Alternating Weak Automata from Universal Trees

TL;DR

This paper presents an improved translation method from alternating parity automata to alternating weak automata, significantly reducing state blow-up and potentially enabling faster algorithms for solving parity games.

Contribution

It introduces a quasi-polynomial translation based on universal trees, improving upon previous methods and impacting the complexity of solving parity games.

Findings

State blow-up is quasi-polynomial, polynomial with logarithmic priorities.

The translation improves upon previous exponential and quasi-polynomial methods.

Potential for faster algorithms in solving parity games if the translation is efficiently constructive.

Abstract

An improved translation from alternating parity automata on infinite words to alternating weak automata is given. The blow-up of the number of states is related to the size of the smallest universal ordered trees and hence it is quasi-polynomial, and only polynomial if the asymptotic number of priorities is logarithmic in the number of states. This is an exponential improvement on the translation of Kupferman and Vardi (2001) and a quasi-polynomial improvement on the translation of Boker and Lehtinen (2018). Any slightly better such translation would (if---like all presently known such translations---it is efficiently constructive) lead to algorithms for solving parity games that are asymptotically faster in the worst case than the current state of the art (Calude, Jain, Khoussainov, Li, and Stephan, 2017; Jurdzi\'nski and Lazi\'c, 2017; and Fearnley, Jain, Schewe, Stephan, and Wojtczak, 2017), and hence it would yield a significant breakthrough.

Equations

$$\mu (v)=\{\begin{array}{ll}2i-1& \text{ if v∈Si},\\ 2i& \text{ if v∈Ri},\end{array}$$

$$\begin{gathered} Q\times \{2,4,\dots ,2n\} \\ \cup \left({\pi }^{-1}(0)\times \{1,3,\dots ,2n-1\}\right)\cup \{\mathtt{reject}\}; \end{gathered}$$

$$(q^{\prime },i)\vee (q^{\prime },i-1)\vee \cdots \vee (q^{\prime },1)$$

$$⟨m_{d-1},m_{d-3},\dots ,m_{\ell }⟩{|}_p=\{\begin{array}{ll}⟨m_{d-1},m_{d-3},\dots ,m_{\ell }⟩& \text{ for p≤ℓ},\\ ⟨m_{d-1},m_{d-3},\dots ,m_{p+1}⟩& \text{ for even p>ℓ},\\ ⟨m_{d-1},m_{d-3},\dots ,m_p⟩& \text{ for odd p>ℓ}.\end{array}$$

$$\mu (v_i){|}_p\ge \mu (v_{i+1}){|}_p\ge \mu (v_{i+2}){|}_p\ge \dots .$$

$$\left(Q^{(d)}\times \{⟨⟩\}\right)\cup \left(Q^{(<d)}\times \{⟨x_{i_1}⟩,⟨x_{i_2}⟩,\dots ,⟨x_{i_m}⟩\}\right)\cup \bigcup _{i=1}^k{\Gamma }_i\cup \{\mathtt{reject}\}.$$

$$\bigvee \left\{(q^{\prime },t^{\prime }):t{|}_{\pi (q)}\ge t^{\prime }{|}_{\pi (q)}\right\},$$

Full text

Alternating Weak Automata from Universal Trees

Laure Daviaud

  

Marcin Jurdziński

  

Karoliina Lehtinen

1 Introduction

The influential class of regular languages of infinite words (often called the $\omega$-regular languages) is defined to consist of all the languages of infinite words that are recognized by finite non-deterministic Büchi automata. The theory of $\omega$-regular languages is quite well understood. In particular, it is known that deterministic Büchi automata are not sufficiently expressive to recognize all the $\omega$-regular languages, but deterministic automata with the so-called parity acceptance conditions are, and that the class of $\omega$-regular languages is closed under complementation. Effective constructions for determinization and complementation of Büchi automata are important tools both in theory and in applications, and both are known 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 alternate between non-deterministic and universal transitions [21, 16]. It turns out that alternating parity automata are no more expressive than non-deterministic Büchi automata, and hence neither allowing alternation, nor the richer parity acceptance conditions, increase expressiveness; this testifies to the robustness of the class of $\omega$-regular languages. On the other hand, alternation increases the expressive power of automata with the so-called weak acceptance conditions: non-deterministic weak automata are not expressive enough to recognize all $\omega$-regular languages, but alternating weak automata are. The weak acceptance conditions are significant due to their applications in logic [21] and thanks to their favourable algorithmic properties [16].

Given that alternating weak automata are expressive enough to recognize all the $\omega$-regular languages, a natural question is whether, and to what degree, alternating weak automata are less succinct than alternating Büchi or alternating parity automata. Another way of stating this question is what blow-up in the number of states is sufficient or required for translations from alternating parity or alternating Büchi automata to alternating weak automata. The first upper bound for the blow-up of a translation from alternating Büchi to alternating weak automata was doubly exponential, obtained by combining a doubly-exponential determinization construction [7] and a linear translation from deterministic parity automata to weak alternating automata [21, 19]. This has been improved considerably by Kupferman and Vardi who have given a quadratic translation from alternating Büchi to alternating weak automata [15], and then they have generalized it to a translation from alternating parity automata with $n$ states and $d$ priorities to alternating weak automata, whose blow-up is $n^{d+O(1)}$, i.e., exponential in the number of priorities in the parity acceptance 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 of states in an automaton is interesting from the algorithmic point of view. For example, the algorithmic problems of checking emptiness of non-deterministic parity automata on infinite trees, of model checking for the modal $\mu$-calculus, of solving two-player parity games, and of checking emptiness of alternating parity automata on infinite words over a one-letter alphabet, are all polynomial-time equivalent. Since checking emptiness of alternating weak automata on words over a one-letter alphabet can be done in linear time, it follows that a translation from alternating parity automata to alternating weak automata implies an algorithm for solving parity games whose complexity matches the blow-up of the number of states in the translation.

