Making Streett Determinization Tight
aa r X i v : . [ c s . F L ] J un Making Streett Determinization Tight
Cong Tian
ICTT and ISN LaboratoryXidian UniversityXi’an, 710071, P.R. [email protected]
Wensheng Wang
ICTT and ISN LaboratoryXidian UniversityXi’an, 710071, P.R. [email protected]
Zhenhua Duan
ICTT and ISN LaboratoryXidian UniversityXi’an, 710071, P.R. [email protected]
Abstract —Optimal determinization construction of Streett au-tomata is an important research problem because it is indis-pensable in numerous applications such as decision problemsfor tree temporal logics, logic games and system synthesis.This paper presents a transformation from nondeterministicStreett automata (NSA) with n states and k Streett pairs toequivalent deterministic Rabin transition automata (DRTA) with n n ( n !) n states, O ( n n ) Rabin pairs for k = ω ( n ) and n n k nk states, O ( k nk ) Rabin pairs for k = O ( n ) . This improves the stateof the art Streett determinization construction with n n ( n !) n + states, O ( n ) Rabin pairs and n n k nk n ! states, O ( nk ) Rabin pairs,respectively. Moreover, deterministic parity transition automata(DPTA) are obtained with n ( n + − n !) n + states, n ( n + priorities for k = ω ( n ) and n ( k + − n ! k nk states, n ( k + priorities for k = O ( n ) , which improves the best construction with n n ( k + n ( k + ( n ( k + − states, n ( k + priorities. Further, weprove a lower bound state complexity for determinization con-struction from NSA to deterministic Rabin (transition) automatai.e. n n ( n !) n for k = ω ( n ) and n n k nk for k = O ( n ) , which matches thestate complexity of the proposed determinization construction.Besides, we put forward a lower bound state complexity fordeterminization construction from NSA to deterministic parity(transition) automata i.e. Ω ( n log n ) for k = ω ( n ) and Ω ( nk log nk ) for k = O ( n ) , which is the same as the state complexity of theproposed determinization construction in the exponent. Keywords —Streett automata, Rabin automata, determiniza-tion, state complexity, lower bound.
I. I
NTRODUCTION
Streett automata [1] are nearly the same as B¨uchi automata[2] except for the acceptance condition. They are exponen-tially more succinct than B¨uchi automata in encoding infinitebehaviors of systems [4]. As a result, Streett automata havean advantage in modeling behaviors of concurrent and reactivesystems [5].Determinization is one of the fundamental notions in au-tomata theory. Given a nondeterministic automaton A , deter-minization of A is the construction of another deterministicautomaton B that recognizes the same language as A does.As for Streett automata, determinization constructions havebeen investigated for decades. In 1992, Safra introduced thefirst determinization construction for nondeterministic Streettautomata (NSA) by using an innovative data structure knownas Streett Safra trees [3]. The states of the resulting deter-ministic automata are not sets of states, but tree structures.Safra’s construction transforms a NSA with n states and k Streett pairs into a deterministic Rabin automaton (DRA) with12 n ( k + n n ( k + n ( k + ( n ( k + n ( k + states and n ( k +
1) Rabin pairs. In 2007, Piterman [9] presented a tighter constructionvia compact Streett Safra trees which are obtained by usinga dynamic naming technique throughout the Streett Safra treeconstruction. With compact Streett Safra trees, a NSA can betransformed into an equivalent deterministic parity automaton(DPA) with 2 n n ( k + n ( k + ( n ( k + n ( k + n ( k + n n ( n !) n + states and O ( n ) Rabin pairs for k = ω ( n ),and n n k n ( k + n ! states and O ( nk ) Rabin pairs for k = O ( n )[5], [6]. Their construction is based on another data struc-ture, namely, µ -Safra trees for Streett determinization , whichreduces the redundancy of index labels and utilizes a batch-mode naming scheme.As for the state lower bound of Streett determinization, ithas also been investigated. For a NSA with n states and k Streett pairs, Cai and Zhang proved a lower bound of Streettcomplementation which is 2 Ω ( n log n + nk log k ) states for k = O ( n )and 2 Ω ( n log n ) states for k = ω ( n ) [8]. It indicates that thelower bound state complexity for determinization constructionfrom NSA to DR(T)A is no smaller than (maybe very closeto) 2 Ω ( n log n + nk log k ) for k = O ( n ) and 2 Ω ( n log n ) for k = ω ( n ).Besides, for the lower bound state complexity for determiniza-tion construction from NSA to deterministic Streett (transition)automata (DS(T)A) or DP(T)A, a result was given in [12], [13]with 2 Ω ( n log n ) states. Later, Yan [15] obtained the same resultvia full automata technique. Since then, the lower bound statecomplexity 2 Ω ( n log n ) for determinization construction fromNSA to DS(T)A or DP(T)A has never improved. There is agap between the upper and lower bounds state complexity fordeterminization construction from NSA to DR(T)A, DS(T)A,or DP(T)A. Therefore, it is interesting to make the statecomplexity for Streett determinization construction tight ortighter.In this paper, we reconstruct µ -Safra trees as H-Safra treesfor Streett determinization by changing the name on eachnode of the tree. As a consequence, an improved constructionof DRTA is obtained with state complexity being n n ( n !) n for k = ω ( n ), and n n k nk for k = O ( n ). Then, LIR-H-Safratrees for Streett determinization are presented by adding laterintroduction records , which records the generation order ofeach node, to H-Safra trees. Based on LIR-H-Safra trees,n improved construction of DPTA is obtained with statecomplexity being 3( n ( n + − n !) n + for k = ω ( n ), and3( n ( k + − n ! k nk for k = O ( n ). We prove the lower boundstate complexity for determinization construction from NSAto DR(T)A by the language game namely L -game [14] whichmatches the state complexity of the proposed determinizationconstruction by H-Safra trees. Moreover, an improved lowerbound state complexity 2 Ω ( n log n ) for k = ω ( n ) and 2 Ω ( nk log nk ) for k = O ( n ) for determiniztion construction from NSA toDP(T)A is proposed based on L -game. It is the same asthe determinization construction by LIR-H-Safra trees in theexponent.The rest of the paper is organized as follows. The nextsection briefly introduces automata over infinite words. InSection III, Cai and Zhang’s NSA-to-DRA determinizationbased on µ -Safra trees is revisited. Our new data structures, H-Safra trees for Streett determinization and LIR-H-Safra treesfor Streett determinization, are presented in Section IV. Inthe sequel, the improved NSA-to-DRTA and NSA-to-DPTAdeterminization constructions are presented in Section V.Section VI studies the lower bound of the determinizationconstruction. II. A UTOMATA
Let Σ be a finite set of symbols called an alphabet. Aninfinite word α is an infinite sequence of symbols from Σ . Σ ω isthe set of all infinite words over Σ . We present α as a function α : N → Σ , where N is the set of non-negative integers. Thus, α ( i ) denotes the letter appearing at the i th position of the word.In general, Inf ( α ) denotes the set of symbols from Σ whichoccur infinitely often in α . Formally, Inf ( α ) = { σ ∈ Σ | ∃ ω n ∈ N : α ( n ) = σ } . Note that ∃ ω n ∈ N means that there existinfinitely many n in N . Definition 1 (Automaton) . An automaton over Σ is a tuple A = ( Σ , Q , δ, Q , λ ), where Q is a non-empty, finite set ofstates, Q ⊆ Q is a set of initial states, δ ⊆ Q × Σ × Q is atransition relation, and λ is an acceptance condition.A run ρ of an automaton A on an infinite word α is aninfinite sequence ρ : N → Q such that ρ (0) ∈ Q and for all i ∈ N , ( ρ ( i ) , α ( i ) , ρ ( i + ∈ δ . A is said to be deterministic if Q is asingleton, and for any ( q , σ, q ′ ) ∈ δ , there exists no ( q , σ, q ′′ ) ∈ δ such that q ′′ , q ′ , and nondeterministic otherwise. Similarto infinite words, Inf ( ρ ) denotes the set of states from Q whichoccur infinitely often in ρ . Formally, Inf ( ρ ) = { q | ∃ ω n ∈ N : ρ ( n ) = q } .Several acceptance conditions are studied in literature. Wepresent three of them here: • Streett, where λ = {h G , B i , h G , B i , . . . , h G k , B k i} with G i , B i ⊆ Q . ρ is accepted iff for all 1 ≤ i ≤ k , we havethat Inf ( ρ ) ∩ G i , ∅ or Inf ( ρ ) ∩ B i = ∅ . • Rabin, where λ = {h A , R i , h A , R i , . . . , h A k , R k i} with A i , R i ⊆ Q . ρ is accepted iff for some 1 ≤ i ≤ k , we havethat Inf ( ρ ) ∩ A i , ∅ and Inf ( ρ ) ∩ R i = ∅ . • Parity, where λ = { λ , λ , . . . , λ k } with λ ∪ λ ∪ . . . ∪ λ k = Q . ρ is accepted iff the minimal index i for which Inf ( ρ ) ∩ λ i , ∅ is even.An automaton accepts a word if it has an accepting run onit. The accepted language of an automaton A , denoted by L ( A ),is the set of words that A accepts.We denote the different types of automata by three letteracronyms in { D , N } × { S , R , P } × { A } . The first letter standsfor the branching mode of the automaton (deterministic ornondeterministic); the second letter stands for the acceptancecondition type (Streett, Rabin, or parity); and the third letterindicates automata. While acceptance condition of an ordinaryautomaton is defined on states, the acceptance condition of atransition automaton is defined on transitions of the automaton.Accordingly, with respect to each type of ordinary automata,we also have its transition version.III. D ETERMINIZATION VIA µ -S AFRA T REES FOR S TREETT
