Alternating Weak Automata from Universal Trees
aa r X i v : . [ c s . F L ] J u l Alternating Weak Automata from Universal Trees
Laure Daviaud
City, University of London, [email protected]
Marcin Jurdziński
University of Warwick, [email protected]
Karoliina Lehtinen
University of Liverpool, [email protected]
Abstract
An improved translation from alternating parity automata on infinite words to alternating weakautomata is given. The blow-up of the number of states is related to the size of the smallest universalordered trees and hence it is quasi-polynomial, and it is polynomial if the asymptotic number ofpriorities is at most logarithmic in the number of states. This is an exponential improvementon the translation of Kupferman and Vardi (2001) and a quasi-polynomial improvement on thetranslation of Boker and Lehtinen (2018). Any slightly better such translation would (if—like allpresently known such translations—it is efficiently constructive) lead to algorithms for solving paritygames that are asymptotically faster in the worst case than the current state of the art (Calude,Jain, Khoussainov, Li, and Stephan, 2017; Jurdziński and Lazić, 2017; and Fearnley, Jain, Schewe,Stephan, and Wojtczak, 2017), and hence it would yield a significant breakthrough.
Theory of computation → Formal languages and automata the-ory; Theory of computation → Algorithmic game theory
Keywords and phrases alternating automata, weak automata, Büchi automata, parity automata,parity games, universal trees
Funding
This work has been supported by the EPSRC grants EP/P020992/1 and EP/P020909/1(Solving Parity Games in Theory and Practice).
Acknowledgements
We thank Moshe Vardi for encouraging us to bring the state-space blow-upof alternating parity to alternating weak automata translation in line with the state-of-the-artcomplexity of solving parity games.
The influential class of regular languages of infinite words (often called the ω -regular lan-guages) is defined to consist of all the languages of infinite words that are recognized byfinite non-deterministic Büchi automata. The theory of ω -regular languages is quite wellunderstood. In particular, it is known that deterministic Büchi automata are not sufficientlyexpressive to recognize all the ω -regular languages, but deterministic automata with the so-called parity acceptance conditions are, and that the class of ω -regular languages is closedunder complementation. Effective constructions for determinization and complementationof Büchi automata are important tools both in theory and in applications, and both areknown to require exponential blow-ups of the numbers of states in the worst case.For applications in logic, it is natural to enrich automata models by the ability to altern-ate between non-deterministic and universal transitions [21, 16]. It turns out that alternatingparity automata are no more expressive than non-deterministic Büchi automata, and henceneither allowing alternation, nor the richer parity acceptance conditions, increase expressive-ness; this testifies to the robustness of the class of ω -regular languages. On the other hand, Alternating Weak Automata from Universal Trees alternation increases the expressive power of automata with the so-called weak acceptanceconditions: non-deterministic weak automata are not expressive enough to recognize all ω -regular languages, but alternating weak automata are. The weak acceptance conditions aresignificant due to their applications in logic [21] and thanks to their favourable algorithmicproperties [16].Given that alternating weak automata are expressive enough to recognize all the ω -regular languages, a natural question is whether, and to what degree, alternating weakautomata are less succinct than alternating Büchi or alternating parity automata. Anotherway of stating this question is what blow-up in the number of states is sufficient or requiredfor translations from alternating parity or alternating Büchi automata to alternating weakautomata. The first upper bound for the blow-up of a translation from alternating Büchito alternating weak automata was doubly exponential, obtained by combining a doubly-exponential determinization construction [7] and a linear translation from deterministicparity automata to weak alternating automata [21, 19]. This has been improved consid-erably by Kupferman and Vardi who have given a quadratic translation from alternatingBüchi to alternating weak automata [15], and then they have generalized it to a translationfrom alternating parity automata with n states and d priorities to alternating weak auto-mata, whose blow-up is n d + O (1) , i.e., exponential in the number of priorities in the parityacceptance condition [14].Understanding the exact trade-off between the complexity of the acceptance condition—weak, Büchi, or parity, the latter measured by the number of priorities—and the number ofstates in an automaton is interesting from the algorithmic point of view. For example, thealgorithmic problems of checking emptiness of non-deterministic parity automata on infinitetrees, of model checking for the modal µ -calculus, of solving two-player parity games, andof checking emptiness of alternating parity automata on infinite words over a one-letteralphabet, are all polynomial-time equivalent. Since checking emptiness of alternating weakautomata on words over a one-letter alphabet can be done in linear time, it follows thata translation from alternating parity automata to alternating weak automata implies analgorithm for solving parity games whose complexity matches the blow-up of the number ofstates in the translation.The first quasi-polynomial translation from alternating parity