The first quasi-polynomial translation from alternating parity automata to alternating weak automata was given recently by Boker and Lehtinen [1]. They have used the register technique, developed by Lehtinen [17] for parity games, to provide a translation from alternating parity automata with $n$ states and $d$ priorities to alternating parity automata with $n^{\Theta(\log(d/\log n))}$ states and $\Theta(\log n)$ priorities; combined with the exponential translation of Kupferman and Vardi [14], this yields an alternating parity to alternating weak translation whose blow-up of the number of states is $n^{\Theta(\log n\cdot\log(d/\log n))}$.

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 parity games due to Calude, Jain, Khoussainov, Li, and Stephan [2]—can be used to further reduce the state-space blow-up in the translation from alternating parity automata to alternating weak 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 to the size of the smallest $(n,d/2)$-universal trees [5], which is polynomial in $n$ if $d=O(\log n)$ and it is $n^{\lg(d/\lg n)+O(1)}$ if $d=\omega(\log n)$. When combined with Kupferman and Vardi’s quadratic translation of alternating Büchi to alternating weak automata [15], we get the composite blow-up of the form $n^{O(\log(d/\log n))}$, down from Boker and Lehtinen’s blow-up of $\left(n^{\Theta(\log(d/\log n))}\right)^{\log n}$.

The necessary size of the state-space blow-up when going from alternating parity automata to alternating weak automata is wide open: the best known lower bound is $\Omega(n\log n)$ [15], closely related to the $2^{\Omega(n\log n)}$ lower bound on Büchi complementation [20], while the best upper bounds are quasi-polynomial. On the other hand, the blow-up for the parity to weak translation that we obtain nearly matches the current state-of-the-art quasi-polynomial upper bounds on the complexity of solving parity games [2, 11, 9]. It follows that any significant improvement over our translation would lead to a breakthrough improvement in the complexity of solving parity games.

The exponential translation from alternating parity automata to alternating weak automata due to Kupferman and Vardi [14] is done by a rather involved induction on the number of priorities. For an automaton with $d$ priorities, it goes through a sequence of $d$ intermediate automata of a generalized type, which they call parity-weak alternating automata. In contrast, our construction is significantly more streamlined and transparent; in particular, it avoids introducing a new class of hybrid parity-weak automata. We first establish a hierarchical decomposition of runs of alternating parity automata as a generalization of the decomposition 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 alternating Büchi automaton, which is a parity automaton with just $2$ priorities. Our work is yet another application of the recently introduced notion of universal trees [11, 5]. Such applications typically focus on algorithms for solving games [11, 6, 5, 3]; our work is the first whose primary focus is on automata.

In addition to universal trees, we use a notion of lazy progress measure. Unlike the standard parity progress measures, which can be recognised by safety automata but require an explicit bound to be known on the number of successive occurrences of odd priorities, lazy progress measures are recognised by Büchi automata and can deal with finite but unbounded numbers of occurrences of successive odd priorities. This Büchi automaton is similar to (but more concise than) the automaton used to characterise parity tree automata that recognise co-Büchi recognisable tree languages [18], itself a generalisation of automata used to decide the weak definability of tree languages given as Büchi automata [22, 4].

A similar concept to our lazy progress measures was already introduced by Klarlund for complementation of Büchi and Streett automata on words [12]. Klarlund indeed proves a result 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, which clearly 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 proof is self-contained. We suspect that the opaqueness of Klarlund’s paper [12] may have been responsible for attracting less attention and shallower absorption by the research community than it deserves. In particular, some of the techniques and results that he presents there have 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 reception and appreciation of Klarlund’s work.

2 Alternating automata

For a finite set $X$, we write $\mathcal{B}^{+}(X)$ for the set of positive Boolean formulas over $X$. We say that a set $Y\subseteq X$ satisfies a formula $\varphi\in\mathcal{B}^{+}(X)$ if $\varphi$ evaluates to $\mathtt{true}$ when all variables in $Y$ are set to $\mathtt{true}$ and all variables in $X\setminus Y$ are set to $\mathtt{false}$. For example, the sets $\{\,x,y\,\}$ and $\{\,x,z\,\}$ satisfy the positive Boolean formula $x\wedge(y\vee z)$, but the set $\{\,y,z\,\}$ does not. An alternating automaton has a finite set $Q$ of states, an initial state $q_{0}\in Q$, a finite alphabet $\Sigma$, and a transition function $\delta:Q\times\Sigma\to\mathcal{B}^{+}(Q)$. Alternating automata allow to combine both non-deterministic and universal transitions; disjunctions in transition formulas model the non-deterministic choices and conjunctions model the universal choices.

We consider alternating automata as acceptors of infinite words. Whether infinite sequences of states in runs of such automata are accepting or not is determined by an acceptance condition. Here, we consider parity, Büchi, co-Büchi, weak, and safety acceptance conditions. In a parity condition, given by a state priority function $\pi:Q\to\{\,0,1,2,\dots,d\,\}$ for some positive even integer $d$, an infinite sequence of states is accepting if the largest state priority that occurs infinitely many times is even. Büchi conditions are a special case of parity conditions in which all states have priorities $1$ or $2$, and co-Büchi conditions are parity conditions in which all states have priorities [math] or $1$.

Let the transition graph of an alternating automaton have an edge $(q,r)\in Q\times Q$ if $r$ occurs in $\delta(q,a)$ for some letter $a\in\Sigma$. We say that a parity automaton has a weak acceptance condition if it is stratified: in every cycle in the transition graph, all states have the same priority. Weak conditions are a special case of both Büchi and co-Büchi conditions in the following sense: if the transition graph of a parity automaton is stratified, then every infinite path in the transition graph satisfies each of the following three conditions if and only if it satisfies the other two:

• •

the parity condition $\pi:Q\to\{\,0,1,2,\dots,d\,\}$;

• •

the co-Büchi condition $\pi^{\prime}:Q\to\{\,0,1\,\}$;

• •

the Büchi condition $\pi^{\prime\prime}:Q\to\{\,1,2\,\}$;

where $\pi^{\prime}(q)=\pi(q)\bmod 2$ and $\pi^{\prime\prime}(q)=2-\pi^{\prime}(q)$ for all $q\in Q$.