This section revisits the determinization construction via µ -Safra trees for Streett [5]. For any positive integer m ∈ N , weuse [ m ] to denote the set { , , . . . , m } . A. µ -Safra Trees for Streett Determinization µ -Safra trees for Streett determinization, presented by Caiand Zhang in 2012 [5], are obtained from Streett Safra trees[3]. A µ -Safra tree for Streett determinization is a labelledordered tree. A tree is ordered just if the nodes are partiallyordered by older-than relation. Compared with Streett Safratrees, the characteristic of µ -Safra trees for Streett determiniza-tion is a batch-mode naming scheme M b for nodes.For an ordered tree, a leaf corresponds to a left spine . A leftspine is a maximal path τ , τ , . . . , τ m such that τ m is a leaf,for any i ∈ { , . . . , m } , τ i is the left-most child of τ i − , and τ , called the head of the left spine, is not a left-most childof its parent [5]. We arrange all left spines with consecutiveintegers starting from 1 as names of left spines. Each node ison exactly one left spine. For the sibling nodes, the name of theleft spine, which contains the left-most sibling, is smaller thanthe others. With this basis, every node can be named uniquely.Nodes in a left spine named ls , from the head downwards, areassigned continuously increasing names, starting from ls . Rule 1 (Batch-mode naming scheme M b ) . If a node τ belongsto the left spine named ls , and τ is the i -th node in ls , the nameof τ is ls . i , i.e. M b ( τ ) = ls . i [5]. Definition 2 ( Cover and
Mini [5], [7]) . For a NSA S = ( Σ , Q , Q , δ, λ ) with | Q | = n and k Streett pairs λ = {h G , B i , h G , B i , . . . , h G k , B k i} . Let β be a subset of [ k ], and G β = S i ∈ β G i , where G i is the first element of the i -th Streett pair h G i , B i i . Then, Cover maps 2 [ k ] to 2 [ k ] such that Cover ( β ) = { j ∈ [ k ] | G j ⊆ G β } Mini also maps 2 [ k ] to 2 [ k ] such that j ∈ Mini ( β ) if, and onlyif, j ∈ [ k ] \ Cover ( β ) and ∀ j ′ ∈ [ k ] \ Cover ( β ) , [ j ′ , j → ( G j ′ ∪ G β G j ∪ G β )] , (1)2 j ′ ∈ [ k ] \ Cover ( β ) , [ j ′ < j → ( G j ′ ∪ G β , G j ∪ G β )] . (2) Example 1.
For a NSA with n = k = Q = { q , q , q } ,and the first elements of the four Streett pairs are G = { q , q } , G = { q } , G = { q , q } , and G = { q } . Let β = { } . We have G β = G = { q , q } . Obviously, G ⊆ G β and G ⊆ G β , whichinfers to Cover ( β ) = { , } .Further, we have [ k ] \ Cover ( β ) = { , } . For j = j ′ = G j ′ ∪ G β = { q , q , q } and G j ∪ G β = { q , q , q } ,which satisfies Conditions (1) and (2). Thus, 1 ∈ Mini ( β ).For j = j ′ =
1, we also have G j ′ ∪ G β = { q , q , q } and G j ∪ G β = { q , q , q } . Obviously, Condition (2) is violatedsince ( j ′ = < ( j = < Mini ( β ). As a result, Mini ( β ) = { } . Definition 3 ( µ -Safra tree for Streett determinization [5]) . Fix a NSA S = ( Σ , Q , Q , δ, λ ) with | Q | = n and k Streettpairs λ = {h G , B i , h G , B i , . . . , h G k , B k i} . A µ -Safra tree forStreett determinization of the NSA S is a labeled ordered tree h T o , V , l , h , M b , E , F , stor i , where T o is an ordered tree, and • V is the set of all nodes in T o . • l : V → Q is a state label of nodes with subsets of Q .The label of every node is equal to the union of its sons.The labels of two siblings are disjoint. • h : V → [ k ] is an index label, which annotates every nodewith a set of indices from [ k ]. The root is annotated by[ k ]. The annotation of every node is contained in that ofits parent and it misses at most one element from theannotation of the parent. Every node that is not a leafhas at least one son with strictly smaller annotation. Inaddition, each leaf τ l satisfies h ( τ l ) = ∅ or Mini ([ k ] − h ( τ l )) = ∅ , where Mini is defined in Definition 2 fordetermining the index labels of nodes. • M b : V → [ n ] . [ µ + µ = min( n , k ), assigns eachnode a unique name by the batch-mode naming scheme. • E , F ⊆ V are two disjoint subsets of V . They are used todefine the Rabin acceptance condition. • stor is an additional structural ordering on nodes. Forevery non-root node τ , let j ( τ ) = max { ( h ( τ p ) ∪ { } ) − h ( τ ) } where τ p is the parent of τ . stor means that for any twosiblings τ and τ ′ , τ ′ is placed to the right of τ if, andonly if, j ( τ ) > j ( τ ′ ), or j ( τ ) = j ( τ ′ ) and τ is older than τ ′ .The following lemma has been proved in [5]. Lemma 1.