automata to alternatingweak automata was given recently by Boker and Lehtinen [1]. They have used the registertechnique, developed by Lehtinen [17] for parity games, to provide a translation from altern-ating parity automata with n states and d priorities to alternating parity automata with n Θ(log( d/ log n )) states and Θ(log n ) priorities; combined with the exponential translation ofKupferman and Vardi [14], this yields an alternating parity to alternating weak translationwhose blow-up of the number of states is n Θ(log n · log( d/ log n )) .The main result reported in this paper is that another technique—universal trees [11, 5],also developed to elucidate the recent major advance in the complexity of solving paritygames due to Calude, Jain, Khoussainov, Li, and Stephan [2]—can be used to further reducethe state-space blow-up in the translation from alternating parity automata to alternatingweak automata. We give a translation from alternating parity automata with n states and d priorities to alternating Büchi automata, whose state-space blow-up is proportional tothe size of the smallest ( n, d/ n if d = O (log n )and it is n lg( d/ lg n )+ O (1) if d = ω (log n ). When combined with Kupferman and Vardi’squadratic translation of alternating Büchi to alternating weak automata [15], we get thecomposite blow-up of the form n O (log( d/ log n )) , down from Boker and Lehtinen’s blow-upof (cid:0) n Θ(log( d/ log n )) (cid:1) log n . . Daviaud, M. Jurdziński and K. Lehtinen 3 The necessary size of the state-space blow-up when going from alternating parity auto-mata to alternating weak automata is wide open: the best known lower bound is Ω( n log n ) [15],closely related to the 2 Ω( n log n ) lower bound on Büchi complementation [20], while the bestupper bounds are quasi-polynomial. On the other hand, the blow-up for the parity to weaktranslation that we obtain nearly matches the current state-of-the-art quasi-polynomial up-per bounds on the complexity of solving parity games [2, 11, 9]. It follows that any significantimprovement over our translation would lead to a breakthrough improvement in the com-plexity of solving parity games.The exponential translation from alternating parity automata to alternating weak auto-mata due to Kupferman and Vardi [14] is done by a rather involved induction on the numberof priorities. For an automaton with d priorities, it goes through a sequence of d interme-diate automata of a generalized type, which they call parity-weak alternating automata.In contrast, our construction is significantly more streamlined and transparent; in particu-lar, it avoids introducing a new class of hybrid parity-weak automata. We first establish ahierarchical decomposition of runs of alternating parity automata as a generalization of thedecomposition of runs of alternating co-Büchi automata due to Kupferman and Vardi [15],and then we use the recently introduced universal trees [11, 5] to construct an alternatingBüchi automaton, which is a parity automaton with just 2 priorities. Our work is yet anotherapplication of the recently introduced notion of universal trees [11, 5]. Such applicationstypically focus on algorithms for solving games [11, 6, 5, 3]; our work is the first whoseprimary focus is on automata.In addition to universal trees, we use a notion of lazy progress measure. Unlike thestandard parity progress measures, which can be recognised by safety automata but requirean explicit bound to be known on the number of successive occurrences of odd priorities, lazyprogress measures are recognised by Büchi automata and can deal with finite but unboundednumbers of occurrences of successive odd priorities. This Büchi automaton is similar to (butmore concise than) the automaton used to characterise parity tree automata that recogniseco-Büchi recognisable tree languages [18], itself a generalisation of automata used to decidethe weak definability of tree languages given as Büchi automata [22, 4].A similar concept to our lazy progress measures was already introduced by Klarlund forcomplementation of Büchi and Streett automata on words [12]. Klarlund indeed proves aresult that is equivalent to one of our key lemmas on parity progress measures. Our proof,however, is more constructive, and it explicitly provides a hierarchical decomposition, whichclearly describes the structure of accepting run dags of parity word automata. Moreover—unlike Klarlund’s proof, which relies on the result about Rabin measures [13]—our proofis self-contained. We suspect that the opaqueness of Klarlund’s paper [12] may have beenresponsible for attracting less attention and shallower absorption by the research communitythan it deserves. In particular, some of the techniques and results that he presents therehave been rediscovered and refined by various authors, often much later [15, 14, 10, 11, 5],including this work. We hope that our paper will help a wider and more thorough receptionand appreciation of Klarlund’s work. For a finite set X , we write B + ( X ) for the set of positive Boolean formulas over X . We saythat a set Y ⊆ X satisfies a formula ϕ ∈ B + ( X ) if ϕ evaluates to true when all variablesin Y are set to true and all variables in X \ Y are set to false . For example, the sets { x, y } and { x, z } satisfy the positive Boolean formula x ∧ ( y ∨ z ), but the set { y, z } does not. An Alternating Weak Automata from Universal Trees alternating automaton has a finite set Q of states , an initial state q ∈ Q , a finite alphabet Σ, and a transition function δ : Q × Σ → B + ( Q ). Alternating automata allow to combineboth non-deterministic and universal transitions; disjunctions in transition formulas