We say that a state is absorbing if its only successor in the transition graph is itself. A parity automaton has a safety acceptance condition if all of its states have priority [math], except for the additional absorbing $\mathtt{reject}$ state that has priority $1$. An automaton with a safety acceptance condition is stratified, and hence safety conditions are a special case of weak conditions.

Whether an infinite word $w=w_{0}w_{1}w_{2}\cdots\in\Sigma^{\omega}$ is accepted or rejected by an alternating automaton $\mathcal{A}$ is determined by the winner of the following acceptance game $\mathcal{G}(\mathcal{A},w)$. The set of positions in the game is the set $Q\times\mathbb{N}$ and the two players, Alice and Elvis, play in the following way. The initial position is $(q_{0},0)$; for every current position $(q_{i},i)$, first Elvis chooses a subset $P$ of $Q$ that satisfies $\delta(q_{i},w_{i})$, then Alice picks a state $q_{i+1}\in P$, and $(q_{i+1},i+1)$ becomes the next current position. Note that Elvis can be thought of making the non-deterministic choices and Alice can be thought of making the universal choices in the transition function of the alternating automaton. This interaction of Alice and Elvis yields an infinite sequence of states $q_{0},q_{1},q_{2},\dots$, and whether Elvis is declared the winner or not is determined by whether the sequence is accepting according to the acceptance condition of the automaton. Acceptance games for parity, Büchi, co-Büchi, and weak conditions are parity games, which are determined [8]: in every acceptance game, either Alice or Elvis has a winning strategy. We say that an infinite word $w\in\Sigma^{\omega}$ is accepted by an alternating automaton $\mathcal{A}$ if Elvis has a winning strategy in the acceptance game $\mathcal{G}(\mathcal{A},w)$, and otherwise it is rejected.

A run dag of an alternating automaton $\mathcal{A}$ on an infinite word $w$ is a directed acyclic graph $G=(V,E,\rho:V\to Q)$, where $V\subseteq Q\times\mathbb{N}$ is the set of vertices; successors (according to the directed edge relation $E$) of every vertex $(q,i)$ are of the form $(q^{\prime},i+1)$; the following conditions hold:

• •

$(q_{0},0)\in V$,

• •

for every $(q,i)\in V$, the Boolean formula $\delta(q,w_{i})$ is satisfied by the set of states $p$, such that $(p,i+1)$ is a successor of $(q,i)$;

and $\rho$ projects vertices onto the first component. Note that every vertex in a run dag has a successor, and hence every maximal path is infinite. We say that a run dag of an automaton $\mathcal{A}$ is accepting if the sequence of states on every infinite path in the run dag is accepting according to the accepting condition of $\mathcal{A}$. The positional determinacy theorem for parity games [8] implies that an infinite word $w$ is accepted by an alternating automaton $\mathcal{A}$ with a parity (or Büchi, co-Büchi, weak, or safety) condition if and only if there is an accepting run dag of $\mathcal{A}$ 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 be partitioned into sets $L_{0},L_{1},L_{2},\dots$, such that every edge goes from some layer $L_{i}$ to the next layer $L_{i+1}$. We define the width of a layered dag with an infinite number of layers $L_{0},L_{1},L_{2},\dots$ to be $\liminf_{i\to\infty}|L_{i}|$. Note that the width of a run dag of an alternating automaton is trivially upper-bounded by the number of states of the automaton.

3 From co-Büchi and Büchi to weak

In this section we summarize the results of Kupferman and Vardi [15] who have given translations from alternating co-Büchi and Büchi automata to alternating weak automata with only a quadratic blow-up in the state space. We recall the decomposition of co-Büchi accepting run dags of Kupferman and Vardi in detail because it motivates and prepares the reader for our generalization of their result to accepting parity run dags. Our main technical result is a translation from alternating parity automata to alternating Büchi automata with only a quasi-polynomial blow-up in the state space, but the ultimate goal is a quasi-polynomial translation from parity to weak automata. Therefore, we also recall how Kupferman and Vardi use their quadratic co-Büchi to weak translation in order to obtain a quadratic Büchi to weak translation.

3.1 Co-Büchi progress measures

The main technical concept that underlies Kupferman and Vardi’s [15] translation from alternating co-Büchi automata to alternating weak automata is that of a ranking function for accepting run dags of alternating co-Büchi automata. As Kupferman and Vardi themselves point 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 certifying parity conditions is very well developed [8, 12, 13, 10, 11, 5] and our main goal in this paper is to use a version of parity progress measures to give a simplified, streamlined, and improved translation from alternating parity to alternating weak automata.

Let $G=(V,E,\pi:V\to\{\,0,1\,\})$ be a layered dag with vertex priorities [math] or $1$, and in which every vertex has a successor. Note that all run dags of an alternating co-Büchi automaton are such layered dags and if the automaton has $n$ states then the width of the run dag is at most $n$. (Observe, however, that while formally the third component $\rho:V\to Q$ in a run dag maps vertices to states, here we instead consider the labeling $\pi:V\to\{\,0,1\,\}$ that labels vertices by the priorities of the states $\pi(v)=\pi(\rho(v))$.)

A co-Büchi progress measure [8, 13, 10] is a mapping $\mu:V\to M$, where $(M,\leq)$ is a well-ordered set, such that for every edge $(v,u)\in E$, we have

if $\pi(v)=0$ then $\mu(v)\geq\mu(u)$, 2. 2.

if $\pi(v)=1$ then $\mu(v)>\mu(u)$.

