwi´nski’s Representation for Countable Languages.Definition 8.2. Define T fin Σ( { x ,...,x n } ) as the set of finite trees over alphabet Σ ∪ { x , . . . , x n } where the internal nodes are Σ-labeled, and the leaves are { x , . . . , x n } -labeled.Let τ ∈ T fin Σ( { x ,...,x n } ) be a finite tree, and let t , . . . , t n ∈ T ω Σ be infinite binary treesover alphabet Σ. We define τ [ t /x , . . . , t n /x n ] as the infinite tree which is obtained from τ by grafting t i on leaves labeled by x i .For a set M ⊆ T fin Σ( { x ,...,x n } ) , we define M [ t /x , . . . , t n /x n ] := S τ ∈ M τ [ t /x , . . . , t n /x n ]. Theorem 8.3 (D. Niwi´nski [15]) . Let L be a countable regular tree language over alphabet Σ . Then there is a finite set of trees { t , . . . , t n } such that the following hold: (1) For each tree t ∈ L and a tree branch π , there is a node v ∈ π and a number ≤ i ≤ n such that t ≥ v = t i . (2) There is a regular finite tree language M ⊆ T fin Σ( { x ,...,x n } ) such that L = M [ t /x , . . . , t n /x n ] . The following lemma strengthen item (2) of Theorem 8.3 by adding another conditionon M , implying a unique representation of each tree in L : Lemma 8.4. Let L be a countable regular tree language over alphabet Σ , and let { t , . . . , t n } be a finite set of trees as in Theorem 8.3. Then there is a regular finite trees language M ⊆ T fin Σ( { x ,...,x n } ) such that L = M [ t /x , . . . , t n /x n ] , and for each t ∈ L there is a unique finite tree τ ∈ M such that t = τ [ t /x , . . . , t n /x n ] .Proof. For each tree t ∈ L , let g ( t ) be the tree which is obtained from t by changingthe label of each node v ∈ { l, r } ∗ where t ≥ v = t i to x i , and removing all descendants of { x , . . . , x n } -labeled node. Claim 8.4.1. g ( t ) is finite for all t ∈ L .Proof. Assume, for the sake of contradiction, that there is t ∈ L such that the set of nodes U ⊆ { l, r } ∗ of g ( t ) is infinite. The number of children of each node in U is bounded by2, and therefore, by K¨onig’s Lemma, there is a tree branch π such that ∀ v ∈ π : v ∈ U .Therefore, by definition of g ( t ), we conclude that t ≥ v = t i for each v ∈ π and 1 ≤ i ≤ n - acontradiction to item (1) of Theorem 8.3. (cid:4) Notice that for each t ∈ L we obtain g ( t )[ t /x , . . . , t n /x n ] = t , and therefore g isinjective. Hence, L = M [ t /x , . . . , t n /x n ] where M := { g ( t ) | t ∈ L } . We will show that M is a regular language of finite trees.It is easy to see that for each t ∈ L and finite tree τ ∈ T fin Σ( { x ,...,x n } ) , τ = g ( t ) iff thefollowing conditions hold: • t = τ [ t /x , . . . , t n /x n ] • t ≥ v = t i for each node v in τ which is not a leaf, and for each 1 ≤ i ≤ n .Since both conditions could be formulated in MSO, we conclude that M is MSO-definable, and therefore regular. MBIGUITY HIERARCHY OF REGULAR INFINITE TREE LANGUAGES 25 Proof of Proposition 8.1. Let L be a countable regular tree language over alphabetΣ. We will show that L can be accepted by an unambiguous PTA.By Lemma 8.4, there is a regular finite tree language M ⊆ T fin Σ( { x ,...,x n } ) and regularinfinite trees t , . . . , t n such that L = M [ t /x , . . . , t n /x n ]. Additionally, for each t ∈ L there is a unique τ ∈ M such that t = τ [ t /x , . . . , t n /x n ].Each infinite tree t i : { l, r } ∗ → Σ is regular, and therefore definable by a Moore ma-chine M i = ( { l, r } , Σ , Q i , q iI , δ Mi , out Mi ). Let A i := ( Q i , Σ , q iI , δ i , F i ) where F i := Q i , and( q, a, q , q ) ∈ δ i iff q = δ ( q, l ), q = δ ( q, r ) and a = out Mi ( q ). It is easy to verify that A i is unambiguous, and L ( A i ) = { t i } . M is regular and therefore can be accepted by anunambiguous FTA B = ( Q B , Σ ∪ { x , . . . , x n } , q B I , δ B ).We use these automata to construct a PTA A := ( Q, Σ , Q I , δ, C ), by: • Q := ∪ ≤ i ≤ n Q i ∪ Q B • q iI := { q B I } ∪ { q iI | ( q B I , x i ) ∈ δ B }• δ is the union of the following: – { ( q, a, q , q ) ∈ δ B | a ∈ Σ } (all transitions of B on inner nodes) – ∪ ≤ i ≤ n δ i – { ( q, a, q iI , q jI ) | ∃ ( q, a, q , q ) ∈ δ B : ( q , x i ) ∈ δ B and ( q , x j ) ∈ δ B } – { ( q, a, q , q jI ) | ∃ ( q, a, q , q ) ∈ δ B : ( q , x j ) ∈ δ B } – { ( q, a, q iI , q ) | ∃ ( q, a, q , q ) ∈ δ B : ( q , x i ) ∈ δ B }• C ( q ) := ( C i ( q ) ∃ i : q ∈ Q i L ( A ) = M [ t /x , . . . , t n /x n ] = L .We will show that A is unambiguous. For each accepting computation φ ∈ ACC ( A , t ),define a set of nodes U φ := { u ∈ { l, r } ∗ | ∀ v < u : φ ( v ) ∈ Q B } . It is easy to see that U φ is downward closed. Assume towards contradiction that U φ is infinite - by K¨onig Lemma, U φ contains an infinite tree branch π . By definition of U φ all states in φ ( π ) are in Q B , andtherefore colored by 1. That is a contradiction to φ being an accepting computation.Define a labeled finite tree t φ : U φ → Σ ∪ { x , . . . , x n } by: t φ := ( x i ∃ i : φ ( u ) = q iI t ( u ) otherwiseBy definition of t φ we obtain t = t φ [ t /x , . . . , t n /x n ], and by definition of B we concludethat t φ ∈ M .Assume, for the sake of contradiction, that A is ambiguous. Therefore, there is a tree t ∈ L and two distinct accepting computations φ , φ ∈ ACC ( A , t ). A i is deterministic foreach 1 ≤ i ≤ n , and therefore φ = φ iff t φ = t φ . We conclude that t φ [ t /x , . . . , t n /x n ] = t φ [ t /x , . . . , t n /x n ] for t φ , t φ ∈ M - a contradiction to the uniqueness property of M .9. Conclusion and Open Questions We proved that the ambiguity hierarchy is strict for regular languages over infinite trees.We proved that countable regular languages are unambiguous.A natural question is whether the ambiguity degree is decidable. However, this is nota trivial matter. In [3] some partial solutions for variants of the problem whether a givenlanguage is unambiguous are provided. A less ambitious task is to develop techniques for computing degrees of ambiguityand compute the degree of ambiguity of some natural languages. Let Σ := { c, a } and L ∃ ∞ a := { t ∈ T ω Σ | there are infinitely many a -labeled nodes in t } . L ∃ ω a := { t ∈ T ω Σ | there is a branch with infinitely many a -labeled nodes in t } . L a −∞ antichain := { t ∈ T ω Σ | the set of a -labeled nodes in t contain an infinite antichain } . All these languages areregular. There are (Moore) reductions from L ∃ a to these languages, hence they are notfinitely ambiguous. We believe that their ambiguity degree is uncountable, but we wereunable to prove this.We provided sufficient conditions for a language to be not finitely ambiguous and for alanguage to have uncountable degree of ambiguity.In particular, we proved that the degree of ambiguity of the complement of a count-able regular language is ℵ or 2 ℵ , and provided natural examples of such languages withcountable degree of ambiguity. We proved that the degree of ambiguity of the complementof a finite regular language is ℵ Yet, it is open whether the degree of ambiguity of thecomplement of countable regular languages is ℵ . References [1] Andr Arnold. Rational omega-languages are non-ambiguous. Theor. Comput. Sci. , 26:221–223, 09 1983.[2] Vince B´ar´any, Lukasz Kaiser, and Alex Rabinovich. Expressing cardinality quantifiers in monadicsecond-order logic over trees. Fundamenta Informaticae , 100(1-4):1–17, 2010.