modelthe non-deterministic choices and conjunctions model the universal choices.We consider alternating automata as acceptors of infinite words. Whether infinite se-quences of states in runs of such automata are accepting or not is determined by an ac-ceptance condition . Here, we consider parity , Büchi , co-Büchi , weak , and safety acceptanceconditions. In a parity condition, given by a state priority function π : Q → { , , , . . . , d } for some positive even integer d , an infinite sequence of states is accepting if the largeststate priority that occurs infinitely many times is even. Büchi conditions are a special caseof parity conditions in which all states have priorities 1 or 2, and co-Büchi conditions areparity conditions in which all states have priorities 0 or 1.Let the transition graph of an alternating automaton have an edge ( q, r ) ∈ Q × Q if r occurs in δ ( q, a ) for some letter a ∈ Σ. We say that a parity automaton has a weak acceptance condition if it is stratified : in every cycle in the transition graph, all states havethe same priority. Weak conditions are a special case of both Büchi and co-Büchi conditionsin the following sense: if the transition graph of a parity automaton is stratified, then everyinfinite path in the transition graph satisfies each of the following three conditions if andonly if it satisfies the other two:the parity condition π : Q → { , , , . . . , d } ;the co-Büchi condition π ′ : Q → { , } ;the Büchi condition π ′′ : Q → { , } ;where π ′ ( q ) = π ( q ) mod 2 and π ′′ ( q ) = 2 − π ′ ( q ) for all q ∈ Q .We say that a state is absorbing if its only successor in the transition graph is itself. Aparity automaton has a safety acceptance condition if all of its states have priority 0, exceptfor the additional absorbing reject state that has priority 1. An automaton with a safetyacceptance condition is stratified, and hence safety conditions are a special case of weakconditions.Whether an infinite word w = w w w · · · ∈ Σ ω is accepted or rejected by an alternatingautomaton A is determined by the winner of the following acceptance game G ( A , w ). Theset of positions in the game is the set Q × N and the two players, Alice and Elvis, playin the following way. The initial position is ( q , current position ( q i , i ), firstElvis chooses a subset P of Q that satisfies δ ( q i , w i ), then Alice picks a state q i +1 ∈ P , and( q i +1 , i +1) becomes the next current position. Note that Elvis can be thought of making thenon-deterministic choices and Alice can be thought of making the universal choices in thetransition function of the alternating automaton. This interaction of Alice and Elvis yieldsan infinite sequence of states q , q , q , . . . , and whether Elvis is declared the winner or notis determined by whether the sequence is accepting according to the acceptance conditionof the automaton. Acceptance games for parity, Büchi, co-Büchi, and weak conditions areparity games, which are determined [8]: in every acceptance game, either Alice or Elvis hasa winning strategy. We say that an infinite word w ∈ Σ ω is accepted by an alternatingautomaton A if Elvis has a winning strategy in the acceptance game G ( A , w ), and otherwiseit is rejected .A run dag of an alternating automaton A on an infinite word w is a directed acyclicgraph G = ( V, E, ρ : V → Q ), where V ⊆ Q × N is the set of vertices; successors (accordingto the directed edge relation E ) of every vertex ( q, i ) are of the form ( q ′ , i + 1); the followingconditions hold:( q , ∈ V , . Daviaud, M. Jurdziński and K. Lehtinen 5 for every ( q, i ) ∈ V , the Boolean formula δ ( q, w i ) is satisfied by the set of states p , suchthat ( p, i + 1) is a successor of ( q, i );and ρ projects vertices onto the first component. Note that every vertex in a run daghas a successor, and hence every maximal path is infinite. We say that a run dag of anautomaton A is accepting if the sequence of states on every infinite path in the run dag isaccepting according to the accepting condition of A . The positional determinacy theorem forparity games [8] implies that an infinite word w is accepted by an alternating automaton A with a parity (or Büchi, co-Büchi, weak, or safety) condition if and only if there is anaccepting run dag of A on w . In other words, run dags are compact representations of(positional) winning strategies for Elvis in the acceptance games.Run dags considered here are a special case of layered dags , whose vertices can bepartitioned into sets L , L , L , . . . , such that every edge goes from some layer L i to thenext layer L i +1 . We define the width of a layered dag with an infinite number of lay-ers L , L , L , . . . to be lim inf i →∞ | L i | . Note that the width of a run dag of an alternatingautomaton is trivially upper-bounded by the number of states of the automaton. In this section we summarize the results of Kupferman and Vardi [15] who have given trans-lations from alternating co-Büchi and Büchi automata to alternating weak automata withonly a quadratic blow-up in the state space. We recall the decomposition of co-Büchi accept-ing run dags of Kupferman and Vardi in detail because it motivates and prepares the readerfor our generalization of their result to accepting parity run dags. Our main technical resultis a translation from alternating parity automata to alternating Büchi automata with onlya quasi-polynomial blow-up in the state space, but the ultimate goal is a quasi-polynomialtranslation from parity to weak automata. Therefore, we also recall how Kupferman andVardi use their quadratic co-Büchi to weak translation in order to obtain a quadratic Büchito weak translation.