It is elementary to argue that existence of a co-Büchi progress measure on a graph is sufficient for every infinite path in the graph satisfying the co-Büchi condition. Importantly, it is also necessary, which can be, for example, deduced from the proof of positional determinacy for parity games due to Emerson and Jutla [8]. In other words, co-Büchi progress measures are witnesses 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 and infinitary condition, it suffices to verify them locally by checking a simple inequality between the 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 as run dags, the well-ordered sets of labels that are needed to certify co-Büchi conditions may be of unbounded (and possibly infinite) size. In order to overcome this disadvantage, and to enable automata-theoretic uses of progress measure certificates, Klarlund has proposed the following concept of lazy progress measures [12]. A lazy (co-Büchi) progress measure is a mapping $\mu:V\to M$, where $(M,\leq)$ is a well-ordered set and $L\subset M$ is the set of lazy-progress elements, and such that:

for every edge $(v,u)\in E$, we have $\mu(v)\geq\mu(u)$; 2. 2.

if $\pi(v)=1$ then $\mu(v)\in L$; 3. 3.

on every infinite path in $G$, there are infinitely many vertices $v$ such that $\mu(v)\not\in L$.

It is elementary to prove the following proposition.

Proposition 3.1**.**

*If a graph has a lazy co-Büchi progress measure then all infinite paths in it satisfy the co-Büchi condition. *

The following converse establishes the attractiveness of lazy co-Büchi progress measures for certifying the co-Büchi conditions on layered dags of bounded width, and hence for certifying accepting run dags of alternating co-Büchi automata.

Lemma 3.2** (Klarlund [12]).**

If all infinite paths in a layered dag $(V,E,\pi:V\to\{\,0,1\,\})$ satisfy the co-Büchi condition and the width of the dag is at most $n$, then there is a lazy co-Büchi progress measure $\mu:V\to M$, where $M=\{\,1,2,\dots,2n\,\}$ and $L=\{\,2,4,6,\dots,2n\,\}$.

Proof 3.3**.**

We summarize a proof given by Kupferman and Vardi [15] that provides an explicit decomposition of the accepting run dag into (at most) $2n$ parts from which a lazy co-Büchi progress measure can be straightforwadly defined. The proof by Klarlund [12] is more succinct, 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 a vertex $v$ whose all descendants (i.e., vertices to which there is a—possibly empty—path from $v$) have priority [math]; call such vertices $1$-safe in $G_{1}=G$. Indeed, otherwise it would be easy to construct an infinite path with infinitely many occurrences of vertices of priority $1$.

Let $S_{1}$ be the set of all the $1$-safe vertices in $G_{1}$, and let $G^{\prime}_{1}=G_{1}\setminus S_{1}$ be the layered dag obtained from $G_{1}$ by removing all vertices in $S_{1}$. Note that there is an infinite path in the subgraph of $G_{1}$ induced by $S_{1}$, and hence the width of $G^{\prime}_{1}$ is strictly smaller than the width of $G_{1}$.

Let $R_{1}$ be the set of all vertices in $G^{\prime}_{1}$ that have only finitely many descendants; call such vertices transient in $G^{\prime}_{1}$. Let $G_{2}$ be the the layered dag obtained from $G^{\prime}_{1}$ by removing all vertices in $R_{1}$. Since $G_{2}$ is a subgraph of $G^{\prime}_{1}$, the width of $G_{2}$ is strictly smaller than the width of $G_{1}$. Moreover, $G_{2}$ shares the key properties with $G_{1}$: every vertex has a successor and hence all the maximal paths are infinite, and all infinite paths satisfy the co-Büchi condition.

By applying the same procedure to $G_{2}$ that we have described for $G_{1}$ above, we obtain the set $S_{2}$ of $1$-safe vertices in $G_{2}$ and the set $R_{2}$ of vertices transient in $G^{\prime}_{2}$, and the layered dag $G_{3}$—obtained from $G_{2}$ by removing all vertices in $S_{2}\cup R_{2}$—has the width that is strictly smaller than that of $G_{2}$. We can continue in this fashion until the graph $G_{k+1}$, for some $k\geq 1$, is empty. Since the width of $G$ is at most $n$, and the widths of graphs $G_{1},G_{2},\dots,G_{k+1}$ are strictly decreasing, it follows that $k\leq n$.

We define $\mu:V\to\{\,1,2,\dots,2n\,\}$ by:

[TABLE]

and note that it is routine to verify that if we let $L=\{\,2,4,\dots,2n\,\}$ be the set of lazy-progress elements then $\mu$ is a lazy co-Büchi progress measure.

3.2 From co-Büchi and Büchi to weak

In this section we present a proof of the following result.

Theorem 3.4** (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 $O(n^{2})$ states. *

Proof 3.5**.**

It suffices to argue that, given an alternating co-Büchi automaton $\mathcal{A}=(Q,q_{0},\Sigma,\delta,\pi:Q\to\{\,0,1\,\})$ with $n$ states, we can design an alternating weak automaton with $O(n^{2})$ states that guesses and certifies a dag run of $\mathcal{A}$ together with a lazy co-Büchi progress measure on it as described in Lemma 3.2. First we construct a safety automaton $\mathcal{S}$ with $O(n^{2})$ states that simulates the automaton $\mathcal{A}$ while guessing a lazy co-Büchi progress measure and verifying conditions 1) and 2) of its definition. Condition 3) will be later handled by turning the safety automaton $\mathcal{S}$ into a weak automaton $\mathcal{W}$ by appropriately assigning odd or even priorities to all states in $\mathcal{S}$. We split the design of $\mathcal{W}$ into those two steps so that we can better motivate and explain the generalized constructions in Section 5.

The safety automaton $\mathcal{S}$ has the following set of states:

[TABLE]

its initial state is $(q_{0},2n)$; and its transition function $\delta^{\prime}$ is obtained from the transition function $\delta$ of $\mathcal{A}$ in the following way: for every state $(q,i)$, and for every $a\in\Sigma$, the formula \delta^{\prime}\big{(}(q,i),a\big{)} is obtained from $\delta(q,a)$ by replacing every occurrence of state $q^{\prime}\in Q$ by the disjunction (i.e., a non-deterministic choice)

[TABLE]

where every occurrence $(q^{\prime},j)$ for which $\pi(q^{\prime})=1$ and $j$ is odd stands for the state $\mathtt{reject}$.

