On the Succinctness of Alternating Parity Good-for-Games Automata
Udi Boker, Denis Kuperberg, Karoliina Lehtinen, Michał Skrzypczak
OOn the Succinctness of Alternating ParityGood-for-Games Automata (Full Version ∗ ) Udi Boker
Interdisciplinary Center (IDC) Herzliya, [email protected]
Denis Kuperberg
CNRS, LIP, École Normale Supérieure, Lyon, [email protected]
Karoliina Lehtinen
University of Liverpool, United [email protected]
Michał Skrzypczak
Institute of Informatics, University of Warsaw, [email protected]
Abstract
We study alternating parity good-for-games (GFG) automata, i.e., alternating parity automata whereboth conjunctive and disjunctive choices can be resolved in an online manner, without knowledge ofthe suffix of the input word still to be read.We show that they can be exponentially more succinct than both their nondeterministic anduniversal counterparts. Furthermore, we present a single exponential determinisation procedure andan
Exptime upper bound to the problem of recognising whether an alternating automaton is GFG.We also study the complexity of deciding “half-GFGness”, a property specific to alternatingautomata that only requires nondeterministic choices to be resolved in an online manner. We showthat this problem is
PSpace -hard already for alternating automata on finite words.
Theory of computation → Logic and verification
Keywords and phrases
Good for games, history-determinism, alternation
Funding
Udi Boker : Israel Science Foundation grant 1373/16
Karoliina Lehtinen : This project has received funding from the European Union’s Horizon 2020research and innovation programme under the Marie Skłodowska-Curie grant agreement No 892704.
Good-for-games (GFG) automata were first introduced in [11] as a tool for solving the synthesisproblem. The equivalent notion of history-determinism was introduced independently in [7]in the context of regular cost functions. Intuitively, a nondeterministic automaton is GFGif nondeterminism can be resolved on the fly, only with knowledge of the input word readso far. GFG automata can be seen as an intermediate formalism between deterministicand nondeterministic ones, with advantages from both worlds. Indeed, like deterministicautomata, GFG automata enjoy good compositional properties—useful for solving gamesand composing automata and trees—and easy inclusion checks [3]. Like nondeterministicautomata, they can be exponentially more succinct than deterministic automata [16].In recent years, much effort has gone into understanding various properties of nondetermin-istic GFG automata, for instance their relationship with deterministic automata [3, 16, 4, 15], ∗ This is the full version of the paper of the same name published at FSTTCS 2020 a r X i v : . [ c s . F L ] S e p On the Succinctness of Alternating Parity Good-for-Games Automata (Full Version) applications in probabilistic model checking [14] and synthesis of LTL, µ -calculus and context-free properties [12, 18], decision procedures for GFGness [19, 16, 2], minimisation [1], andlinks with recent advances in parity games [9].Alternating GFG automata are a natural generalisation of nondeterministic GFG auto-mata that enjoy the same compositional properties as nondeterministic GFG automata, whileproviding more flexibility. As we show in the present work, for some languages alternatingGFG parity automata can also be exponentially more succinct, allowing for better synthesisprocedures. Indeed, two-player games with winning conditions given by alternating GFGautomata are solvable in quasipolynomial time, via a linear reduction to parity games, whilefor winning conditions given by arbitrary alternating automata, solving games requiresdeterminisation and has therefore double-exponential complexity.Alternating GFG automata were introduced independently by Colcombet [8] and Quirl [22]while a form of alternating GFG automata with requirements specific to counters were alsoconsidered in [17], as a tool to study cost functions on infinite trees. Boker and Lehtinenstudied the expressiveness and succinctness of alternating GFG automata in [5], showingthat theyare not more succinct than DFAs on finite words,are as expressive as deterministic ones of the same acceptance condition on infinite words,and can be determinised with a 2 θ ( n ) size blowup for the Büchi and coBüchi conditions.Many questions about GFG alternating automata were left open, in particular whetherthere exists a doubly exponential succinctness gap between alternating GFG and deterministicautomata, and the complexity of deciding whether an alternating parity automaton is GFG. Succinctness of alternating GFG automata.
We show that there is a single exponentialgap between alternating parity GFG automata and deterministic ones, thereby answeringa question left open in [5]. This is in contrast to general alternating automata, for whichdeterminisation incurs a double-exponential size increase. However, we also show that altern-ating GFG automata can present exponential succinctness compared to both nondeterministicand universal GFG automata. This means that alternating GFG automata can be used toreduce the complexity of solving some games with complex acceptance conditions.
Recognising GFG automata.
We give an
Exptime upper bound to the problem of decidingwhether an alternating parity automaton is GFG, matching the known upper bound forrecognising nondeterministic parity GFG automata.We also study the complexity of deciding “half-GFGness”, i.e., whether the nondetermin-ism (or universality) of an automaton is GFG. This property guarantees that compositionwith games preserves the winner for one of the players. We show that already on finitewords, this problem is
PSpace -hard, and it is in
Exptime for alternating Büchi automata.This shows that a
PTime algorithm for deciding GFGness must exploit the subtle interplaybetween nondeterminism and universality, and cannot be reduced to checking independentlywhether each of them is GFG.
Roadmap
We begin with some definitions, after which, in Section 3, we define alternating GFGautomata, study their succinctness and the complexity of deciding half-GFGness, that is,whether the nondeterminism within an alternating automaton is GFG. Section 4 provides asingle-exponential determinisation procedure for alternating GFG parity automata. Section 5 . Boker, D. Kuperberg, K. Lehtinen, M. Skrzypczak 3 shows that GFGness of alternating parity automata is in
Exptime , using the determinisationof the previous section. Throughout the paper, we provide high-level proof sketches, withdetailed technical developments in the appendix.
Words and automata. An alphabet Σ is a finite nonempty set of letters. A finite (resp.infinite) word u = u . . . u k ∈ Σ ∗ (resp. w = w w . . . ∈ Σ ω ) is a finite (resp. infinite) sequenceof letters from Σ. A language is a set of words, and the empty word is written (cid:15) . We denotea set { i, i + 1 , . . . , j } of integers by [ i, j ].An alternating word automaton is a tuple A = (Σ , Q, ι, δ, α ), where: Σ is an alphabet; Q is a finite nonempty set of states; ι ∈ Q is an initial state; δ : Q × Σ → B + ( Q ) is a transitionfunction where B + ( Q ) is the set of positive Boolean formulas ( transition conditions ) over Q ; and α , on which we elaborate below, is either an acceptance condition or a transitionlabelling on top of which an acceptance condition is defined. For a state q ∈ Q , we denote by A q the automaton that is derived from A by setting its initial state ι to q .An automaton A is nondeterministic (resp. universal) if all its transition conditions aredisjunctions (resp. conjunctions), and it is deterministic if all its transition conditions arejust states. We represent the transition function of nondeterministic and universal automataas δ : Q × Σ → Q , and of a deterministic automaton as δ : Q × Σ → Q . A transition of anautomaton is a triple ( q, a, q ) ∈ Q × Σ × Q , sometimes also written q a −→ q .We denote by b δ ⊆ B + ( Q ) the set of all subformulas of formulas in the image of δ , i.e., allthe Boolean formulas that “appear” somewhere in the transition function of A . Acceptance conditions.
There are various acceptance (winning) conditions, defined withrespect to the set of transitions that a path of A visits infinitely often. (Notice that atransition condition allows for many possible transitions.) We later formally define acceptanceof a word w by A in terms of games, and consider a path of A on a word w as a play in thatgame. For nondeterministic automata, a “run” coincides with a “path”.Some of the acceptance conditions are defined on top of a labelling of the transitionsrather than directly on the transitions. In particular, in the parity condition, we have α : Q × Σ × Q → Γ, where Γ ⊆ N is a finite set of priorities and a path is accepting if andonly if the highest priority seen infinitely often on it is even.The Büchi and coBüchi conditions are special cases of the parity condition with Γ = { , } and Γ = { , } , respectively. When speaking of Büchi and coBüchi automata, we often referto α as the set of “accepting transitions”, namely the transitions that are mapped to 2 in theBüchi case and to 0 in the coBüchi case. The weak condition is a special case of both theBüchi and coBüchi conditions, in which every path eventually remains in the same priority.The Rabin and Streett conditions are more involved, yet defined directly on the set T oftransitions. A Rabin condition is a set { ( B , G ) , ( B , G ) , . . . , ( B k , G k ) } , with B i , G i ⊆ T ,and a path ρ is accepting iff for some i ∈ [1 , k ], we have that the set inf( ρ ) of transitionsthat are visited infinitely often in ρ satisfies (inf( ρ ) ∩ B i = ∅ and inf( ρ ) ∩ G i = ∅ ). A Streettcondition is dual: a set { ( B , G ) , ( B , G ) , . . . , ( B k , G k ) } , with B i , G i ⊆ Q , whereby a path ρ is accepting iff for all i ∈ [1 , k ], we have (inf( ρ ) ∩ B i = ∅ or inf( ρ ) ∩ G i = ∅ ). Acceptance is defined in the literature with respect to either states or transitions; for technical reasonswe prefer to work with acceptance on transitions.