The main technical concept that underlies Kupferman and Vardi’s [15] translation fromalternating co-Büchi automata to alternating weak automata is that of a ranking function foraccepting run dags of alternating co-Büchi automata. As Kupferman and Vardi themselvespoint out, ranking functions can be seen as equivalent to Klarlund’s progress measures [12].We will adopt Klarlund’s terminology because the theory of progress measures for certifyingparity conditions is very well developed [8, 12, 13, 10, 11, 5] and our main goal in this paperis to use a version of parity progress measures to give a simplified, streamlined, and improvedtranslation from alternating parity to alternating weak automata.Let G = ( V, E, π : V → { , } ) be a layered dag with vertex priorities 0 or 1, andin which every vertex has a successor. Note that all run dags of an alternating co-Büchiautomaton are such layered dags and if the automaton has n states then the width of the rundag is at most n . (Observe, however, that while formally the third component ρ : V → Q in a run dag maps vertices to states, here we instead consider the labeling π : V → { , } that labels vertices by the priorities of the states π ( v ) = π ( ρ ( v )).)A co-Büchi progress measure [8, 13, 10] is a mapping µ : V → M , where ( M, ≤ ) is awell-ordered set, such that for every edge ( v, u ) ∈ E , we have if π ( v ) = 0 then µ ( v ) ≥ µ ( u ), if π ( v ) = 1 then µ ( v ) > µ ( u ). Alternating Weak Automata from Universal Trees
It is elementary to argue that existence of a co-Büchi progress measure on a graph is sufficientfor every infinite path in the graph satisfying the co-Büchi condition. Importantly, it is alsonecessary, which can be, for example, deduced from the proof of positional determinacy forparity games due to Emerson and Jutla [8]. In other words, co-Büchi progress measures arewitnesses for the property that all infinite paths in a graph satisfy the co-Büchi condition.The appeal of such witnesses stems from the property that while certifying a global andinfinitary condition, it suffices to verify them locally by checking a simple inequality betweenthe labels of the source and the target of each edge in the graph.The disadvantage of progress measures as above is that on graphs of infinite size, such asrun dags, the well-ordered sets of labels that are needed to certify co-Büchi conditions maybe of unbounded (and possibly infinite) size. In order to overcome this disadvantage, andto enable automata-theoretic uses of progress measure certificates, Klarlund has proposedthe following concept of lazy progress measures [12]. A lazy (co-Büchi) progress measure is a mapping µ : V → M , where ( M, ≤ ) is a well-ordered set and L ⊂ M is the set of lazy-progress elements, and such that: for every edge ( v, u ) ∈ E , we have µ ( v ) ≥ µ ( u ); if π ( v ) = 1 then µ ( v ) ∈ L ; on every infinite path in G , there are infinitely many vertices v such that µ ( v ) L .It is elementary to prove the following proposition. ◮ Proposition 1.
If a graph has a lazy co-Büchi progress measure then all infinite paths init satisfy the co-Büchi condition.
The following converse establishes the attractiveness of lazy co-Büchi progress measures forcertifying the co-Büchi conditions on layered dags of bounded width, and hence for certifyingaccepting run dags of alternating co-Büchi automata. ◮ Lemma 2 (Klarlund [12]) . If all infinite paths in a layered dag ( V, E, π : V → { , } ) satisfy the co-Büchi condition and the width of the dag is at most n , then there is a lazyco-Büchi progress measure µ : V → M , where M = { , , . . . , n } and L = { , , , . . . , n } . Proof.