In other words, the safety automaton $\mathcal{S}$ can be thought of as consisting of $2n$ copies $\mathcal{A}_{2n},\mathcal{A}_{2n-1},\dots,\mathcal{A}_{1}$ of $\mathcal{A}$, with the non-accepting states $\pi^{-1}(1)$ removed from the odd-indexed copies $\mathcal{A}_{2n-1},\mathcal{A}_{2n-3},\dots,\mathcal{A}_{1}$, and in whose acceptance games, Elvis always has the choice to stay in the current copy of $\mathcal{A}$ or to move to one of the lower-indexed copies of $\mathcal{A}$. Since the transitions of the safety automaton $\mathcal{S}$ always respect the transitions of the original co-Büchi automaton $\mathcal{A}$, an accepting run dag of $\mathcal{S}$ yields a run dag of $\mathcal{A}$ (obtained from the first components of the states $(q,i)$) and a labelling of its vertices by numbers in $\{\,1,2,\dots,2n\,\}$ (obtained from the second components of the states $(q,i)$). It is routine to verify that the design of the state set and of the transition function of the safety automaton $\mathcal{S}$ guarantees that the latter labelling satisfies conditions 1) and 2) of the definition of a lazy co-Büchi progress measure, where the set of lazy-progress elements is $\{\,2,4,\dots,2n\,\}$.

By setting the state priority function $\pi^{\prime}:(q,i)\mapsto i+1$ for all non-$\mathtt{reject}$ states in $\mathcal{S}$, and $\pi^{\prime}:\mathtt{reject}\mapsto 1$, we obtain from $\mathcal{S}$ an automaton $\mathcal{W}$ whose acceptance condition is weak because—by design—the transition function is non-increasing w.r.t. the state priority function. One can easily verify that the addition of this weak acceptance condition to $\mathcal{S}$ allows the resulting automaton $\mathcal{W}$, for every input word, to guess and verify a lazy progress measure—if one exists—on a run dag of automaton $\mathcal{A}$ on the input word, while $\mathcal{W}$ rejects the input word otherwise. This completes our summary of the proof of Theorem 3.4.

Corollary 3.6** (Kupferman and Vardi [15]).**

There is a translation that given an alternating Büchi automaton with $n$ states yields an equivalent alternating weak automaton with $O(n^{2})$ states.

The argument of Kupferman and Vardi is simple and it exploits the ease with which alternating automata can be complemented. Given an alternating Büchi automaton $\mathcal{A}$ with $n$ states, first complement it with no state space blow-up, obtaining an alternating co-Büchi automaton with $n$ states, next use the translation from Theorem 3.4 to obtain an equivalent alternating weak automaton with $O(n^{2})$ states, and finally complement the latter again with no state space blow-up, hence obtaining an alternating weak automaton that is equivalent to $\mathcal{A}$ and that has $O(n^{2})$ states.

4 Lazy parity progress measures

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-closed set 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-descendant terminology to describe relative positions of nodes in a tree. For example, $\langle\rangle$ is the root; node $\langle x,y\rangle$ is the child of the node $\langle x\rangle$ that is reached from it via the branching direction $y$; node $\langle x,y\rangle$ is the parent of node $\langle x,y,z\rangle$; nodes $\langle x,y\rangle$ and $\langle x,y,z\rangle$ are descendants of nodes $\langle\rangle$ and $\langle x\rangle$; nodes $\langle\rangle$ and $\langle x\rangle$ are ancestors of nodes $\langle x,y\rangle$ and $\langle x,y,z\rangle$; and a node is a leaf if it does not have any children. All nodes in a well-ordered tree are well-ordered by the lexicographic order that is induced by the well-order on the branching directions; for example, we have $\langle x\rangle<\langle x,y\rangle$, and $\langle x,y,z\rangle<\langle x,w\rangle$ if $y<w$. We define the depth of a node to be the number of elements in the eponymous sequence, the height of a tree to be the 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, and the labels are nodes in a well-ordered tree. A tree labelling of a graph with vertex priorities that do not exceed a positive even integer $d$ is a mapping from vertices of the graph to nodes in a well-ordered tree of height at most $d/2$. We write $\langle m_{d-1},m_{d-3},\dots,m_{\ell}\rangle$, for some odd $\ell$, $1\leq\ell<d$, to denote such nodes. We say that such a node has an (odd) level $\ell$ and an (even) level $\ell-1$, and the root $\langle\rangle$ has level $d$. Moreover, for every priority $p$, $0\leq p\leq d$, we define the $p$-truncation $\langle m_{d-1},m_{d-3},\dots,m_{\ell}\rangle|_{p}$ in the following way:

[TABLE]

We then say that a tree labelling $\mu$ of a graph $G=(V,E)$ with vertex priorities $\pi:V\to\{\,0,1,2,\dots,d\,\}$ is a parity progress measure if the following progress condition holds for every edge $(v,u)\in E$:

if $\pi(v)$ is even then $\mu(v)|_{\pi(v)}\geq\mu(u)|_{\pi(v)}$; 2. 2.

if $\pi(v)$ is odd then $\mu(v)|_{\pi(v)}>\mu(u)|_{\pi(v)}$.

It is well-known that satisfaction of such local conditions on every edge in a graph is sufficient for 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 determinacy of parity games due to Emerson and Jutla [8]. In other words, parity progress measures are witnesses for the property that all infinite paths in a graph satisfy the parity condition. Like for the simpler co-Büchi condition, their appeal stems from the property that they certify conditions that are global and infinitary by verifying conditions that are local to every edge in the graph.

Similar to the simpler co-Büchi progress measures, parity progress measures may unfortunately require unbounded or even infinite well-ordered trees to certify parity conditions on infinite graphs, and hence we consider lazy parity progress measures, also inspired by Klarlund’s pioneering work [12]. A lazy tree is a well-ordered tree with a distinguished subset of its nodes called lazy nodes. For convenience, we assume that only leaves may be lazy and the root never is.