On the Succinctness of Alternating Parity Good-for-Games Automata (Full Version)
Sizes and types of automata.
The size of A is the maximum of the alphabet size, thenumber of states, the transition function length, which is the sum of the transition conditionlengths over all states and letters, and the acceptance condition’s index, which is 1 for weak,Büchi and coBüchi, | Γ | for parity, and k for Rabin and Street.We sometimes abbreviate automata types by three-letter acronyms in { D, N, U, A } × {
F,W, B, C, P, R, S } × {
A,W } . The first letter stands for the transition mode, the second for theacceptance condition, and the third indicates that the automaton runs on finite or infinitewords. For example, DPW stands for a deterministic parity automaton on infinite words. Games and strategies.
Some of our technical proofs use standard concepts of an arena, agame, a winning strategy, etc. For the sake of completeness, we provide precise mathematicaldefinitions of these objects in Appendix A. Here we will just overview the involved concepts.First, we work with two-player games of perfect information, where the players are Eveand Adam. These games are played on graphs (called arenas). Most of the considered gamesare of infinite duration and their winning condition is expressed in terms of the infinitesequences of edges taken during the play. We invoke results of determinacy (one of theplayers has a winning strategy), as well as of positional determinacy (one of the players hasa strategy that depends only on the last position of the play). R a A R a × A vv a Q = { q , q , q } δ ( q , a ) = ( q ∧ q ) ∨ ( q ∧ q ) δ ( q , a ) = q ∧ q δ ( q , a ) = q ∨ q α ( q , a, q ) = 3 α ( q , a, q ) = 2 α ( q , a, q ) = 6 α ( q , a, q ) = 4 α ( q , a, q ) = 5 α ( q , a, q ) = 3 α ( q , a, q ) = 5 v, q v , q v, q v , q v, q v , q v , q , a ,( q ∧ q ) ∨ ( q ∧ q ) v , q , a , q ∧ q v , q , a , q ∧ q v , q , a , q ∧ q v , q , a , q ∨ q v , q , a , q v , q , a , q v , q , a , q v , q , a , q v , q , a , q v , q , a , q v , q , a , q ( a , ) ( a , ) ( a , ) ( a , ) ( a , ) ( a , ) ( a , ) Figure 1
A one-step arena over a letter a ∈ Σ, obtained as a product of a simple arena R a with the alternating parity automaton A . In this example v is controlled by Eve and v by Adam.The transitions with no label are labelled by (cid:15) . Diamond-shaped positions belong to Eve andsquare-shaped positions belong to Adam. . Boker, D. Kuperberg, K. Lehtinen, M. Skrzypczak 5 q q q q q q q q q q q q q q q q q q q q q q q q Figure 2
The four possible boxes corresponding to Eve’s choices in the one-step arena of Figure 1.(All edges should be labelled with a , which we omit for better readability.) Model-checking games.
To represent the semantics of an alternating automaton A , wetreat the Boolean formulas that appear in the transition conditions of A as games. Moreprecisely, given a letter a ∈ Σ we represent the transition conditions q δ ( q, a ) ∈ B + ( Q ) asthe one-step arena over a (see Figure 1). A play over this arena begins in a state q ∈ Q ; thenthe players go down the formula δ ( q, a ) with Eve resolving disjunctions and Adam resolvingconjunctions; and finally they reach an atom q ∈ Q and the play stops. This means that aplay over the one-step arena over a results in a transition of the form q a −→ q .The language L ( A ) of an alternating automaton A over an alphabet Σ is defined viathe model-checking game , defined for an automaton A and a word w = a a a · · · ∈ Σ ω . A configuration of this game is a state q of A and a position i ∈ ω of w , starting at ( ι, i th round, starting at configuration ( q i , i ), the players play on the one-step arena from q i over a i , resulting in a transition q i a i −→ q i +1 . The next configuration is ( q i +1 , i +1). Theacceptance condition of A becomes the winning condition of this game. A accepts w if Evehas a winning strategy in this game.For technical convenience, we define (in Appendix A) the model-checking game in termsof a synchronised product of the word w (treated as an infinite graph) and the automaton A .Synchronised products turn out to be useful in the analysis of various games presented inthis paper and will be used throughout the technical versions of the proofs, in the appendix. (cid:73) Definition 1.
Given an alternating automaton A , we denote by A the dual automaton: ithas the same alphabet, set of states, and initial state. Its transition conditions δ A ( q, a ) areobtained from those of A by replacing each disjunction ∨ with conjunction ∧ and vice versa.Its acceptance condition is the dual of A s condition. (In parity automata, all priorities areincreased by .) A recognises the complement L ( A ) c of L ( A ) . Boxes.
Another technical concept that we use is that of boxes (see Figure 2), which describeEve’s local strategies for resolving disjunctions within a transition condition. Consider analternating automaton A and a letter a ∈ Σ. Moreover, fix a strategy σ of Eve that resolvesdisjunctions in all the transition conditions δ ( q, a ) for q ∈ Q . Now, the box of A , a , and σ isa subset of Q × Σ × Q and contains a triple ( q, a, q ) iff σ resolves disjunctions of δ ( q, a ) insuch a way that Adam (resolving conjunctions) can reach the atom q . In other words, thisbox contains ( q, a, q ) if there is a play consistent with σ on δ ( q, a ) that reaches the atom q .We use β to denote single boxes and by B A ,a we denote the set of all boxes of A and a , while B A denotes the union S a ∈ Σ B A ,a . We give a more formal definition based on synchronisedproducts in the Appendix, see page 15. (cid:73) Definition 2.
Given a sequence of boxes π = b , b , . . . of an automaton A and a path ρ =( q , a , q ) , ( q , a , q ) , . . . , we say that ρ is a path of π if for every i we have ( q i , a i , q i +1 ) ∈ b i .The sequence π is said to be universally accepting if every path in π is accepting in A . On the Succinctness of Alternating Parity Good-for-Games Automata (Full Version) ∨ start ∧ b, c b ca, b, c b, c b b, c Figure 3
Alternating weak automaton accepting words over { a, b, c } in which a occurs finitelyoften and c occurs infinitely often. Omitted transitions lead to a rejecting sink. Intuitively, a sequence of boxes π as above represents a particular positional strategy σ of Eve in the model-checking game over the word w = a a a . . . In that case, a path of π corresponds to a possible play of this game consistent with σ , and the sequence is universallyaccepting if and only if the strategy is winning. Good-for-games (GFG) nondeterministic automata are automata in which the nondetermin-istic choices can be resolved without looking at the future of the word. For example, consideran automaton that consists of a nondeterministic choice between a component that acceptswords in which a occurs infinitely often and a component that accepts words in which a occursfinitely often. This automaton accepts all words but is not GFG since the nondeterministicchoice of component cannot be resolved without knowing the whole word.To extend this definition to alternating automata, we must look both at its nondeterminismand universality and require that both can be resolved without knowledge of the future. Thefollowing letter games capture this intuition. (cid:73) Definition 3 (Letter games [5]) . Given an alternating automaton A , Eve’s letter gameproceeds at each turn from a state q of A , starting from the initial state of A , as follows:Adam chooses a letter a ,Adam and Eve play on the one-step arena over a from q to a new state q , where Everesolves disjunctions and Adam conjunctions.A play of the letter game thus generates a word w and a path ρ of A on w . Eve wins thisplay if either w / ∈ L ( A ) or ρ is accepting in A .Adam’s letter game is similar, except that Eve chooses letters and Adam wins if either w ∈ L ( A ) or the path ρ is rejecting. A more formal definition is given in Appendix B. (cid:73)
Definition 4 (GFG automata [5]) . An automaton A is ∃ -GFG if Eve wins her letter game;it is ∀ -GFG if Adam wins his letter game. Finally, A is GFG if it is both ∃ - GFG and ∀ - GFG . As shown in [5, Theorem 8], an automaton A is GFG if and only if it is indeed “good forplaying games”, in the sense that its product with every game whose winning condition is L ( A ) preserves the winner of the game. (cid:73) Example 5.