We summarize a proof given by Kupferman and Vardi [15] that provides an explicitdecomposition of the accepting run dag into (at most) 2 n parts from which a lazy co-Büchiprogress measure can be straightforwadly defined. The proof by Klarlund [12] is moresuccinct, but the former is more constructive and hence more transparent.Observe that if all infinite paths satisfy the co-Büchi condition then there must be avertex v whose all descendants (i.e., vertices to which there is a—possibly empty—pathfrom v ) have priority 0; call such vertices 1 -safe in G = G . Indeed, otherwise it would beeasy to construct an infinite path with infinitely many occurrences of vertices of priority 1.Let S be the set of all the 1-safe vertices in G , and let G ′ = G \ S be the layered dagobtained from G by removing all vertices in S . Note that there is an infinite path in thesubgraph of G induced by S , and hence the width of G ′ is strictly smaller than the widthof G .Let R be the set of all vertices in G ′ that have only finitely many descendants; call suchvertices transient in G ′ . Let G be the the layered dag obtained from G ′ by removing allvertices in R . Since G is a subgraph of G ′ , the width of G is strictly smaller than thewidth of G . Moreover, G shares the key properties with G : every vertex has a successorand hence all the maximal paths are infinite, and all infinite paths satisfy the co-Büchicondition.By applying the same procedure to G that we have described for G above, we obtainthe set S of 1-safe vertices in G and the set R of vertices transient in G ′ , and the layered . Daviaud, M. Jurdziński and K. Lehtinen 7 dag G —obtained from G by removing all vertices in S ∪ R —has the width that is strictlysmaller than that of G . We can continue in this fashion until the graph G k +1 , for some k ≥
1, is empty. Since the width of G is at most n , and the widths of graphs G , G , . . . , G k +1 are strictly decreasing, it follows that k ≤ n .We define µ : V → { , , . . . , n } by: µ ( v ) = ( i − v ∈ S i , i if v ∈ R i , and note that it is routine to verify that if we let L = { , , . . . , n } be the set of lazy-progress elements then µ is a lazy co-Büchi progress measure. ◭ In this section we present a proof of the following result. ◮ Theorem 3 (Kupferman and Vardi [15]) . There is a translation that given an alternating co-Büchi automaton with n states yields an equivalent alternating weak automaton with O ( n ) states. Proof.
It suffices to argue that, given an alternating co-Büchi automaton A = ( Q, q , Σ , δ, π : Q → { , } ) with n states, we can design an alternating weak automaton with O ( n ) statesthat guesses and certifies a dag run of A together with a lazy co-Büchi progress measure onit as described in Lemma 2. First we construct a safety automaton S with O ( n ) states thatsimulates the automaton A while guessing a lazy co-Büchi progress measure and verifyingconditions 1) and 2) of its definition. Condition 3) will be later handled by turning the safetyautomaton S into a weak automaton W by appropriately assigning odd or even priorities toall states in S . We split the design of W into those two steps so that we can better motivateand explain the generalized constructions in Section 5.The safety automaton S has the following set of states: Q × { , , . . . , n } ∪ (cid:0) π − (0) × { , , . . . , n − } (cid:1) ∪ { reject } ;its initial state is ( q , n ); and its transition function δ ′ is obtained from the transitionfunction δ of A in the following way: for every state ( q, i ), and for every a ∈ Σ, the formula δ ′ (cid:0) ( q, i ) , a (cid:1) is obtained from δ ( q, a ) by replacing every occurrence of state q ′ ∈ Q by thedisjunction (i.e., a non-deterministic choice)( q ′ , i ) ∨ ( q ′ , i − ∨ · · · ∨ ( q ′ ,
1) (1)where every occurrence ( q ′ , j ) for which π ( q ′ ) = 1 and j is odd stands for the state reject .In other words, the safety automaton S can be thought of as consisting of 2 n cop-ies A n , A n − , . . . , A of A , with the non-accepting states π − (1) removed from the odd-indexed copies A n − , A n − , . . . , A , and in whose acceptance games, Elvis always has thechoice to stay in the current copy of A or to move to one of the lower-indexed copies of A .Since the transitions of the safety automaton S always respect the transitions of the ori-ginal co-Büchi automaton A , an accepting run dag of S yields a run dag of A (obtainedfrom the first components of the states ( q, i )) and a labelling of its vertices by numbersin { , , . . . , n } (obtained from the second components of the states ( q, i )). It is routineto verify that the design of the state set and of the transition function of the safety auto-maton S guarantees that the latter labelling satisfies conditions 1) and 2) of the definition ofa lazy co-Büchi progress measure, where the set of lazy-progress elements is { , , . . . , n } . Alternating Weak Automata from Universal Trees
By setting the state priority function π ′ : ( q, i ) i + 1 for all non- reject states in S ,and π ′ : reject