[3] Marcin Bilkowski and Michal Skrzypczak. Unambiguity and uniformization problems on infinite trees.In Simona Ronchi Della Rocca, editor, Computer Science Logic 2013 (CSL 2013), CSL 2013, September2-5, 2013, Torino, Italy , volume 23 of LIPIcs , pages 81–100. Schloss Dagstuhl - Leibniz-Zentrum f¨urInformatik, 2013.[4] Arnaud Carayol and Christof L¨oding. MSO on the infinite binary tree: Choice and order. In Interna-tional Workshop on Computer Science Logic , pages 161–176. Springer, 2007.[5] Arnaud Carayol, Christof L¨oding, Damian Niwinski, and Igor Walukiewicz. Choice functions and well-orderings over the infinite binary tree. Open Mathematics , 8(4):662–682, 2010.[6] Thomas Colcombet. Unambiguity in automata theory. In International Workshop on DescriptionalComplexity of Formal Systems , pages 3–18. Springer, 2015.[7] E Allen Emerson and Charanjit S Jutla. Tree automata, mu-calculus and determinacy. In FoCS , vol-ume 91, pages 368–377. Citeseer, 1991.[8] Yuri Gurevich and Leo Harrington. Trees, automata, and games. In Proceedings of the fourteenth annualACM symposium on Theory of computing , pages 60–65, 1982.[9] Yuri Gurevich and Saharon Shelah. Rabin’s uniformization problem 1. The Journal of Symbolic Logic ,48(4):1105–1119, 1983.[10] Yo-Sub Han, Arto Salomaa, and Kai Salomaa. Ambiguity, nondeterminism and state complexity offinite automata. Acta Cybernetica , 23(1):141–157, 2017.[11] Jozef Jir´asek, Galina Jir´askov´a, and Juraj ˇSebej. Operations on unambiguous finite automata. In Inter-national Conference on Developments in Language Theory , pages 243–255. Springer, 2016.[12] Ernst Leiss. Succinct representation of regular languages by boolean automata. Theoretical computerscience , 13(3):323–330, 1981.[13] Hing Leung. Descriptional complexity of nfa of different ambiguity. International Journal of Foundationsof Computer Science , 16(05):975–984, 2005.[14] Robert McNaughton. Testing and generating infinite sequences by a finite automaton. Information andcontrol , 9(5):521–530, 1966.[15] Damian Niwi´nski. On the cardinality of sets of infinite trees recognizable by finite automata. In An-drzej Tarlecki, editor, Mathematical Foundations of Computer Science 1991 , pages 367–376, Berlin,Heidelberg, 1991. Springer Berlin Heidelberg.[16] D. Perrin and J.´E. Pin. Infinite Words: Automata, Semigroups, Logic and Games . ISSN. ElsevierScience, 2004. MBIGUITY HIERARCHY OF REGULAR INFINITE TREE LANGUAGES 27 [17] Michael O Rabin. Decidability of second-order theories and automata on infinite trees. Transactions ofthe american Mathematical Society , 141:1–35, 1969.[18] Alexander Rabinovich and Doron Tiferet. Ambiguity hierarchy of regular infinite tree languages. InJavier Esparza and Daniel Kr´al’, editors, , volume 170 of LIPIcs ,pages 80:1–80:14. Schloss Dagstuhl - Leibniz-Zentrum f¨ur Informatik, 2020.[19] Richard Edwin Stearns and Harry B Hunt III. On the equivalence and containment problems for un-ambiguous regular expressions, regular grammars and finite automata. SIAM Journal on Computing ,14(3):598–611, 1985.[20] Wolfgang Thomas. Automata on infinite objects. In Formal Models and Semantics , pages 133–191.Elsevier, 1990.[21] Boris A. Trakhtenbrot and Ya. M. Barzdin. Finite automata, behavior and synthesis. 1973. Appendix A. Proof of Claim 4.1.4 Claim 4.1.4. Let t be a regular tree such that t / ∈ L ( A ) . Then, Pathfinder has a regularpositional winning strategy in G t , A .Proof. t is regular, and therefore there is a formula ψ t ( σ ) which defines t in the unlabeledfull-binary tree.We will use ψ t ( σ ) to define the following formula P athf inderW ins A ,t ( φ, ST R ), asthe conjunction of the following conditions:(1) ∃ π such that:(a) π is a branch(b) ∀ u ∈ π : ( ST R ( u, φ ( u · l ) , φ ( u · r )) = l ) ↔ u · l ∈ π ) - the Pathfinder moves d . . . d j . . . are consistent with ST R and are along the branch π .