The automaton in Figure 3 accepts the language L of words in which a occursfinitely often and c occurs infinitely often. Here Eve loses her letter game: Adam can play c until Eve takes the transition to the second state, and then play a followed by c ω . Conversely,Eve wins Adam’s letter game: her strategy is to play b , take the transition to the secondstate an keep playing b until Adam takes the transition into the third state, after which . Boker, D. Kuperberg, K. Lehtinen, M. Skrzypczak 7 ∨ start ∧ b, c bca, b, c b, c b Figure 4
Alternating coBüchi ∀ -GFG automaton accepting words over { a, b, c } in which a occursfinitely often and c occurs infinitely often. she plays c once and then b ω . This automaton is neither ∃ -GFG nor ∀ -GFG, and taking itsproduct with games with L as winning condition does not preserve the winner of the game.In contrast, the automaton in Figure 4 is ∀ -GFG but not ∃ -GFG. Indeed, Adam’s winningstrategy in his letter game is to resolve the conjunction from the middle state by alwaysmoving to the right-hand state when Eve plays b . This forces Eve to choose between playing c infinitely many times (in which case, the word is in the language) or letting Adam build arejecting run. Taking its product with one-player games with winning condition L preservesthe winner whenever Eve is the player controlling all positions. However, this is not the casefor one-player games where Adam is the sole player. We show in this section that alternating GFG automata can be more succinct than bothnondeterministic and universal GFG automata. (cid:73)
Lemma 6.
There is a family ( C n ) n ∈ N of alternating GFG { , , } -parity automata of sizelinear in n over a fixed alphabet, such that every nondeterministic GFG parity automatonand universal GFG parity automaton for L ( C n ) is of size Ω( n ) . Proof.
From [16], there is a family ( A n ) n ∈ N of GFG-NCWs with n states over a fixedalphabet Σ, such that every DPW for L n = L ( A n ) is of size 2 Ω( n ) . For every n ∈ N , let B n be the dual of A n , so B n is a UBW accepting L n . We build an APW C n over Σ of size linearin n , by setting its initial state to move to the initial state of A n when reading the letter a ∈ Σ and to the initial state of B n when reading the letter b ∈ Σ. The acceptance conditionof C n is a parity condition with priorities { , , } : accepting transitions of A n are assignedpriority 0, and accepting transitions of B n priority 2. Other transitions have priority 1.The automaton C n is represented below:Observe that L ( C n ) = aL n ∪ bL n , and that C n is GFG: its initial state has only deterministictransitions, and over the A n and B n components, the strategy to resolve the nondeterminismand universality, respectively, follows the strategy to resolve the nondeterminism of A n ,which is guaranteed due to A n ’s GFGness.Consider a GFG UPW E n for L ( C n ), and let q be a state to which E n moves when reading a , according to some strategy that witnesses E n ’s GFGness. Then E qn is a GFG UPW for L n .Its dual is therefore a GFG NPW E n for L n . On the Succinctness of Alternating Parity Good-for-Games Automata (Full Version)
Since A n is a GFG NPW for L n , by [3, Theorem 4] we obtain a DPW for L n of size |A n ||E n | . By choice of L n , this DPW must be of size 2 Ω( n ) , and since A n is of size n , itfollows that E n , and hence E n , must be of size 2 Ω( n ) . By a symmetric argument, every GFGNPW for L ( C n ) must also be of size 2 Ω( n ) . (cid:74) Informally, the language L n above describes a set of threads, of which at least oneeventually satisfies a safety property. Then, the above construction can be understoodas describing a property of reactive systems where, depending on the input, the systemguarantees either that there is a thread that eventually satisfies a safety property, or thatall threads satisfy a liveness (Büchi) property. The GFG alternating automaton can thenbe used to solve in polynomial time games with such languages as winning condition, forexample in the context of synthesis: the product of the game arena and the alternatingautomaton for L n is a parity game with 3 priorities with the same winner as the originalgame. In contrast, a DPW, GFG NPW and GFG UPW for the same language would all beexponentially larger. In order to decide GFGness, it is enough to be able to decide the ∃ -GFG property on theautomaton and its dual. A natural first approach is therefore to study the complexity ofdeciding whether an APW is ∃ -GFG. Yet, we will show that already on finite words, thisproblem is PSpace -hard, while we conjecture that deciding GFGness is in
PTime . (cid:73) Lemma 7.
Deciding whether an AFA is ∃ - GFG is PSpace -hard.
Proof.
We reduce from NFA universality: starting from an NFA A , we build an AFA B based on the dual of A , with an additional non-GFG choice to be resolved by Eve. This AFA B is ∃ -GFG if and only if L ( B ) = ∅ , which happens if and only if L ( A ) = Σ ∗ . We cruciallyuse the fact that B is not necessarily ∀ -GFG.Let A be an NFA over an alphabet Σ = { a, b } and ¯ A its dual. We want to check whether L ( A ) = Σ ∗ . We build an AFA B , as depicted below, by first making Eve guess the secondletter. If her guess is wrong, the automaton proceeds to a rejecting sink state ⊥ . Otherwise,it proceeds to the initial state of ¯ A . The size of B is linear in the size of A .If L ( ¯ A ) = ∅ , then L ( B ) = ∅ , so B is trivially ∃ -GFG. However, if there is some u ∈ L ( ¯ A ),then Adam has a winning strategy in Eve’s letter game on B . This strategy consistsof playing a , then playing the letter that brings Eve to ⊥ , and finally playing u . Theresulting word is in L ( B ) = Σ L ( A ), so this witnesses that B is not ∃ -GFG. We obtain that L ( A ) = Σ ∗ ⇔ L ( ¯ A ) = ∅ ⇔ B is ∃ -GFG, which is the wanted reduction. (cid:74) For Büchi automata, and so in particular for finite words, we can give an
Exptime algorithm for this problem. (cid:73)
Lemma 8.
Deciding whether an ABW is ∃ - GFG is in
Exptime . . Boker, D. Kuperberg, K. Lehtinen, M. Skrzypczak 9 Proof.
It is shown in [5, Lemma 23] that removing alternation from an ABW A using thebreakpoint construction [20] yields an NBW B such that if A is ∃ -GFG then B is GFG.Moreover, the converse also holds: if B is GFG then A is ∃ -GFG, since playing Eve’s lettergame in B is more difficult for Eve than playing it in A . This means that starting from anABW A , we can build an exponential size NBW B via breakpoint construction, and testwhether B is GFG via the algorithm from [2], in time polynomial with respect to B . Overall,this yields an Exptime algorithm deciding whether A is ∃ -GFG. (cid:74) In this section we provide a procedure that, given an alternating GFG parity automaton,produces an equivalent deterministic parity automaton with single-exponentially many states.To do so, we first provide an alternation-removal procedure that preserves GFG status.Then, we apply this procedure to both the input automaton and its complement and usethe GFG strategies in these two automata to determinise the input automaton. Our proofs,in Appendix C, rely on some analysis of when GFG strategies can use the history of the word,rather than the full history of the play (which also includes the choices of how to resolve thenondeterminism and universality), and on the memoryless determinacy of parity games.Our method for going from alternating to nondeterministic automata is similar to thatof Dax and Klaedtke [10]: they take a nondeterministic automaton that recognises theuniversally-accepting words in ( B A ) ω and add nondeterminism that upon reading a letter a ∈ Σ chooses a box in B A over a. Yet in our approach, in order to guarantee that theoutcome preserves GFGness, the intermediate automaton is deterministic. (cid:73) Theorem 9.