For a µ -Safra tree for Streett determinization ofa NSA with n states and k Streett pairs, there are at most n left spines, and each left spine has at most µ + µ = min( n , k ). Therefore, [ n ] . [ µ +
1] node names are sufficient[5].Accordingly, Lemma 2 is easily obtained.
Lemma 2.
The number of nodes in a µ -Safra tree for Streettdeterminization is at most n ( µ +
1) [5].Fig. 1 illustrates a µ -Safra tree for Streett determinizationof a NSA with 5 states, namely, a , b , c , d and e . This µ - Safra tree contains 12 nodes. The state sets shown in nodesare state labels. The batch-mode names and index labels ofnodes are given in red and blue, respectively. There are fourleft spines, i.e. { . , . , . , . } , { . , . , . } , { . , . } and { . , . , . } . a, b, c, d, eb, e c a, db e a, d h = { } h = { } h = { } h = { } h = { } h = { } h = { } b e cc a, d h = { } h = { } h = { } h = { } h = { } Fig. 1. A µ -Safra tree for Streett determinization Along a sequence of µ -Safra tree for Streett determinizationtransformations, there may exist some node whose name ischanged. For instance, when a node moves into another leftspine, the node should be renamed. The renaming scheme isstated by Rule 2 [5]. Rule 2 (Batch-mode renaming scheme) . When a left spine iscreated, nodes in the left spine are assigned names from anunused name bucket. When a left spine is removed, the namebucket of the left spine is recycled. When a left spine ls isgrafted into another left spine ls ′ , the name bucket of ls isrecycled and nodes on ls are renamed as if they were on ls ′ ,originally.In the transformations, the index labels h of the new creatednodes also need to be defined. The index label h of a node τ is a subset of the indices set of all Streett pairs. We willcheck whether all states in l ( τ ) visits the first elements G s ofthese Streett pairs one by one. But there may exist a situationthat some G of a Streett pair h G , B i is contained by another G ′ . If G ′ has been checked, it is redundant to further check G . In order to reduce unnecessary inspections, the functions Mini , which decreases the combination of index labels h , willbe utilized in the determinization construction. In [5], it hasbeen proved that using Mini to select the index labels of thechildren is sound and complete.
B. Construction of µ -Safra Trees for Streett Determinization Fix a NSA S = ( Σ , Q , Q , δ, h G , B i [ k ] ). The initial µ -Safratree for Streett determinization of S is a single-branch (onlya left spine) labelled tree T I . Every node is named by thebatch-mode naming scheme. For each node τ , l ( τ ) = Q and h ( τ ) = h ( τ p ) − max { Mini ([ k ] − h ( τ p )) } . Specially, for the root τ r , h ( τ r ) = [ k ], and the leaf τ l satisfies h ( τ l ) = ∅ or Mini ([ k ] − h ( τ l )) = ∅ . Set E = ∅ and F = ∅ . Given a µ -Safra tree T µ for Streett determinization of S and σ ∈ Σ , we construct anew µ -Safra tree ˆ T µ for Streett determinization, called the σ - successor of T µ , in six steps as follows.3) Update : Set E and F to empty sets and replace the statelabel of every node τ in T µ by S q ∈ l ( τ ) δ ( q , σ ). Call theresultant labelled tree T µ .2) Create siblings : Apply the following transformations tonon-leaf nodes of T µ . Let τ be a node with m children τ , . . . , τ m . Sequentially consider the following cases foreach i ∈ [1 .. m ] from 1 to m .a) If l ( τ i ) ∩ G j ( τ i ) , ∅ , add a child τ ′ to τ with l ( τ ′ ) = l ( τ i ) ∩ G j ( τ i ) and h ( τ ′ ) = h ( τ ) − max { [0 .. j ( τ i )) ∩ ( { } ∪ Mini ([ k ] − h ( τ ))) } , and remove the states in l ( τ i ) ∩ G j ( τ i ) from τ i as well as