A lazy parity progress measure is a tree labelling $\mu$ of a graph $(V,E)$, where the labels are nodes in a lazy tree $T$, such that:

for every edge $(v,u)\in E$, $\mu(v)|_{\pi(v)}\geq\mu(u)|_{\pi(v)}$; 2. 2.

if $\pi(v)$ is odd then node $\mu(v)$ is lazy and its level is at least $\pi(v)$; 3. 3.

on every infinite path in $G$, there are infinitely many vertices $v$, such that $\mu(v)$ is not lazy.

First we establish that existence of a lazy progress measure is sufficient for all infinite paths in a graph to satisfy the parity condition.

Lemma 4.1**.**

If a graph has a lazy parity progress measure then all infinite paths in it satisfy the parity condition.

Proof 4.2**.**

For the sake of contradiction, assume that there is an infinite path $v_{1},v_{2},v_{3},\ldots$ in the graph for which the highest priority $p$ that occurs infinitely often is odd. Let $i\geq 1$ be such that $\pi(v_{j})\leq p$ for all $j\geq i$. By condition 1), we have:

[TABLE]

Let $i\leq i_{1}<i_{2}<i_{3}<\cdots$ be such that $\pi(v_{i_{k}})=p$ for all $k=1,2,3,\dots$. By condition 2), all labels $\mu(v_{i_{k}})$, for $k=1,2,3,\dots$, are lazy and their level in the tree is at least $p$. By condition 3), for infinitely many $k$, $\pi(v_{k})$ is not lazy, so infinitely many of the inequalities in (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 graph to satisfy the parity condition. Moreover, we explicitly quantify the size of a lazy ordered tree 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 simple operation that we call a lazification of a finite ordered tree. If $T$ is a finite ordered tree, then its lazification $\mathrm{lazi}(T)$ is a finite lazy tree that is obtained from $T$ in the following way:

• •

all nodes in $T$ are also nodes in $\mathrm{lazi}(T)$ and they are not lazy;

• •

for every non-leaf node $t$ in $T$, $t$ has extra lazy children in the tree $\mathrm{lazi}(T)$, one smaller and one larger than all the other children, and one in-between every pair of consecutive children.

It is routine to argue that if a tree has $n$ leaves and it is of height at most $h$ then its lazification $\mathrm{lazi}(T)$ has $O(nh)$ nodes and it is also of height $h$.

Theorem 4.3** (Klarlund [12]).**

If all infinite paths in a layered dag satisfy the parity condition and the width of the dag is at most $n$, then there is a lazy parity progress measure whose labels are nodes in a tree that is a lazification of an ordered tree with at most $n$ leaves.

Proof 4.4**.**

Klarlund’s proof [12] is very succinct and it heavily relies on the result of Klarlund and Kozen on Rabin measures [13]. Our proof is not only self-contained but it also is more constructive and transparent. The hierarchical decomposition describes the fundamental structure of accepting run dags of alternating parity automata and it may be of independent interest. The argument presented here is a generalization of the proof of Lemma 3.2—given in Section 3—from co-Büchi conditions to parity conditions.

Consider a layered dag $G=(V,E,\pi)$ where $\pi:V\rightarrow\{\,0,1,2,\dots,d\,\}$. For a priority $p$, $0\leq p<d$, we write $G^{\leq p}$ for the subgraph induced by the vertices whose priority is at most $p$.

We describe the following decomposition of $G$. Let $D$ be the set that consists of all vertices of the top even priority $d$ in $G$, and $R_{0}$ all those vertices in the subgraph $G^{\leq d-1}$ that have finitely many descendants. We say that those vertices are $(d-1)$-transient in $G^{\leq d-1}$. In other words, $D\cup R_{0}$ is the set of vertices from which every path reaches (possibly immediately) a vertex of priority $d$.

Let $G_{1}=G\setminus(D\cup R_{0})$ be the layered dag obtained from $G$ by removing all vertices in $D\cup R_{0}$. Observe that every vertex in $G_{1}$ has at least one successor and hence—unless $G_{1}$ is empty—all maximal paths are infinite. W.l.o.g., assume henceforth that $G_{1}$ is not empty. We argue that there must be a vertex in $G_{1}$ whose all descendants have priorities at most $d-2$; call such vertices $(d-1)$-safe in $G_{1}$. Indeed, otherwise it would be easy to construct an infinite path with infinitely many occurrences of the odd priority $d-1$ and no occurrences of the top even priority $d$.

Let $S_{1}$ be the set of all the $(d-1)$-safe vertices in $G_{1}$. Let $H_{1}$ be the subgraph of $G$ induced by $S_{1}$, let $n_{1}$ be the width of $H_{1}$, and note that $n_{1}>0$. Set $G^{\prime}_{1}=G_{1}\setminus S_{1}$ to be the layered dag obtained from $G_{1}$ by removing all $(d-1)$-safe vertices in $G_{1}$.

Let $R_{1}$ be the set of all vertices in $G^{\prime}_{1}$ that have only finitely many descendants; call such vertices $(d-1)$-transient in $G^{\prime}_{1}$. Finally, let $G_{2}$ be the layered dag obtained from $G^{\prime}_{1}$ by removing all the $(d-1)$-transient vertices in $G^{\prime}_{1}$. Note that the width of $G_{2}$ is smaller than the width of $G_{1}$ by at least $n_{1}>0$.

Unless graph $G_{2}$ is empty, we can now apply the same steps to $G_{2}$ that we have described for $G_{1}$, and obtain:

• •

the set $S_{2}$ of $(d-1)$-safe vertices in $G_{2}$;

• •

the subgraph $H_{2}$ of $G_{2}$ induced by $S_{2}$, which is of width $n_{2}>0$;

• •

the layered dag $G^{\prime}_{2}$, obtained from $G_{2}$ by removing all the vertices in $S_{2}$;

• •

the set $R_{2}$ of $(d-1)$-transient vertices in $G^{\prime}_{2}$;

• •

the layered graph $G_{3}$, obtained from $G_{2}$ by removing all vertices in $S_{2}\cup R_{2}$, and whose width is smaller than the width of $G_{2}$ by at least $n_{2}>0$.