Consider an alternating parity automaton A with n states and index k . Thereexists a nondeterministic parity automaton box ( A ) with O ( nk log nk ) states that is equivalentto A such that if A is GFG then box ( A ) is also GFG. In Section 5, where we discuss decision procedures, we will show that box ( A ) is GFG exactly when A is ∃ -GFG. For now, the rest of this section is devoted to the proof of Theorem 9,of which a detailed version can be found in Appendix C.2. (cid:73) Lemma 10.
Consider an alternating parity automaton A with n states and index k . Thenthere exists a deterministic parity automaton B with O ( nk log nk ) states over the alphabet B A that recognises the set of universally-accepting words for A . If A is a Büchi automaton, then B can also be taken as Büchi, and in general the parity index of the automaton B is linear inthe number of transitions of A . Proof sketch.
We first construct a nondeterministic parity (resp. coBüchi) automaton overthe alphabet B A that recognises the complement of the set of universally-accepting wordsfor A . This automaton is easy to build: it guesses a path that is not accepting, and hasthe dual acceptance condition to A . We then obtain the automaton B by determinising andcomplementing this automaton. (cid:74) We now build the automaton box ( A ) of Theorem 9. It is the same as the automaton B of Lemma 10, except that the alphabet is Σ and the transition function is defined as follows:For every state p of B and a ∈ Σ, we have δ box ( A ) ( p, a ) := S β ∈ B ( A ,a ) δ B ( p, β ).In other words, the automaton box ( A ) reads a letter a , nondeterministically guesses a box β ∈ B A ,a , and follows the transition of B over β . Thus, the runs of box ( A ) over a word w = w w w · · · ∈ Σ ω are in bijection with sequences of boxes ( β i ) i ∈ N such that β i ∈ B A ,w i for all i ∈ N .Fix an infinite word w ∈ Σ ω . Our aim is to prove that w ∈ L ( A ) ⇔ w ∈ L ( box ( A )). (cid:73) Lemma 11.
There exists a bijection between positional strategies of Eve in the acceptancegame of A over w and runs of box ( A ) over w . Moreover, a strategy is winning if and only ifthe corresponding run is accepting. Thus L ( A ) = L ( box ( A )) . (cid:73) Remark 12.
The above alternation-removal procedure also extends to alternating Rabinautomata but fails for alternating Streett automata A : since Streett games are not positionallydetermined for Eve, the acceptance game of A over a word w is not positionally determinedfor Eve. (cid:73) Lemma 13.
For an alternating ∃ - GFG parity automaton A , the automaton box ( A ) is GFG. Intuitively, this is because the construction of box ( A ) preserves the nondeterminism of A . The aim of this section is to prove the following determinisation theorem; see Appendix C.3for a detailed proof. (cid:73)
Theorem 14. If A is an alternating parity GFG automaton then there exists a deterministicparity automaton D that recognises the same language and has size at most exponential inthe size of A . Moreover, the parity index of D is the same as that of A . (cid:73) Remark 15.
Theorem 9 and [3, Theorem 4], which uses an NRW-GFG and its complementNRW-GFG to obtain a DRW, together give an exponential deterministic parity automatonfor L ( A ). However, the index of A might not be preserved. On the other hand, from [5,Theorem 19] we know that there exists a deterministic parity automaton equivalent to A with the same index, but it might have more than exponentially many states. Here we areable to guarantee both the preservation of the index and an exponential upper bound on thesize of the deterministic automaton.Observe that Theorem 9 can be applied both to A and its dual. Therefore, we can fixa pair of nondeterministic GFG parity automata box ( A ) and box ( ¯ A ) that recognise L ( A )and L ( A ) c respectively and are both of size exponential in A . We use the automata A , box ( A ), and box ( ¯ A ) to construct two auxiliary games G ( A ) and G ( A ) .The game G ( A ) proceeds from a configuration consisting of a pair ( p, q ) of states from box ( ¯ A ) and A respectively, starting from their initial states, as follows:Adam chooses a letter a ∈ Σ;Eve chooses a transition p a −→ p in box ( ¯ A );Eve and Adam play on the one-step arena over a from q to a new state q .A play in G ( A ) consists of a run ρ in box ( ¯ A ) and a path ρ in A . It is winning for Eve ifeither ρ is accepting in box ( ¯ A ) (in which case w / ∈ L ( A )), or ρ is accepting in A .If A is ∃ -GFG and box ( ¯ A ) is GFG, Eve has a winning strategy in G ( A ) consisting ofbuilding a run in box ( ¯ A ) using her GFG strategy in box ( ¯ A ) and a path in A using her ∃ -GFG strategy in A . This guarantees that if w ∈ L ( A ) then the path in A is accepting,and otherwise the run in box ( ¯ A ) is accepting. . Boker, D. Kuperberg, K. Lehtinen, M. Skrzypczak 11 We then argue that as the winning condition of G ( A ) is a Rabin condition, Eve also hasa winning strategy that is positional in A , that is, which only depends on the history of theword and the current position. (Interestingly, the question of whether Eve can resolve thenondeterminism in a class of alternating GFG automata with only the knowledge of the wordread so far does not tightly correspond to whether the acceptance condition of this class ismemoryless. For example, it does hold for the generalised-Büchi condition, though it is notmemoryless.) See Appendix C for details. (cid:73) Remark 16.
There is some magic here: both the GFG strategies of Eve in A and in box ( ¯ A )may require exponential memory, yet, when she needs to satisfy the disjunction of the twoconditions, no more memory is needed. In a sense, the states of A provide the memory for box ( ¯ A ) and the states of box ( ¯ A ) provide the memory for A .The game G ( A ) is similar, except that Adam is given control of box ( A ) and Eve is incharge of letters. This time Adam wins a play, consisting of a run of box ( A ) and a path in A , if either the path of A is rejecting or the run of box ( A ) is accepting.Accordingly, if A is GFG, then he can win by using the ∃ -GFG strategy in box ( A )and the ∀ -GFG strategy in A . Then if w ∈ L ( A ), the run in box ( A ) is accepting, and oth-erwise the path of A is rejecting. As before, he also has a positional winning strategy in G ( A ).We are now ready to build the deterministic automaton from a GFG APW A , usingpositional winning strategies σ and τ for Eve and Adam in G ( A ) and G ( A ), respectively.Let D be the automaton with states of the form ( q, p , p ), with q a state of A , p a stateof box ( A ) and p a state of box ( ¯ A ). A transition of D over a moves to ( q , p , p ) such thatmoving from ( q, p ) to ( q , p ) is consistent with τ ; and moving from ( q, p ) to ( q , p ) isconsistent with σ . The acceptance condition of D is inherited from A . (cid:73) Lemma 17.
For a GFG APW A and D built as above, L ( A ) = L ( D ) . (cid:73) Remark 18.
To extend this construction to an alternating GFG Rabin automaton A , wewould need to remove alternations from both A and its dual while preserving GFGness.However, the dual is a Streett automaton, for which we cannot invoke positional determinacy. We use the development of the last section to show that deciding whether an APW is GFGis in
Exptime . This matches the best known upper bound for the same problem on NPW.The main result of this section is the following theorem; its proof is in Appendix D.1. (cid:73)
Theorem 19.
There exists an
Exptime algorithm that takes as input an alternating parityautomaton A and decides whether A is GFG. The idea is to construct the (exponential size) NPWs box ( A ) and box ( ¯ A ) for L ( A ) and L ( A ) c respectively, which are GFG if and only if A is ∃ -GFG and ∀ -GFG respectively.Then, it remains to check whether both are indeed GFG. Since we don’t have a polynomialprocedure to check this, instead, we will build a game which Eve wins if and only if both areindeed GFG, and which we can solve in exponential time with respect to the size of A .First, we observe the following reciprocal of Lemma 13. (cid:73) Lemma 20. If box ( A ) is GFG then A is ∃ - GFG . Proof.