1, we obtain from S an automaton W whose acceptance condition isweak because—by design—the transition function is non-increasing w.r.t. the state priorityfunction. One can easily verify that the addition of this weak acceptance condition to S allows the resulting automaton W , for every input word, to guess and verify a lazy progressmeasure—if one exists—on a run dag of automaton A on the input word, while W rejectsthe input word otherwise. This completes our summary of the proof of Theorem 3. ◭◮ Corollary 4 (Kupferman and Vardi [15]) . There is a translation that given an alternatingBüchi automaton with n states yields an equivalent alternating weak automaton with O ( n ) states. The argument of Kupferman and Vardi is simple and it exploits the ease with whichalternating automata can be complemented. Given an alternating Büchi automaton A with n states, first complement it with no state space blow-up, obtaining an alternating co-Büchiautomaton with n states, next use the translation from Theorem 3 to obtain an equivalentalternating weak automaton with O ( n ) states, and finally complement the latter again withno state space blow-up, hence obtaining an alternating weak automaton that is equivalentto A and that has O ( n ) states. Before we introduce lazy parity progress measures , we recall the definition of (standard)parity progress measures [11, 5]. We define a well-ordered tree to be a finite prefix-closedset of sequences of elements of a well-ordered set. We call such sequences nodes of the tree,and their components are branching directions . We use the standard ancestor-descendantterminology to describe relative positions of nodes in a tree. For example, hi is the root ;node h x, y i is the child of the node h x i that is reached from it via the branching direction y ;node h x, y i is the parent of node h x, y, z i ; nodes h x, y i and h x, y, z i are descendants of nodes hi and h x i ; nodes hi and h x i are ancestors of nodes h x, y i and h x, y, z i ; and a node is a leaf if it does not have any children. All nodes in a well-ordered tree are well-ordered bythe lexicographic order that is induced by the well-order on the branching directions; forexample, we have h x i < h x, y i , and h x, y, z i < h x, w i if y < w . We define the depth of anode to be the number of elements in the eponymous sequence, the height of a tree to bethe maximum depth of a node, and the size of a tree to be the number of its nodes.Parity progress measures assign labels to vertices of graphs with vertex priorities, andthe labels are nodes in a well-ordered tree. A tree labelling of a graph with vertex prioritiesthat do not exceed a positive even integer d is a mapping from vertices of the graph to nodesin a well-ordered tree of height at most d/
2. We write h m d − , m d − , . . . , m ℓ i , for some odd ℓ ,1 ≤ ℓ < d , to denote such nodes. We say that such a node has an (odd) level ℓ and an (even)level ℓ −
1, and the root hi has level d . Moreover, for every priority p , 0 ≤ p ≤ d , we definethe p -truncation h m d − , m d − , . . . , m ℓ i| p in the following way: h m d − , m d − , . . . , m ℓ i| p = h m d − , m d − , . . . , m ℓ i for p ≤ ℓ, h m d − , m d − , . . . , m p +1 i for even p > ℓ, h m d − , m d − , . . . , m p i for odd p > ℓ. We then say that a tree labelling µ of a graph G = ( V, E ) with vertex priorities π : V →{ , , , . . . , d } is a parity progress measure if the following progress condition holds for everyedge ( v, u ) ∈ E : . Daviaud, M. Jurdziński and K. Lehtinen 9 if π ( v ) is even then µ ( v ) | π ( v ) ≥ µ ( u ) | π ( v ) ; if π ( v ) is odd then µ ( v ) | π ( v ) > µ ( u ) | π ( v ) .It is well-known that satisfaction of such local conditions on every edge in a graph is sufficientfor every infinite path in the graph satisfying the parity condition [10, 11]. Less obviously,it is also necessary, which can be, again, deduced from the proof of positional determinacyof parity games due to Emerson and Jutla [8]. In other words, parity progress measures arewitnesses for the property that all infinite paths in a graph satisfy the parity condition. Likefor the simpler co-Büchi condition, their appeal stems from the property that they certifyconditions that are global and infinitary by verifying conditions that are local to every edgein the graph.Similar to the simpler co-Büchi progress measures, parity progress measures may unfor-tunately require unbounded or even infinite well-ordered trees to certify parity conditionson infinite graphs, and hence we consider lazy parity progress measures, also inspired byKlarlund’s pioneering work [12]. A lazy tree is a well-ordered tree with a distinguished sub-set of its nodes called lazy nodes . For convenience, we assume that only leaves may be lazyand the root never is.A lazy parity progress measure is a tree labelling µ of a graph ( V, E ), where the labelsare nodes in a lazy tree T , such that: for every edge ( v, u ) ∈ E , µ ( v ) | π ( v ) ≥ µ ( u ) | π ( v ) ; if π ( v ) is odd then node µ ( v ) is lazy and its level is at least π ( v ); on every infinite path in G , there are infinitely many vertices v , such that µ ( v ) is notlazy.First we establish that existence of a lazy progress measure is sufficient for all infinitepaths in a graph to satisfy the parity condition. ◮ Lemma 5.
If a graph has a lazy parity progress measure then all infinite paths in it satisfythe parity condition.
Proof.