We can continue in this fashion, obtaining graphs $G_{1},G_{2},\dots,G_{k+1}$, until the graph $G_{k+1}$, for some $k\geq 0$, is empty. Since the width of $G$ is at most $n$ and the widths of the graphs $G_{1},G_{2},\dots,G_{k}$ are positive (unless $k=0$), we have that $k\leq n$ and $\sum_{i=1}^{k}n_{i}\leq n$.

The proces described above yields a hierarchical decomposition of the layered dag; we now define—by induction on $d$—the tree that describes the shape of this decomposition. We then argue that the lazification of this tree provides the set of labels in a lazy parity progress measure.

In the base case $d=0$, the shape of the decomposition is the well-ordered tree $T$ of height $h=0/2=0$ with only a root node $\langle\rangle$. It is straightforward to see that the function that maps every vertex onto the root is a (lazy) progress measure.

For $d\geq 2$, note that all vertices in dags $H_{1},H_{2},\dots,H_{k}$ have priorities at most $d-2$. By the inductive hypothesis, there are trees $T_{1},T_{2},\dots,T_{k}$, of heights at most $h-1=(d-2)/2$ and with at most $n_{1},n_{2},\dots,n_{k}$ leaves, respectively, which are the shapes of the hierarchical decompositions of dags $H_{1},H_{2},\dots,H_{k}$, respectively.

We now construct the finite ordered tree $T$ of height at most $h=d/2$ that is the shape of the hierarchical decomposition of $G$: let $T$ consist of the root node $\langle\rangle$ that has $k$ children, which are the roots of the subtrees $T_{1},T_{2},\dots,T_{k}$, in that order. Note that the number of leaves of $T$ is at most $\sum_{i=1}^{k}n_{i}\leq n$. Consider the following mapping from vertices in the graph onto nodes in the lazification $\mathrm{lazi}(T)$ of tree $T$:

• •

vertices in set $D$ are mapped onto the root of $\mathrm{lazi}(T)$;

• •

vertices in transient sets $R_{0},R_{1},R_{2},\dots,R_{k}$ are mapped onto the lazy children of the root of $\mathrm{lazi}(T)$: those in $R_{0}$ onto the smallest lazy child, those in $R_{1}$ onto the lazy child between the roots of $T_{1}$ and $T_{2}$, etc.;

• •

vertices in subgraphs $H_{1},H_{2},\dots,H_{k}$ are inductively mapped onto the appropriate nodes in the lazy subtrees of $\mathrm{lazi}(T)$ 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 a lazy parity progress measure. Condition 3) is ensured by the fact that the root of $T$ is not lazy and by the inductive hypothesis. Recall that every infinite path satisfies the parity condition, 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 the path contains only vertices in one of the sets $S_{i}$ and we can use the inductive hypothesis. *

5 From parity to Büchi via universal trees

In this section we complete the proof of the main technical result of the paper, which is a quasi-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 and universal trees [11, 5], and the state space blow-up of the translation is merely quadratic in the size of the smallest universal tree. Nearly tight quasi-polynomial upper and lower bounds have 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 at most logarithmic in the number of states, then the state space blow-up of our translation is only polynomial.

Theorem 5.1**.**

There is a translation that given an alternating parity automaton with $n$ states and $d$ priorities yields an equivalent alternating weak automaton whose number of states is polynomial if $d=O(\log n)$ and it is $n^{O\left(\lg(d/\lg n)\right)}$ if $d=\omega(\log n)$.

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 tree of height at most $h$ and with at most $n$ leaves can be isomorphically embedded into it. In such 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-preserving way—onto the children of its image. In order to upper-bound the size of the blow-up in our parity to weak translation, we rely on the following upper bound on the size of the smallest universal trees.

Theorem 5.2** (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 $h=\omega(\log n)$.

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 5.1, we establish the following lemma that provides a translation from alternating parity automata to alternating Büchi automata whose state-space blow-up is tightly linked to the size of universal trees.

Lemma 5.3**.**

There is a translation that given an alternating parity automaton with $n$ states and $d$ 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 $(n,d/2)$-universal ordered tree $U$.

Note that Theorem 5.1 follows from Lemma 5.3 by Theorem 5.2 and Corollary 3.6.

Proof 5.4**.**

(of Lemma 5.3) Given an alternating parity automaton $\mathcal{A}=(Q,q_{0},\Sigma,\delta,\pi:Q\to\{\,0,1,\ldots,d\,\})$ with $n$ states, we now design an alternating Büchi automaton that guesses and certifies a dag run of $\mathcal{A}$ 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 $\mathcal{A}$ while guessing a lazy parity progress measure and verifying conditions 1) and 2) of its definition. Condition 3) will be later handled by turning the safety automaton into a Büchi automaton 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 $\mathcal{B}_{T}$ from any lazy well-ordered tree $T$, and then we argue that the alternating parity automaton $\mathcal{A}$ is equivalent to the alternating Büchi automaton $\mathcal{B}_{\mathrm{lazi}(U)}$, for every $(n,d/2)$-universal tree $U$.

Let $T$ be a lazy tree of width $n$ and height $d/2$. The construction is by induction on $d$. The safety automaton $\mathcal{S}_{T}$ 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 $\mathcal{S}_{T}$ is $\left(Q\times\{\langle\rangle\}\right)\cup\{\,\mathtt{reject}\,\}$.

• •

Otherwise, let $\langle x_{1}\rangle,\langle x_{2}\rangle,\ldots,\langle x_{k}\rangle$ be the children of the root, and $1\leq i_{1}<i_{2}<\ldots<i_{m}\leq k$ are the indices of its leaves that are lazy.