Assume that box ( A ) is GFG and consider a strategy witnessing this. Such a strategycan be easily turned into a function σ : Σ + → B A that, given a word w ∈ L ( A ) produces a universally accepting word of boxes of A . Now, due to the definition of a box, each suchbox defines a positional strategy of Eve in the respective one-step game. This allows us toconstruct a winning strategy of Eve in the letter game over A . (cid:74) Thus, A is GFG if and only if both box ( A ) and box ( ¯ A ) are GFG. To decide this, weconsider a game G where Adam plays letters and Eve produces runs of the automata box ( A ) and box ( ¯ A ) in parallel. The winning condition of G requires that at least one ofthe constructed runs must be accepting.Now, each sequence of letters given by Adam belongs either to the language of box ( A ) orto box ( ¯ A ) and therefore, a winning strategy of Eve in G must comprise of two strategieswitnessing GFGness of both box ( A ) and box ( ¯ A ). Dually, if both box ( A ) and box ( ¯ A ) areGFG then Eve wins G by playing the two strategies in parallel.It remains to show that G is solvable in Exptime . Its winning condition is a disjunction ofparity conditions, with index linear in the number of transitions of A . This winning conditionis recognised by a deterministic parity automaton of exponential size with polynomial index.To solve G , we take its product with this deterministic automaton that recognises its winningcondition, and solve the resulting parity game with an algorithm that is polynomial in thesize of the game whenever, like here, the number of priorities is logarithmic in the size of thegame, for instance [6]. Details of this construction and its complexity are in Appendix D.1. The results obtained in this work shed new light on where alternating GFG automataresemble nondeterministic ones, and where they differ. Overall, our results show thatallowing GFG alternations add succinctness without significantly increasing the complexityof determinisation nor decision procedures.In particular, we show that alternating parity GFG automata can be exponentially moresuccinct than any equivalent nondeterministic GFG automata, yet this succinctness does notbecome double exponential when compared to deterministic automata, answering a questionfrom [5]. Some further succinctness problems are left open here, such as the possibility of adoubly exponential gap between alternating GFG automata of stronger acceptance conditionsand deterministic ones, as well as between ∃ -GFG parity automata and deterministic ones.We also show that the interplay between the two players can be used to decide whetheran automaton is GFG without deciding ∃ -GFG and ∀ -GFG separately, yielding an Exptime algorithm. This matches the current algorithms for deciding GFGness on non-deterministicautomata. Bagnol and Kuperberg conjectured that GFGness is
PTime decidable for non-deterministic parity automata of fixed index [2]; we extend this conjecture to alternatingautomata.It then becomes interesting to ask how to build an alternating automaton GFG. Indeed,Henzinger and Piterman [11] proposed a transformation of nondeterministic automata intoGFG automata, which, despite in some cases leading to a deterministic automaton, is,conceptually, a much simpler procedure than determinisation. Indeed, in many examples ofnon-GFG automata, adding transitions suffices to obtain a GFG one. We leave finding sucha procedure for alternating automata as future work.
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Appendix
A Appendix of Section 2
In this section of the appendix we provide the remaining technical definitions from Section 2that are used in the proofs.
Games.
A Σ -arena is a directed (finite or infinite) graph with nodes (positions) split into E -labelled positions of Eve and A -labelled positions of Adam, where the edges (transitions)are labelled by elements of Σ t { (cid:15) } . The role of (cid:15) is to mark edges that have no influence onthe winner of a play, e.g., edges allowing players to resolve some Boolean formula.We represent such an arena as R = ( V, X, V E , V A ), where V is its set of positions; X ⊆ V × (cid:0) Σ t { (cid:15) } (cid:1) × V its transitions; V E ⊆ V the E -positions; and V A = V \ V E the A -positions.Notice that the definition allows more than one transition between a pair of positions(such transitions needs to have distinct labels). We will require that each infinite pathcontains infinitely many Σ-labelled transitions. An arena might be rooted at an initialposition v ι ∈ V . We say that a position v is terminal if there is no outgoing transition from v (i.e. no element of X of the form ( v, a, v )). If we don’t say that an arena is partial then it isassumed that there are no terminal positions.If R is a (partial) Σ-arena and V ⊆ V is a set of positions, then R (cid:22) V is the sub-arena of R defined as the restriction of R to the positions in V , namely for P ∈ { E, A } , the P -positionsof R (cid:22) V are V P := V P ∩ V , and its transitions are X := X ∩ ( V × (Σ ∪ { (cid:15) } ) × V ). We saythat two (partial) Σ-arenas R = ( V, X, V E , V A ) and R = ( V , X , V E , V A ) are isomorphic ifthere exists a bijection i : V → V that preserves the membership in V P / V P , for P ∈ { E, A } ,and sets of transitions X / X .A partial play in R is a path in R , i.e., an element π = v e v e . . . of V · (cid:0) X · V ) ∗ ∪ (cid:0) V · X (cid:1) ω ,where for every i we have e i = ( v i , a i , v i +1 ). Such a partial play is said to begin in v . A partialplay is a play if either it is infinite or the last position v i is terminal.A game is a Σ-arena together with a winning condition W ⊆ Σ ω . An infinite play π issaid to be winning for Eve in the game if the sequence of Σ-labels ( a i ) i ∈ N of the transitionsalong π form a word in W . Else π is winning for Adam. Games with some class X of winningconditions (e.g., the parity condition) are called X games (e.g., parity games).A strategy for Eve (resp. Adam) is a function τ : V · (cid:0) X · V (cid:1) ∗ → X that maps a history v e v . . . e i − v i , i.e. a finite prefix of a play in R , to a transition e i whenever v i belongsto V E (resp. to V A ). A partial play v e v e . . . agrees with a strategy τ for Eve (Adam)if whenever v i ∈ V E (resp. in V A ), we have e i = τ ( v e v . . . e i − v i ). A strategy for Eve(Adam) is winning from a position v ∈ V if all plays beginning in v that agree with it arewinning for Eve (Adam). We say that a player wins the game from a position v ∈ V if theyhave a winning strategy from v . If the game is rooted at v ι , we say that a player wins thegame if they win from v ι .A strategy is positional if its value depends only on the last position, i.e., τ ( v e · · · e i − v i )depends only on v i . In that case the strategy of a player P can be represented as a function τ : V P → X .We also define the notion of strategy with memory M for player P . This is a tuple( σ, M, m , upd ) where M is a set of memory states ; m is an initial memory state ; upd : M × X → M is an update function , and σ : M × V P → X is a strategy deciding which moveshould be played, depending only on the current memory state and on the current position. . Boker, D. Kuperberg, K. Lehtinen, M. Skrzypczak 15 Along a play, the memory starts with m , and is updated along every transition according to upd . The general notion of strategy corresponds to M = ( V · X ) ∗ , and positional strategiescorrespond to M being a singleton. A player has a finite-memory winning strategy if thereexists a winning strategy using a finite memory set M . (cid:73) Proposition 21.
Let G and G be two Σ -games with the same winning condition, suchthat the unfoldings of G and G are isomorphic. Then Eve has a winning strategy in G ifand only if she has a winning strategy in G . (cid:73) Proposition 22 ([13]) . Rabin games are positionally determined for Eve. (If Eve hasa winning strategy then she has a positional winning strategy.) (cid:73)
Definition 23 (Synchronised product) . The synchronised product R × A of a (partial) Σ -arena R = ( V, X, V E , V A ) and an alternating automaton A = (Σ , Q, ι, δ, α ) with a set oftransitions T and labelling α : T → Γ is a (partial) Σ × Γ -arena defined as follows. Its set ofpositions is ( V × Q ) ∪ ( V × Q × Σ × b δ ) , and its transitions are defined by: (cid:10) ( v, q ) , (cid:15), ( v , q, a, δ ( q, a )) (cid:11) for ( v, q ) ∈ V × Q and h v, a, v i ∈ X ; (cid:10) ( v, q ) , (cid:15), ( v , q ) (cid:11) for ( v, q ) ∈ V × Q and h v, (cid:15), v i ∈ X ; (cid:10) ( v, q, a, b ) , (cid:15), ( v, q, a, b i ) (cid:11) for ( v, q, a, b ) ∈ V × Q × Σ × b δ with b = b ∨ b or b = b ∧ b and i = 1 , ; (cid:10) ( v, q, a, q ) , (cid:0) a, α ( q, a, q ) (cid:1) , ( v, q ) (cid:11) for ( v, q, a, q ) ∈ V × Q × Σ × b δ with q ∈ Q .The positions belonging to Eve are of the form ( v, q ) where v ∈ V E and of the form ( v, q, a, b ∨ b ) . The remaining ones belong to Adam. If R has an initial position v ι then theinitial position of the product is ( v ι , ι ) . We implicitly assume that the arena only contains vertices that are reachable from V × Q .(They need not be reachable from an initial position of R and an initial state of A , but fromsome position of R and state of A .)We will sometimes consider longer products, like ( R × A ) × A , where R is a Σ-arena andboth automata A and A are over the alphabet Σ. Assume that A and A have transitionslabelled in sets Γ and Γ respectively. Notice that in that case the arena R × A is formallya Σ × Γ -arena. Thus, to make the above formula precise, we treat the automaton A asan automaton over the alphabet Σ × Γ and assume that it ignores the second component ofthe letters read. One-step arenas.