For the sake of contradiction, assume that there is an infinite path v , v , v , . . . inthe graph for which the highest priority p that occurs infinitely often is odd. Let i ≥ π ( v j ) ≤ p for all j ≥ i . By condition 1), we have: µ ( v i ) | p ≥ µ ( v i +1 ) | p ≥ µ ( v i +2 ) | p ≥ . . . . (2)Let i ≤ i < i < i < · · · be such that π ( v i k ) = p for all k = 1 , , , . . . . By condition 2),all labels µ ( v i k ), for k = 1 , , , . . . , are lazy and their level in the tree is at least p . Bycondition 3), for infinitely many k , π ( v k ) is not lazy, so infinitely many of the inequalitiesin (2) must be strict, which contradicts the well-ordering of the tree T . ◭ Now we argue that existence of lazy parity progress measure is also necessary for a graphto satisfy the parity condition. Moreover, we explicitly quantify the size of a lazy orderedtree the labels from which are sufficient to give a lazy progress measure for a layered dag,as a function of the width of the dag. Before we do that, however, we introduce a simpleoperation that we call a lazification of a finite ordered tree. If T is a finite ordered tree, thenits lazification lazi( T ) is a finite lazy tree that is obtained from T in the following way:all nodes in T are also nodes in lazi( T ) and they are not lazy;for every non-leaf node t in T , t has extra lazy children in the tree lazi( T ), one smallerand one larger than all the other children, and one in-between every pair of consecutivechildren.It is routine to argue that if a tree has n leaves and it is of height at most h then itslazification lazi( T ) has O ( nh ) nodes and it is also of height h . ◮ Theorem 6 (Klarlund [12]) . If all infinite paths in a layered dag satisfy the parity conditionand the width of the dag is at most n , then there is a lazy parity progress measure whoselabels are nodes in a tree that is a lazification of an ordered tree with at most n leaves. Proof.
Klarlund’s proof [12] is very succinct and it heavily relies on the result of Klarlundand Kozen on Rabin measures [13]. Our proof is not only self-contained but it also is moreconstructive and transparent. The hierarchical decomposition describes the fundamentalstructure of accepting run dags of alternating parity automata and it may be of independentinterest. The argument presented here is a generalization of the proof of Lemma 2—givenin Section 3—from co-Büchi conditions to parity conditions.Consider a layered dag G = ( V, E, π ) where π : V → { , , , . . . , d } . For a priority p ,0 ≤ p < d , we write G ≤ p for the subgraph induced by the vertices whose priority is atmost p .We describe the following decomposition of G . Let D be the set that consists of allvertices of the top even priority d in G , and R all those vertices in the subgraph G ≤ d − that have finitely many descendants. We say that those vertices are ( d − -transient in G ≤ d − . In other words, D ∪ R is the set of vertices from which every path reaches (possiblyimmediately) a vertex of priority d .Let G = G \ ( D ∪ R ) be the layered dag obtained from G by removing all verticesin D ∪ R . Observe that every vertex in G has at least one successor and hence—unless G is empty—all maximal paths are infinite. W.l.o.g., assume henceforth that G is notempty. We argue that there must be a vertex in G whose all descendants have prioritiesat most d −
2; call such vertices ( d − -safe in G . Indeed, otherwise it would be easy toconstruct an infinite path with infinitely many occurrences of the odd priority d − d .Let S be the set of all the ( d − G . Let H be the subgraph of G induced by S , let n be the width of H , and note that n >
0. Set G ′ = G \ S to be thelayered dag obtained from G by removing all ( d − G .Let R be the set of all vertices in G ′ that have only finitely many descendants; callsuch vertices ( d − G ′ . Finally, let G be the layered dag obtained from G ′ by removing all the ( d − G ′ . Note that the width of G is smallerthan the width of G by at least n > G is empty, we can now apply the same steps to G that we have describedfor G , and obtain:the set S of ( d − G ;the subgraph H of G induced by S , which is of width n > G ′ , obtained from G by removing all the vertices in S ;the set R of ( d − G ′ ;the layered graph G , obtained from G by removing all vertices in S ∪ R , and whosewidth is smaller than the width of G by at least n > G , G , . . . , G k +1 , until the graph G k +1 ,for some k ≥
0, is empty. Since the width of G is at most n and the widths of the graphs G , G , . . . , G k are positive (unless k = 0), we have that k ≤ n and P ki =1 n i ≤ n .The proces described above yields a hierarchical decomposition of the layered dag; wenow define—by induction on d —the tree that describes the shape of this decomposition. Wethen argue that the lazification of this tree provides the set of labels in a lazy parity progressmeasure.In the base case d = 0, the shape of the decomposition is the well-ordered tree T ofheight h = 0 / hi . It is straightforward to see that the function . Daviaud, M. Jurdziński and K. Lehtinen 11 that maps every vertex onto the root is a (lazy) progress measure.For d ≥
2, note that all vertices in dags H , H , . . . , H k have priorities at most d −