For $i\notin\{\,i_{1},i_{2},\ldots,i_{m}\,\}$, let $T_{i}$ be the lazy subtrees of $T$ of height at most $d/2-1$ rooted in $\langle x_{i}\rangle$. By induction, for all $i$, we obtain an alternating Büchi automaton that is obtained from the lazy tree $T_{i}$ and from the alternating parity automaton $\mathcal{A}$ restricted to the states of priority up to $d-2$. Let $\Omega_{i}$ denote its set of non-$\mathtt{reject}$* states. They are pairs consisting of an element of $Q$ and of a node in a tree of height $d/2-1$; the latter is a sequence $\langle m_{d-3},m_{d-5},\dots,m_{\ell}\rangle$ of at most $d/2-1$ branching directions. Let $\Gamma_{i}$ be the set consisting of the pairs $\left(q,\langle x_{i},m_{d-3},m_{d-5},\dots,m_{\ell}\rangle\right)$ for $\left(q,\langle m_{d-3},m_{d-5},\dots,m_{\ell}\rangle\right)\in\Omega_{i}$. Set $Q^{(d)}$ (resp. $Q^{(<d)}$) the set of states of priority $d$ (resp. $<d$) in $\mathcal{A}$. The states of $\mathcal{S}_{T}$ are defined as:*

[TABLE]

The initial state is $(q_{0},t)$ where $t$ is the largest tuple such that $(q_{0},t)$ is a state. Let us now define the transition function: for every state $(q,t)$, and for every $a\in\Sigma$, the formula \delta^{\prime}\big{(}(q,t),a\big{)} is obtained from $\delta(q,a)$ by replacing every occurrence of state $q^{\prime}\in Q$ by the disjunction (i.e., a non-deterministic choice)

[TABLE]

where every occurrence $(q^{\prime},t^{\prime})$ which is not in the set of states stands for the state $\mathtt{reject}$.

In other words, the safety automaton $\mathcal{S}_{T}$ can be thought of as consisting of copies of $\mathcal{A}$, for each node of the tree $T$, in whose acceptance games Elvis always has the choice to stay in the current copy of $\mathcal{A}$ or to move to one of a smaller node with respect to the priority of the current state. Since the transitions of the safety automaton $\mathcal{S}_{T}$ always respect the transitions of the original parity automaton $\mathcal{A}$, an accepting run dag of $\mathcal{S}_{T}$ yields a run dag of $\mathcal{A}$ (obtained from the first components of the states $(q,t)$) and a labelling of its vertices by nodes in $T$ (obtained from the second components of the states $(q,t)$). It is routine to verify that the design of the state set and of the transition function of the safety automaton $\mathcal{S}_{T}$ guarantees that the latter labelling satisfies condition 1) and 2) of the definition of a lazy parity progress measure.

In order to obtain the Büchi automaton $\mathcal{B}_{T}$ from the safety automaton $\mathcal{S}_{T}$, it suffices to appropriately assign priorities $1$ and $2$ to all states: we let the state $\mathtt{reject}$ and all states $(q,t)$ such that $t$ is a lazy node in tree $T$ have priority $1$, and we let all states $(q,t)$ such that $t$ is not a lazy node in tree $T$ have priority $2$. Note that this ensures that a run of $\mathcal{B}_{T}$ is accepting if and only if the tree labelling of a run dag of $\mathcal{A}$ that the underlying safety automaton $\mathcal{S}_{T}$ guesses—in the second component of its states—satisfies condition 3) of the definition of a lazy parity progress measure.

We now argue that if $U$ is an $(n,d/2)$-universal tree then the alternating Büchi automaton $\mathcal{B}_{\mathrm{lazi}(U)}$ is equivalent to the alternating parity automaton $\mathcal{A}$. Firstly, all words accepted by $\mathcal{B}_{T}$ for any finite lazy ordered tree $T$ are also accepted by $\mathcal{A}$. This is because—as we have argued above—every accepting run dag of any such automaton $\mathcal{B}_{T}$ yields both a run dag of $\mathcal{A}$ (in the first state components) and a lazy parity progress measure on it (in the second state components), and the latter certifies that the former is accepting.

It remains to argue that every word accepted by $\mathcal{A}$ is also accepted by $\mathcal{B}_{\mathrm{lazi}(U)}$. By Theorem 4.3, for every accepting run dag of $\mathcal{A}$, there is a lazy progress measure whose labels are nodes in a tree $\mathrm{lazi}(T)$, where $T$ is an ordered tree of height at most $d/2$ and with at most $n$ leaves. It is routine to verify that if an ordered tree can be isomorphically embedded in another, then the same holds for their lazifications. By $(n,d/2)$-universality of $U$, it follows that $\mathrm{lazi}(T)$ can be isomorphically embedded in $\mathrm{lazi}(U)$. Therefore, for every word on which there is an accepting run dag of $\mathcal{A}$, the automaton $\mathcal{B}_{\mathrm{lazi}(U)}$ has the capacity to guess the run dag of $\mathcal{A}$ and to guess and certify a lazy progress measure on it.

In order to conclude the $O(ndL_{U})$ upper bound on the number of states of $\mathcal{B}_{\mathrm{lazi}(U)}$, it suffices to observe that if the number of leaves in an ordered tree $T$ of height $h$ is $L$ then the number of nodes in $\mathrm{lazi}(T)$ is $O(hL)$.

6 Open questions

Our use of universal trees to turn alternating parity automata into Büchi automata, like Boker and Lehtinen’s [1] register technique, does not exploit alternations (although the further Büchi to weak translation does): all transitions that are not copied from the original automaton are non-deterministic. Can universal and non-deterministic choices be combined to further improve these translations? Can the long-standing $\Omega(n\log n)$ lower bound [20] be improved, for example by a combination of the full-automata technique of Yan [23] and the recent lower bound techniques for non-deterministic safety separating automata based on universal trees [5] and universal graphs [3]?