(2) ∃ σ : ψ t ( σ ) and at least one of the following holds:(a) ∃ v ∈ π such that ( φ ( v ) , σ ( v ) , φ ( v · l ) , φ ( v · r )) / ∈ δ - the Automaton move at ( v, φ ( v ))is invalid.(b) The maximal color which C assigns infinitely often to states in φ ( π ) is odd. Claim A.1. P athf inderW ins A ,t ( φ, ST R ) holds for a positional strategy ST R of Pathfinderand a computation φ of A on a tree t ′ iff the play s of ST R against str φ in G t , A is winningfor Pathfinder. Proof. By definition of G t , A , Pathfinder wins if either Automaton makes an invalid move(condition 2a) or the maximal color which is assigned infinitely often to the positions in π s is odd. Since all Pathfinder positions have color 0, this is equivalent to the maximal colorassigned infinitely often to Automaton positions being odd.Let s = e , d , e , d , . . . , e i , d i , . . . . Notice that by condition 1, there is a unique branch π such that π = v , . . . v i , . . . where v i = d . . . d i − . By Claim 4.1.1, we have φ ( v i ) = q i ,where the i -th position of Automaton in π s is ( v i , q i ). Since C G ( v i , q i ) = C ( q i ), we concludethat the maximal color which C assigns infinitely often to states in φ ( π ) is odd iff themaximal color which C G assigns infinitely often to positions in π s is odd. This is assuredby condition 2b. (cid:4) Let W inningStrategy t , A ( ST R ) := ∀ φ such that the following holds: • If there is t such that φ is an accepting computation of A on t , then: – P athf inderW ins A ,t ( φ, ST R ) holdsRecalling that the set of all computation of A is MSO-definable, we conclude that W inningStrategy t , A ( ST R )is MSO-definable in the unlabeled full-binary tree. Claim A.2. W inningStrategy t , A ( ST R ) holds for a positional strategy ST R of Pathfinderiff ST R is a positional winning strategy of Pathfinder. Proof. ⇒ : By Claim A.1, ST R wins in G t , A against each positional strategy of Automaton.Assume, for the sake of contradiction, that is a non-positional strategy str ′ of automatonwhich wins against ST R . Then by positional determinacy of parity games, we concludethat there is a positional strategy str ′′ which wins against ST R - a contradiction. ⇐ : Follow immediately from Claim A.1. (cid:4) t / ∈ L ( A ) and therefore by Claim 4.1.1(3), Automaton does not have a positional win-ning strategy. From positional determinacy of parity games we conclude that Pathfinder MBIGUITY HIERARCHY OF REGULAR INFINITE TREE LANGUAGES 29 has a positional winning strategy. Therefore, there is a strategy ST R ′ which satisfies W inningStrategy t , A ( ST R ) in the unlabeled full-binary tree.Therefore, W inningStrategy t , A ( ST R ) defines a non-empty tree language over alpha-bet Q × Q → { l, r } . By Rabin’s basis Theorem, we conclude that there is a regular tree [ ST R in this language, and by Claim A.2 we conclude that [ ST R is a positional winningstrategy for Pathfinder in G t , A . (cid:4) Remark (Logic Free Proof of Claim 4.1.4). One can reduce a membership game for aregular tree t to a game on a finite graph. By positional determinacy Theorem, Pathfinderwill have a positional winning strategy in the reduced game. From this strategy a regularwinning strategy in G t , A for Pathfinder is easily constructed. This work is licensed under the Creative Commons Attribution License. To view a copy of thislicense, visit https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/