all its descendants.b) If l ( τ i ) ∩ G j ( τ i ) = ∅ and l ( τ i ) ∩ B j ( τ i ) , ∅ , add a child τ ′ to τ with l ( τ ′ ) = l ( τ i ) ∩ B j ( τ i ) and h ( τ ′ ) = h ( τ i ), andremove the states in l ( τ i ) ∩ B j ( τ i ) from τ i as well as allits descendants.Call the resultant labelled tree T µ .3) Horizontal merge : For any two siblings τ and τ ′ in T µ and any state q ∈ l ( τ i ) ∩ l ( τ i ′ ), if j ( τ ) < j ( τ ′ ), or j ( τ ) = j ( τ ′ ) and τ is older than τ ′ , then remove q from τ ′ andall its descendants. Remove nodes with empty state labeland add their names, if defined, to E . Call the resultantlabelled tree T µ .4) Vertical merge : For each non-leaf τ in T µ , if all childrenare annotated by h ( τ ), then remove all the children andtheir descendants. Add the name of τ to F . Call theresultant labelled tree T µ .5) Rename : Rename nodes whose names are defined in T µ according to Rule 2 and add nodes that are renamed to E , which results in T µ .6) Create children : Repeat the following procedure until nonew nodes can be added: For each leaf τ in T µ such that h ( τ ) , ∅ and Mini ([ k ] − h ( τ )) , ∅ , add to τ a new child τ ′ . Set l ( τ ′ ) = l ( τ ), h ( τ ′ ) = h ( τ ) − max { Mini ([ k ] − h ( τ )) } .Then name nodes whose names are undefined accordingto the batch-mode naming scheme. The resultant labelledtree is denoted as ˆ T µ .ˆ T µ is a µ -Safra tree for Streett determinization.Thus, given a NSA S = ( Σ , Q , Q , δ, h G , B i [ k ] ), by applyingthe above six-step procedure recursively until no new µ -Safratrees can be created, an associated DRA DR = ( Σ , Q DR , T µ I ,δ DR , λ DR ) can be constructed. Here, Q DR is the set of µ -Safratrees for Streett determinization of S , T µ I is the initial µ -Safra tree for Streett determinization, δ DR is the µ -Safra-tree-Streett transition relation (i.e. T µ σ −→ ˆ T µ whenever ˆ T µ is the σ -successor of T µ ), and λ DR = { ( A τ , R τ ) , . . . , ( A τ k , R τ k ) } (where k ≥
1) is the Rabin acceptance condition. For each i , the node τ i is given by its name, A τ i is the set of µ -Safra trees forStreett determinization (node τ i belongs to F ), and R τ i the setof µ -Safra trees for Streett determinization (node τ i belongsto E ).Given an input ω -word α : ω → Σ , we call the sequence Π = T µ T µ T µ T µ . . . of µ -Safra trees for Streett determinizationsuch that T µ = T µ I , and for all i ∈ ω , T µ i + is the α ( i )-successorof T µ i , the µ -Safra Streett trace of the NSA S over α . Weview the µ -Safra Streett trace of S over α as the run of the DRA DR over α . Then we say that α is accepted by the DRA if there exists i ∈ { , . . . , k } such that Inf ( Π ) ∩ A τ i , ∅ and Inf ( Π ) ∩ R τ i = ∅ . Theorem 3 (Cai and Zhang [5], [6]) . Given a NSA S with nstates and k Streett pairs, a DRA with n n ( n !) n + states, O ( n ) Rabin pairs for k = ω ( n ) , and n n k n ( k + n ! states, O ( nk ) Rabinpairs for k = O ( n ) can be constructed that recognizes thelanguage L ( S ) . By deleting the two sets E and F of each µ -Safra tree inthe Streett determinization and recording the accepting andrejecting nodes throughout each transition, a DRTA can beconstructed. Corollary 4.
Given a NSA S with n states and k Streett pairs,a DRTA with n n ( n !) n + states, O ( n ) Rabin pairs for k = ω ( n ) ,and n n k nk n ! states, O ( nk ) Rabin pairs for k = O ( n ) can beconstructed that recognizes the language L ( S ) . IV. H-S
AFRA T REES AND
LIR-H-S
AFRA T REES FOR S TREETT D ETERMINIZATION
This section presents two new data structures, called H-Safra trees and LIR-H-Safra trees for Streett determinization.