For a letter a ∈ Σ, we denote by R a a partial Σ-arena consisting of twovertices v and v (it does not matter which of the players controls them), and one transition( v, a, v ). Then, for an automaton A = (Σ , Q, ι, δ, α ), the product R a × A is a partial arena,in which the players should resolve their choices in the formulas δ ( q, a ) for all the possiblestates q ∈ Q . We call it the one-step arena of A over a . Such an arena contains one positionof the form ( v, q ) for each state q ∈ Q ; a set of non-terminal positions of the form ( v , q, a, ψ )for some q ∈ Q and ψ ∈ b δ ; and one terminal position of the form ( v , q ) for each state q ∈ Q .(See Figure 1.) Boxes.
In the later exposition, we will be interested in the combinatorial structure ofpossible strategies of Eve over one-step arenas R a × A . For a positional strategy σ of Eve ina game of the form R a × A , we define the box of A , a , and σ , denoted by β ( A , a, σ ), as therelation that is a subset of Q × Σ × Q and contains a triple ( q, a, q ) iff there exists a play in R a × A that is consistent with σ , starting in ( v, q ) and ending in ( v , q ). We further definefor every a ∈ Σ, the set B ( A ,a ) = { β ( A , a, σ ) | σ is a positional strategy of Eve } . Finally, let B A := S a ∈ Σ B ( A ,a ) . Notice that | B A | ≤ | Q × Σ × Q | . When speaking of an arbitrary box, wemean any non-empty relation β ⊆ Q × Σ × Q where all the letters a appearing on the middlecomponent are equal.Figure 2 represents B ( A ,a ) for the automaton A of Figure 1: Since there are two bin-ary-choice positions of Eve in the corresponding one-step arena, there are four distinctpositional strategies of Eve, which give the four possible boxes. They correspond to Evechoosing respectively LL, LR, RL, RR, where L stands for a left choice and R for a rightchoice in each of her two binary-choice positions. (cid:73) Proposition 24.
Consider a letter a and an automaton A with states Q and transitionfunction δ . Then there is a bijection between B ( A ,a ) and the positional strategies of Eve inthe one-step arena of A and a . Acceptance of a word by an automaton.
We define the acceptance directly in terms of themodel-checking (acceptance/membership) game, which happens to be exactly the productof the automaton with a path-like arena describing the input word. More precisely, givena word w ∈ Σ ω , the model-checking game is defined as the product R w × A , where thearena R w consists of an infinite path ω , of which all positions belong to Eve (although itdoes not matter); the transitions are of the form h i, w i , i +1 i ; the initial position is 0; and thewinning condition is based on the winning condition of A (the Σ-component of the labelsis ignored). We say that A accepts w if Eve has a winning strategy in the model-checkinggame R w × A . The language of an automaton A , denoted by L ( A ), is the set of words thatit accepts (recognises).Notice that for each i ∈ N , the sub-arena of R w × A with positions in (cid:0) { i } × Q (cid:1) ∪ (cid:0) { i +1 } × Q × Σ × B + ( Q ) (cid:1) ∪ (cid:0) { i +1 } × Q (cid:1) is isomorphic to the one-step arena of A over w i . B Appendix of Section 3 (cid:73)
Definition 25 (A formalisation of Definition 3) . Let R A, Σ be the Σ -arena consisting of asingle position v that belongs to Adam and the set of transitions X of the form ( v, a, v ) foreach letter a ∈ Σ (see Figure 5). The arena R E, Σ is the same except that v belongs to Eve.Notice that the products R A, Σ × A and R E, Σ × A are both labelled by Σ × Γ , where Σ isthe alphabet of A and Γ is A ’s labelling, on top of which its acceptance condition is defined.Thus, the winning condition of games defined on these arenas can depend on a sequence oflabels of the form ( a i , γ i ) i ∈ N . Then, Eve’s letter game is played over R A, Σ × A , where Evewins if: ( a i ) i ∈ N / ∈ L ( A ) or the sequence ( γ i ) i ∈ N satisfies the acceptance condition of A .Dually, Adam’s letter game is played over R E, Σ × A , where Adam wins if: ( a i ) i ∈ N ∈ L ( A ) or the sequence ( γ i ) i ∈ N violates the acceptance condition of A . C Appendix of Section 4
This section provides the technical details of the determinisation procedure in Section 4. Westart with some technical analysis of the types of histories needed to win letter games. . Boker, D. Kuperberg, K. Lehtinen, M. Skrzypczak 17
C.1 Good for Games Automata: Required Histories
We begin by considering an expanded letter game. This will allow us to use a form ofpositional determinacy in letter games. R A, Σ R ∗ A, Σ va b (cid:15)a b aa ab ba bb a b a b a b ... ... ... ... ... Figure 5
The arenas R A, Σ and R ∗ A, Σ , allowing Adam to choose an arbitrary sequence of letters.We define Eve’s letter game for an automaton A over the product of R A, Σ × A , and her expandedletter game over R ∗ A, Σ × A ; She wins a play if the word generated by Adam is not in L ( A ) or thepath generated by her (resolving A ’s nondeterminism) and by Adam (resolving A ’s universality)satisfies A ’s acceptance condition. Expanded letter games.
The definition of the letter games (Definition 3) has the importantadvantage of being defined over a finite-arena. Yet, as a result, these games generally do notallow for positional determinacy.We provide below an expanded variant of the letter game that will have same unfoldingas the original one, while being defined over an infinite arena. This will allow Eve to havepositional determinacy in these games for Rabin automata.Let R ∗ A, Σ be the Σ-arena with the set of positions V = Σ ∗ , all belonging to Adam, andthe set of transitions X of the form h w, a, w · a i for each word w ∈ Σ ∗ and letter a ∈ Σ (seeFigure 5). The initial position of this arena is (cid:15) . The arena R ∗ E, Σ is the same, except thatall the positions belong to Eve. We define Eve’s expanded letter game over R ∗ A, Σ × A and Adam’s expanded letter game over R ∗ E, Σ × A with the same winning conditions as in their(non expanded) variants. The following follows directly from Proposition 21. (cid:73) Proposition 26.
For every automaton A , the expanded letter games for A have the samewinners as the (standard) letter games for A . History Requirement.
Although R ∗ A, Σ and R ∗ E, Σ are trees, the arenas of the expandedletter games are directed acyclic graphs as there can exist two distinct paths from the initialposition to a given position ( w, q ). Thus, a priori, a winning strategy of a player of such agame might need some history of a play. However, as expressed by the following theorem, itis not the case. (cid:73) Theorem 27. If A is a Rabin (or parity) automaton then Eve’s expanded letter game ispositionally determined for Eve. Proof.