2. Bythe inductive hypothesis, there are trees T , T , . . . , T k , of heights at most h − d − / n , n , . . . , n k leaves, respectively, which are the shapes of the hierarchicaldecompositions of dags H , H , . . . , H k , respectively.We now construct the finite ordered tree T of height at most h = d/ G : let T consist of the root node hi that has k children,which are the roots of the subtrees T , T , . . . , T k , in that order. Note that the number ofleaves of T is at most P ki =1 n i ≤ n . Consider the following mapping from vertices in thegraph onto nodes in the lazification lazi( T ) of tree T :vertices in set D are mapped onto the root of lazi( T );vertices in transient sets R , R , R , . . . , R k are mapped onto the lazy children of theroot of lazi( T ): those in R onto the smallest lazy child, those in R onto the lazy childbetween the roots of T and T , etc.;vertices in subgraphs H , H , . . . , H k are inductively mapped onto the appropriate nodesin the lazy subtrees of lazi( T ) that are rooted in the k non-lazy children of the root.It is easy to verify that this mapping satisfies conditions 1) and 2) of the definition of alazy parity progress measure. Condition 3) is ensured by the fact that the root of T is notlazy and by the inductive hypothesis. Recall that every infinite path satisfies the paritycondition, thus the highest priority p seen infinitely often on a given path is even. If p = d ,the path visits infinitely often vertices labelled by the root of T . Otherwise, eventually thepath contains only vertices in one of the sets S i and we can use the inductive hypothesis. ◭ In this section we complete the proof of the main technical result of the paper, which is aquasi-polynomial translation from alternating parity automata to alternating weak automata.The main technical tools that we use to design our translation are lazy progress measures anduniversal trees [11, 5], and the state space blow-up of the translation is merely quadratic inthe size of the smallest universal tree. Nearly tight quasi-polynomial upper and lower boundshave recently been given for the size of the smallest universal trees [11, 5] and, in particular,they imply that if the number of priorities in a family of alternating parity automata is atmost logarithmic in the number of states, then the state space blow-up of our translation isonly polynomial. ◮ Theorem 7.
There is a translation that given an alternating parity automaton with n states and d priorities yields an equivalent alternating weak automaton whose number ofstates is polynomial if d = O (log n ) and it is n O (lg( d/ lg n )) if d = ω (log n ) . Before we proceed to prove the theorem, we recall the notion of universal ordered trees.An ( n, h ) -universal (ordered) tree [5] is an ordered tree, such that every finite ordered treeof height at most h and with at most n leaves can be isomorphically embedded into it. Insuch an embedding, the root of the tree must be mapped onto the root of the universal tree,and the children of every node must be mapped—injectively and in an order-preservingway—onto the children of its image. In order to upper-bound the size of the blow-up in ourparity to weak translation, we rely on the following upper bound on the size of the smallestuniversal trees. ◮ Theorem 8 (Jurdziński and Lazić [11]) . For all positive integers n and h , there is an ( n, h ) -universal tree with at most quasi-polynomial number of leaves. More specifically, the number of leaves is polynomial in n if h = O (log n ) , and it is n lg( h/ lg n )+ O (1) if h = ω (log n ) . We also note that Czerwiński et al. [5] have subsequently proved a nearly-matching quasi-polynomial lower bound, hence establishing that the smallest universal trees have quasi-polynomial size.In order to prove Theorem 7, we establish the following lemma that provides a translationfrom alternating parity automata to alternating Büchi automata whose state-space blow-upis tightly linked to the size of universal trees. ◮ Lemma 9.
There is a translation that given an alternating parity automaton with n statesand d priorities yields an equivalent alternating Büchi automaton whose number of states is O ( ndL U ) where L U is the number of leaves in an ( n, d/ -universal ordered tree U . Note that Theorem 7 follows from Lemma 9 by Theorem 8 and Corollary 4.
Proof. (of Lemma 9) Given an alternating parity automaton A = ( Q, q , Σ , δ, π : Q →{ , , . . . , d } ) with n states, we now design an alternating Büchi automaton that guessesand certifies a dag run of A together with a lazy parity progress measure on it. As for the co-Büchi to weak case, we first construct a safety automaton that simulates the automaton A while guessing a lazy parity progress measure and verifying conditions 1) and 2) of itsdefinition. Condition 3) will be later handled by turning the safety automaton into a Büchiautomaton by appropriately assigning priorities 1 or 2 to all states in the safety automaton.Below we give a general construction of an alternating Büchi automaton B T from anylazy well-ordered tree T , and then we argue that the alternating parity automaton A isequivalent to the alternating Büchi automaton B lazi( U ) , for every ( n, d/ U .Let T be a lazy tree of width n and height d/
2. The construction is by induction on d .The safety automaton S T has the following set of states, which are pairs of an element of Q and of a node in T .If d = 0, then the set of states of S T is ( Q × {hi} ) ∪ { reject } .Otherwise, let h x i , h x i , . . . , h x k i be the children of the root, and 1 ≤ i < i < . . .
2. Let Ω i denote its set of non- reject states. They are pairsconsisting of an element of Q and of a node in a tree of height d/ −
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