A. H-Safra Trees for Streett Determinization
As for B¨uchi determinization, Schewe proposes a tightconstruction via history trees which results in an equivalentDRTA [10]. In Schewe’s construction, instead of explicitnames, nodes are implicitly named. This leads to a reduction ofstate complexity. With this motivation, we put forward a newdata structure namely
H-Safra trees for Streett determinization.Compared with µ -Safra trees for Streett determinization, theonly difference is the naming scheme of nodes.For a structural ordered tree with state and index labels (i.e. a µ -Safra tree for Streett determinization without names, E and F ), denoted by T si (Fig. 2 is an example), we give anew naming scheme depending only on the index label h ofnodes, which is expressed by Rule 3. a, b, c, d, eb, e c a, db e a, d h = { } h = { } h = { } h = { } h = { } h = { } h = { } b e cc a, d h = { } h = { } h = { } h = { } h = { } Fig. 2. A structural ordered tree with state and index labels
Rule 3 (Naming scheme M n ) . • For the root τ r , M n ( τ r ) = ǫ ; • for each node τ in the second level, M n ( τ ) = j ( τ ) i + where i = |{ τ ′ | τ ′ is the left sibling of τ , and j ( τ ′ ) = j ( τ ) }| ;4 for any other node τ , M n ( τ ) = M n ( τ p ) . j ( τ ) i + .Utilizing the new naming scheme, we can get a H-Safratree for Streett determinization . Definition 4 (H-Safra trees for Streett determinization) . A H-Safra tree for Streett determinization of a given NSA S = ( Σ , Q , Q , δ, λ ) with n states and k Streett pairs is a pair h T si , M n i where T si is a structural ordered tree with state andindex labels of S , and M n is the new naming scheme.Fig. 3 is a H-Safra tree for Streett determinization obtainedfrom Fig. 2 by using the new naming scheme. Here, the namesof nodes are given in red. For the node τ with l ( τ ) = { a , d } and h ( τ ) = { , } ( j ( τ ) = τ ′ such that j ( τ ′ ) = j ( τ ) = τ is 2 . j = 2 j = 2 j = 2 j = 1 j = 1 . . ǫ a, b, c, d, eb, e c a, db e a, d h = { } h = { } h = { } h = { } h = { } h = { } h = { } b e cc a, d h = { } h = { } h = { } h = { } h = { } . . . . . . . . . . j = 3 j = 3 j = 1 j = 2 j = 1 j = 3 Fig. 3. A H-Safra tree
Obviously, each node in a structural ordered tree with stateand index labels can be uniquely named.The new naming scheme is the core of our determinizationconstruction. Given a NSA, H-Safra trees for Streett deter-minization will be taken as the states of the final DRTA. By thenaming scheme, once the index label h of each node is fixed,the name is also determined, which makes the state complexitydecrease. Lemma 5.
The number of H-Safra trees for Streett deter-minization of a given NSA is equal to the number of structuralordered trees with state and index labels, i.e. µ -Safra trees forStreett determinization without names, E , and F , occurring inthe determinization construction. Proof.
By the naming scheme M n , for each node τ occurringin a structural ordered tree with state and index labels, a uniquename M n ( τ ) is assigned to τ . M n ( τ ) depends on the indexlabel and the position of τ in the tree. Thus, the number ofH-Safra trees for Streett determinization of a NSA is equalto the number of structural ordered trees with state and indexlabels. (cid:3) B. LIR-H-Safra Trees for Streett Determinization
In order to transform a NSA to a DPTA, we need a dynamicnode identification scheme that captures the order in which the nodes are created when constructing the σ -successors.Consequently, the state complexity of the DPTA transform willincrease. Similar to the constructions of Schewe from NBAto DPA [10] and from NPA to DPA [11], the data structurewe shall use is H-Safra trees for Streett determinization with later introduction record (LIR), called LIR-H-Safra trees forStreett determinization . A LIR is a sequence of nodes in theH-Safra tree for Streett determinization according to the orderthe nodes are generated.
Definition 5 (LIR-H-Safra trees for Streett determinization) . Given a NSA S = ( Σ , Q , Q , δ, λ ) with n states and k Streettpairs, a LIR-H-Safra tree for Streett determinization is a pair h H , LIR i where H is a H-Safra tree for Streett determinizationand LIR stores the order in which the nodes of H are created. . . ǫ a, b, c, d, eb, e c a, db e a, d h = { } h = { } h = { } h = { } h = { } h = { } h = { } b e cc a, d h = { } h = { } h = { } h = { } h = { } . . . . . . . . . . LIR= { ǫ, , , . , , . , . , . , . . , . . , . . , . . } Fig. 4. A LIR-H-Safra tree
Fig. 4 is a LIR-H-Safra tree for Streett determinization. TheLIR contains all nodes of the tree such that each node appearsafter its left siblings. Every node in LIR is represented by itsname for simplicity.As for each node τ of a given LIR-H-Safra tree for Streettdeterminization, we introduce an extra notation p ( τ ) to denotethe position of τ in the LIR.V. D ETERMINIZATION VIA
H-S
AFRA T REES AND
LIR-H-S
AFRA T REES FOR S TREETT
This section presents a NSA-to-DRTA determinizationtransform via H-Safra trees and a NSA-to-DPTA determiniza-tion transform via LIR-H-Safra trees.