We will show that the winning condition of this game can be represented as a Rabincondition and invoke Proposition 22. Let D be a deterministic parity automaton recognisingthe complement of the language L ( A ), for a Rabin automaton A over the alphabet Σ. Let Ω : Q D → N be the priority assignment of D (without loss of generality we can assume thatthe states of D bear priorities).Consider the arena R A, Σ that is derived from the arena R ∗ A, Σ by adding to transitionspriorities according to the deterministic of runs of D , that is, by changing the labelling ofevery transition h w, a, w · a i to h w, (Ω D ( q ) , a ) , w · a i , where a ∈ Σ and q is the state of D reached after reading the word w from the initial state of D .Consider the product R A, Σ × A , in which for A s transitions we ignore these additionallabels. The labels of that product are now of the form ( ‘, a, γ ), where ‘ ∈ N is a priority of D , a ∈ Σ, and γ ∈ Γ is a label of A . Notice that when one forgets about the first coordinate ofthe label, the game is equal to R ∗ A, Σ × A . Moreover, given a sequence of labels ( ‘ i , a i , γ i ) i ∈N ,by the choice of D , we know that ( a i ) i ∈N / ∈ L ( A ) if and only if the sequence ( ‘ i ) i ∈N satisfiesthe parity condition.Define the game G over R A, Σ × A , in which Eve wins a play labelled by ( ‘ i , a i , γ i ) i ∈ N if( ‘ i ) i ∈ N satisfies the parity condition of D or ( γ i ) i ∈ N satisfies the Rabin condition of A .Notice that both disjuncts above can be written as Rabin conditions and therefore G ispositionally determined for Eve. Moreover, the choice of D guarantees that the new winningcondition is equivalent to Eve’s condition in her expanded letter game on A —the same playsare winning for Eve in G and her expanded letter game on A . Since the structure of the gameis also preserved, it means that Eve’s expanded letter game on A is positionally determinedfor Eve. (cid:74)(cid:73) Remark 28.
Dually, Adam’s expanded letter game for a Streett automaton is positionallydetermined for Adam.As a consequence of Theorem 27, for alternating ∃ -GFG Rabin automata, a strategy forEve to resolve the nondeterminism may ignore the history of the play, and only consider thehistory of the word read, as is the case for nondeterministic GFG automata.We will now argue that Eve’s positional strategy σ in the expanded letter game on analternating automaton A can be represented as a function σ : Σ + → B A that assigns to eachword wa a box β wa ∈ B ( A ,a ) . Indeed, let ( w · a ) ∈ Σ + and let V wa be the set of positions of R ∗ A, Σ × A of the form ( w, q ), ( w · a, q, a, b ), or ( w · a, q ) for q ∈ Q and b ∈ B + ( Q ). Observethat the partial arena of R ∗ A, Σ × A restricted to V wa is isomorphic to the one-step arena R a × A . Thus, σ provides a positional strategy over this arena, which by Proposition 24can be encoded as a box β wa . More formally, let β wa contain ( q, a, q ), if there is a playconsistent with σ that visits both the positions ( w, q ) and then ( wa, q ).Then, in the next lemma we show that if σ is also winning, then the sequences of boxes β wa only has accepting paths. (cid:73) Definition 29.
Consider an automaton A with states Q and initial state ι , and an infiniteword u = β , β , . . . ∈ ( B A ) ω . We say that a sequence of transitions ρ = ( q i , a i , q i +1 ) i ∈ N is a path of u if q = ι and for every i ∈ N , we have ( q i , a i q i +1 ) ∈ β i . The word u is universallyaccepting for A if each of its paths satisfies the acceptance condition of A . (cid:73) Lemma 30.
Given an alternating ∃ - GFG
Rabin automaton A , there is a positional strategy σ in her expanded letter game on A such that for every word w ∈ L ( A ) the sequence of boxes u = β , β , . . . ∈ ( B A ) ω defined as β i = σ ( w (cid:22) i +1 ) is universally accepting for A . Proof.
Consider words w and u as above. Let ρ = ( q i , a i , q i +1 ) i ∈ N be a path of u . Since thestrategy σ is positional, the definition of σ ( w (cid:22) i +1 ) implies that there exists a single play of . Boker, D. Kuperberg, K. Lehtinen, M. Skrzypczak 19 q q q q q q ∧∨ t t t t Acceptance condition:(Finitely often t or Infinitely often t ) and(Finitely often t or Infinitely often t ) Figure 6
A GFG ASW over a singleton alphabet, for which Eve’s ∃ -GFG strategy cannot onlyremember the prefix of the word read so far, but also some history about the visited states. the expanded letter game that visits all the positions of the form ( w (cid:22) i , q i ) for i ∈ N . Since w ∈ L ( A ), the winning condition of the expanded letter game guarantees that the path ρ must be accepting. (cid:74) Observe that the above arguments do not hold for alternating GFG Streett automata:Since Streett games are not positionally determined for Eve, Eve’s expanded letter game fora Streett automaton is not positionally determined for Eve (an analogous of Theorem 27does not hold). Furthermore, we provide in Figure 6 an example of an alternating GFGStreett automaton, in which Eve cannot resolve her nondeterminism only according to thehistory of the word read. (cid:73)
Proposition 31.
Consider an ∃ - GFG alternating Streett automaton A with transitionconditions in DNF. Then Eve might not have a strategy σ : Σ + → B A satisfying Lemma 30. Proof.
Consider the ASW depicted in Figure 6. It is ∃ -GFG, as witnessed by the strategythat chooses the transition t in q if the last visited state was q and t otherwise. Yet,there is no strategy that only remembers the word read so far, as this only gives the lengthof the word, and cannot help in determining whether the path visited q or q . (cid:74) C.2 Alternation Removal in GFG Parity Automata
This section presents the proof of the following theorem: (cid:73)
Theorem 9.
Consider an alternating parity automaton A with n states and index k . Thereexists a nondeterministic parity automaton box ( A ) with O ( nk log nk ) states that is equivalentto A such that if A is GFG then box ( A ) is also GFG. (cid:73) Lemma 10.
Consider an alternating parity automaton A with n states and index k . Thenthere exists a deterministic parity automaton B with O ( nk log nk ) states over the alphabet B A that recognises the set of universally-accepting words for A . If A is a Büchi automaton, then B can also be taken as Büchi, and in general the parity index of the automaton B is linear inthe number of transitions of A . Proof.
Notice that it is easy to construct a nondeterministic parity automaton S over thealphabet B A that recognises the complement of the set of universally-accepting words for A —it is enough to guess a path that is not accepting, and have the acceptance conditionthat is the dual of A ’s condition. If A is a Büchi automaton, then S is a coBüchi one.Formally, for an alternating parity (resp. Büchi) automaton A = (Σ , Q, ι, δ, α ), we define the nondeterministic parity (resp. coBüchi) automaton S = ( B A , Q, ι, δ S , α ), where α is thedual of α and δ S is defined as follows. For every states q, q ∈ Q and box β ∈ B A , we have q ∈ δ S ( q, β ) iff q a −→ q ∈ β for some a .Now, one can translate S to an equivalent deterministic parity automaton B with2 O ( nk log nk ) states [21] and then complement the acceptance condition of B , getting therequired automaton B .Since nondeterministic coBüchi automata can be determinised into deterministic coBüchiautomata, if A is Büchi, so is B . In general, the parity index of the automaton B is linear inthe number of transitions of A . (cid:74) We now proceed to the construction of the automaton box ( A ) of Theorem 9. It is thesame as the automaton B of Lemma 10, except that the alphabet is Σ and the transitionfunction is defined as follows: For every state p of box ( A ) and a ∈ Σ, we have δ box ( A ) ( p, a ) := ∪ β ∈ B ( A ,a ) δ B ( p, β ).In other words, the automaton box ( A ) reads a letter a , nondeterministically guesses a box β ∈ B A ,a , and follows the transition of B over β . Thus, the runs of box ( A ) over a word w ∈ Σ ω are in bijection between sequences of boxes ( β i ) i ∈ N such that β i ∈ B A ,w i for i ∈ N .Fix an infinite word w ∈ Σ ω . Our aim is to prove that w ∈ L ( A ) ⇔ w ∈ L ( box ( A )). (cid:73) Lemma 11.
There exists a bijection between positional strategies of Eve in the acceptancegame of A over w and runs of box ( A ) over w . Moreover, a strategy is winning if and only ifthe corresponding run is accepting. Thus L ( A ) = L ( box ( A )) . Proof.
Consider a run of box ( A ) over w , and observe that it corresponds to a sequenceof boxes β , . . . . Notice that each box β i corresponds to Eve’s choices in A over w i , andtherefore provides a positional strategy for Eve in the one-step arena R w i × A . The sequenceof these choices provides a positional strategy for Eve in R w × A .Dually, given a positional strategy for Eve in R w × A , one can extract a sequence ofstrategies for Eve in the one-step arenas R w i × A , and each of them corresponds to a box β i . Proposition 24 shows that each path in β , . . . corresponds to a play consistent with theconstructed strategy and vice versa: each play gives rise to a path.Now, a run is accepting if and only if the sequence of boxes is universally accepting, whichmeans exactly that all the plays consistent with the corresponding strategy are winning. (cid:74) We now show that the automaton box ( A ) is also GFG. (cid:73) Lemma 13.