A. Construction of H-Safra Trees for Streett Determinization
Fix a NSA S = ( Σ , Q , Q , δ, h G , B i [ k ] ). The initial H-Safratree for Streett determinization of S is a single-branch labelledtree H I . For each node τ of H I , the state label l ( τ ) = Q andindex label h ( τ ) = h ( τ p ) − max { Mini ([ k ] − h ( τ p )) } . Specially,for the root τ r , h ( τ r ) = [ k ], and the leaf τ l satisfies h ( τ l ) = ∅ or Mini ([ k ] − h ( τ l )) = ∅ . Every node in H I is named by thenew naming scheme.5iven a H-Safra tree H for Streett determinization of S and σ ∈ Σ , we construct a new H-Safra tree ˆ H for Streett deter-minization, called the σ - successor of H , and the signaturessig acc and sig re j of the transition, in six steps as follows.We intuitively illustrate the six steps of construction by anexample. Fig. 5 shows all transitions for an input letter σ from the states in the H-Safra tree for Streett determinizationin Fig. 3. aeb dc G = { a } , B = { b } G = { e, d } , B = { c } G = { c } , B = ∅ Fig. 5. Relevant fragment of a Streett automaton
Step 1: Update : Replace the state label of every node τ in H by S q ∈ l ( τ ) δ ( q , σ ). Call the resultant labelled tree H .Let H be the H-Safra tree for Streett determinization inFig.3 for the NSA whose transition is depicted in Fig.5. Fig.6shows the tree structure H resulting from H after Step 1 ofthe construction procedure. Compared with H , state labels ofall nodes in H are updated. . . ǫ a, b, c, d, eb, c, d e a, c, ec, d b h = { } h = { } h = { } h = { } h = { } h = { } h = { } c, d b h = { } h = { } h = { } h = { } h = { } . . . . . . . . . . ee a, c, ea, c, e Fig. 6. Step 1 of the construction procedure
Step 2: Create siblings : Apply the following transfor-mations to non-leaf nodes of H from the root. Let τ be a nodewith m children τ , . . . , τ m . Sequentially consider the followingtwo cases for each i ∈ [1 .. m ] from 1 to m :a) If l ( τ i ) ∩ G j ( τ i ) , ∅ , add a youngest child τ ′ to τ with l ( τ ′ ) = l ( τ i ) ∩ G j ( τ i ) and h ( τ ′ ) = h ( τ ) − max { [0 .. j ( τ i )) ∩ ( { } ∪ Mini ([ k ] − h ( τ ))) } , and remove the states in l ( τ i ) ∩ G j ( τ i ) from τ i and all its descendants; thenb) if l ( τ i ) ∩ B j ( τ i ) , ∅ , add a youngest child τ ′ to τ with l ( τ ′ ) = l ( τ i ) ∩ B j ( τ i ) and h ( τ ′ ) = h ( τ i ), and remove the statesin l ( τ i ) ∩ B j ( τ i ) from τ i and all the descendants.Note that the names of the new created nodes are not definedcurrently. Then rearrange sibling nodes by the structural or-dering from the second level to the last level.We use a simple example illustrated in Fig.7 to show howthe sibling nodes are rearranged. For the siblings τ , τ , τ and τ in Fig.7 (a), we have j ( τ ) = j ( τ ) = j ( τ ) =
1, and j ( τ ) =
2. We rearrange the siblings according to the valueof j from the largest to the smallest. As for τ and τ with j ( τ ) = j ( τ ), τ is younger than τ , since the later the nodeis generated, the younger it is. It indicates that the relativeorder of nodes with the same j will not change. The resultanttree after structural ordering is shown in Fig.7 (b). Comparedwith Fig.7 (a), the positions of τ and τ , and τ and τ areswapped, respectively. ǫ a, b, c, d, eb, d e h = { } h = { } h = { } h = { } a c h = { } ǫ a, b, c, d, eb, d e h = { } h = { } h = { } h = { } a c h = { } j = 2 j = 3 j = 1 j = 2 j = 3 j = 2 j = 2 j = 1 structural ordering τ τ τ τ τ τ τ τ ( a )( b ) Fig. 7. Structural ordering
After Step 2, the resultant labelled tree, called H , is shownin Fig.8. The nodes without names are new created in thisstep, and every node observes the structural ordering. The statelabels of the nodes in grey will be deleted in Step 3. a, b, c, d, eb, d cb h = { } h = { } h = { } h = { } h = { } h = { } h = { } . . . ǫ a c e ed a h = { } h = { } h = { } h = { } h = { } h = { } h = { } h = { } h = { } h = { } h = { } . . . . . . . . . h = { } Fig. 8. Step 2 of the construction procedure