For an alternating ∃ - GFG parity automaton A , the automaton box ( A ) is GFG. Proof.
Let σ be a positional winning strategy for Eve in her expanded letter game for A (overthe arena R ∗ A, Σ × A ). The proof is based on the construction of the function σ : Σ + → B A ,see the paragraph before Definition 29.Consider the following way of resolving the nondeterminism of box ( A ): after reading w ∈ Σ ∗ , when the next letter a ∈ Σ is provided, the automaton moves to the state δ B ( p, β wa )where β wa = σ ( wa ). Consider an infinite word w ∈ L ( A ) and let β , . . . be the sequence ofboxes used to construct the run of box ( A ) over w . Lemma 30 implies that this sequence isuniversally accepting and therefore, the constructed run of B must also be accepting. (cid:74) C.3 Single-Exponential Determinisation of Alternating Parity GFGAutomata
The aim of this section is to prove the following determinisation theorem. . Boker, D. Kuperberg, K. Lehtinen, M. Skrzypczak 21 (cid:73)
Theorem 14. If A is an alternating parity GFG automaton then there exists a deterministicparity automaton D that recognises the same language and has size at most exponential inthe size of A . Moreover, the parity index of D is the same as that of A . First consider the synchronised product R A, Σ × box ( ¯ A ), which is a game with labels ofthe form Σ × Γ box ( ¯ A ) , where Γ box ( ¯ A ) is the parity condition of box ( ¯ A ). Now, we can treat theautomaton A as an automaton over the alphabet Σ × Γ box ( ¯ A ) that just ignores the secondcomponent of the given letter. Thus, we can define a game G ( A ) = (cid:0) R A, Σ × box ( ¯ A ) (cid:1) × A .Notice that G ( A ) is naturally divided into rounds, between two consecutive positions ofthe form ( v, p, q ), where v is the unique position of R A, Σ , p is a state of box ( ¯ A ), and q isa state of A . Such a round, starting in ( v, p, q ), consists of first Adam choosing a letter a ;then Eve resolving nondeterminism of box ( ¯ A ) from p over a ; and then both players playingthe game corresponding to the transition condition δ ( q, a ) of A .Let the winning condition of G ( A ) say that either the sequence of transitions of box ( ¯ A ) isaccepting or the sequence of transitions of A is accepting. Since A is ∃ -GFG and box ( ¯ A ) isGFG, we know that Eve has a winning strategy in G ( A ): she just plays her GFG strategiesin both automata and is guaranteed to win whether the word produced by Adam is in L ( A )or L ( box ( ¯ A )).As the winning condition of G ( A ) is a disjunction of two Rabin conditions, Eve has apositional winning strategy. Fix such a strategy σ .Now do the same with A and box ( A ) for Adam: define G ( A ) as (cid:0) R E, Σ × box ( A ) (cid:1) × A ,where box ( A ) is the automaton box ( A ) where the transitions are turned from nondeterministicto universal, i.e, we replace ∨ with ∧ .Again, in a round of G ( A ) from a position ( v, p, q ): Eve plays a letter a ; Adam resolvesnondeterminism of box ( A ) (i.e., the universality in its dual); then they both resolve thechoices in A . Let Adam win G if either the play of A is rejecting or the run of box ( A ) isaccepting. Again we can ensure that Adam has a winning strategy in G ( A ), because bothautomata are GFG: he uses the GFG strategy of box ( A ) and the ∀ -GFG strategy over A . Ifthe word given by Eve belongs to L ( A ) then Adam wins by producing an accepting run of box ( A ), otherwise he wins by refuting an accepting run of A . Let τ be his positional winningstrategy in that game.We are now ready to build the deterministic automaton from a GFG APW A , usingpositional winning strategies σ and τ for Eve and Adam in G ( A ) and G ( A ), respectively.Let D be the automaton with states of the form ( q, p , p ), with q a state of A , p a stateof box ( A ) and p a state of box ( ¯ A ). A transition of D over a moves to ( q , p , p ) such that(( q, p ) , ( q , p )) is consistent with τ and (( q, p ) , ( q , p )) is consistent with σ . In other words,when reading a letter a in such a state, the following computations are performed: We simulate the choices made by σ in G ( A ) upon obtaining a from Adam. This waywe know how to resolve nondeterminism of box ( ¯ A ) and what to do with disjunctionsinside A . We simulate the choices made by τ in G ( A ) upon obtaining a from Eve. This way weknow how to resolve nondeterminism in box ( A ) and what to do with conjunctions of A . In the end we proceed to a new state of A and resolved nondeterminism of both box ( A )and box ( ¯ A ).The acceptance condition of D is inherited from A . (cid:73) Lemma 17.
For a GFG APW A and D built as above, L ( A ) = L ( D ) . Proof.
Take a word w ∈ Σ ω . First assume that w ∈ L ( A ). Eve cannot win a play of thegame G with the letters played in R A, Σ coming from w using by the first disjunct of her winning condition, since L ( box ( ¯ A )) = L ( ¯ A ). Thus, all the plays over w consistent with herwinning strategy σ in G must guarantee that the constructed path of A is accepting. Thus,the run of the automaton D over w is accepting.Now assume that w / ∈ L ( A ). Dually, no play of the game G with the letters coming from w can produce an accepting run of box ( A ) over w . Thus, the strategy τ guarantees that thesequence of visited states of A is rejecting. Thus, the run of D over w must be rejecting. (cid:74) D Appendix of Section 5D.1 Proof of Theorem 19
Our aim is to provide an
Exptime algorithm for deciding if a given alternating parityautomaton is GFG.Recall the construction of the two nondeterministic parity automata box ( A ) and box ( ¯ A )for L ( A ) and L ( A ) c respectively, as defined in Section 4.2. We will use these automata todesign a game characterising the fact that A is both ∃ -GFG and ∀ -GFG, i.e, A is just GFG.Recall that the automata box ( A ) and box ( ¯ A ) have exponential number of states in thenumber of states of A . However, due to Lemma 10 their parity index is linear in the numberof transitions of A . Consider the game G = (cid:0) R A, Σ × box ( A ) (cid:1) × box ( ¯ A ), i.e, the game whereAdam plays a letter and Eve replies with two boxes, one of A and the other of A . Let thewinning condition of that game for Eve say that either of the runs of box ( A ) or box ( ¯ A ) mustbe accepting. (cid:73) Lemma 32.
Eve has a winning strategy in G if and only if A is GFG. Proof.
Clearly if A is GFG then both box ( A ) and box ( ¯ A ) are GFG as nondeterministicautomata. Therefore, one can use strategies witnessing their GFGness to construct a singlestrategy for Eve in G . This strategy must be winning, because each word proposed byAdam either belongs to L ( box ( A )) = L ( A ) or to L ( box ( ¯ A )) = L ( A ) c .Now assume that Eve has a winning strategy in G . This strategy consists of twocomponents: one is a strategy in R A, Σ × box ( A ) and the other in R A, Σ × box ( ¯ A ). By thefact that the languages of box ( A ) and box ( ¯ A ) are disjoint, the above components are infact winning strategies in the letter games for box ( A ) and box ( ¯ A ) respectively. Thus, byLemma 20 we know that A is both ∃ -GFG and ∀ -GFG. (cid:74) What remains is to show how to solve the game G in Exptime . Let n be the size of theautomaton A . Our aim is to turn it into a parity game of size exponential in n but witha number of priorities polynomial in n . Then, by invoking for instance [6], we know thatsuch a game can be solved in Exptime . (cid:73) Lemma 33.
Let
Γ = { , . . . , N } be a set of priorities. Then, there exists a deterministicparity automaton of size exponential in N , with a number of priorities polynomial in N thatrecognises the language L of words w ∈ (Γ × Γ) ω that satisfy the parity condition on at leastone coordinate. Proof.
It is a rather standard construction. One possibility is to design a nondeterministicBüchi automaton for L with N states. Then, the standard determinisation procedure [21]applied to this automaton gives a deterministic parity automaton as in the statement. (cid:74) Therefore, we conclude the proof of Theorem 19 by taking a product of the game G00