Step 3: Horizontal merge : For each node τ in H starting from the root, and every state q ∈ l ( τ ), if q also occursin the state label of a sibling τ ′ of τ such that j ( τ ′ ) < j ( τ ), or j ( τ ′ ) = j ( τ ) and τ ′ is older than τ , then remove q from τ aswell as all its descendants. Afterward, for any node τ , remove6 if l ( τ ) = ∅ . A removed node whose name is defined is called rejecting .Let H be the resultant tree. Next, we define sig re j = { τ | τ is the rejecting node occurring in the current tree } , called the rejecting signature of the δ -successor / transition being de-fined. The resulting tree is depicted in Fig. 9 with sig re j = { , . , . , . , . , . . , . . , . . , . . } .In the resultant tree, the state labels of the siblings are pairwisedisjoint and there exists no empty node. Nevertheless, theremay exist a node which is equal to each of its children inindex label. a, b, c, d, eb, d cb h = { } h = { } h = { } h = { } ǫ a ed a h = { } h = { } h = { } h = { } Fig. 9. Step 3 of the construction procedure
Step 4: Vertical merge : For each non-leaf τ in H starting from the root, if the index label of each child is equalto h ( τ ), then remove all the children of τ as well as theirdescendants. The nodes whose descendants have thus beenremoved are called accepting .Let H be the resultant tree. Next define sig acc = { τ | τ is the accepting node occurring in the current tree } , calledthe accepting signature of the δ -successor / transition beingdefined. The resulting tree is depicted in Fig. 10 with sig acc = { } . The state labels of the siblings are pairwise disjoint, andno node is equal to each of its children in index label. Thenames of nodes might not follow the new naming scheme. Thenodes that will be renamed in Step 4 are drawn in red. a, b, c, d, eb, d cb h = { } h = { } h = { } h = { } ǫ a ed h = { } h = { } h = { } Fig. 10. Step 4 of the construction procedure
Step 5: Rename : Rename nodes whose names aredefined in H starting from the root by applying the namingscheme (Rule 3). The nodes which should be renamed are alsorejecting in this step. Add these rejecting nodes to sig re j . As forthis example, sig re j = { , . , . , . , . , . . , . . , . . , . . , } . Call the resultant labelled tree H . Fig. 11 shows the treethat results from Step 5. All nodes observe the naming scheme.Then the resultant tree will spawn in the next step. a, b, c, d, eb, d cb h = { } h = { } h = { } h = { } ǫ a ed h = { } h = { } h = { } Fig. 11. Step 5 of the construction procedure
Step 6: Create children : Repeat the following procedureuntil no new nodes can be added: For each leaf τ in H , if h ( τ ) , ∅ and Mini ([ k ] − h ( τ )) , ∅ , add to τ a new child τ ′ . Set l ( τ ′ ) = l ( τ ) and h ( τ ′ ) = h ( τ ) − { max( Mini ([ k ] − h ( τ ))) } . Thendefine names of the nodes which have not been named by thenew naming scheme yet.The resultant labelled tree is a H-Safra tree for Streettdeterminization, which we call ˆ H . Note that given H and σ ∈ Σ , there are a unique σ -successor ˆ H , sig acc , and sig re j .Fig. 12 shows ˆ H , called the σ - successor of H , obtainedthrough the six steps. Note that states in the resultant DRTAare H-Safra trees for Streett determinization, and the signatures sig acc , sig re j are part of the transition relation of the DRTAtransform. h = { } . h = { } h = { } h = { } a, b, c, d, eb, d cb h = { } h = { } h = { } h = { } ǫ a ed h = { } h = { } h = { } d b aa cc ee h = { } h = { } h = { } h = { } . . . . . . . . . . . . . . Fig. 12. Step 6 of the construction procedure
Based on the six-step procedure, given a NSA S = ( Σ , Q , Q , δ, h G , B i [ k ] ), an equivalent DRTA RT = ( Σ , Q RT , Q RT ,δ RT , λ RT ) can be obtained. Here Q RT is the set of H-Safra treesfor Streett determinization w.r.t S ; Q RT is the initial H-Safratree for Streett determinization; δ RT is a transition relationthat is established during the construction of H-Safra treesfor Streett determinization, consisting of transitions (typically δ ) which are of the form H σ −−→ δ sig ˆ H where δ sig = ( sig acc , sig re j )is the signature of the transition δ , with σ ranging over Σ , and H ranging over Q RT ; and λ RT = { ( A I , R I ) , . . . , ( A Ik , R Ik ) } isthe Rabin acceptance condition. Note that, in each Rabin pair7 A I , R I ), I ranges over the names appearing in the H-Safratrees for Streett determinization. A I is the set of transitionsthrough which node τ with name being I is accepting, while R I is the set of transitions through which node τ with namebeing I is rejecting.Given an input ω -word α : ω → Σ , we call the sequence Π = ( H , α (0) , H ) ( H , α (1) , H )( H , α (2) , H ) . . . of transi-tions where H = H I , and for all i ∈ ω , H i + is the α ( i )-successor of H i , the H-Safra Streett trace of the NSA S over α . We view the H-Safra Streett trace of S over α as the run of the DRTA RT over α . Then we say that α is accepted bythe DRTA if Inf ( Π ) ∩ A Ii , ∅ and Inf ( Π ) ∩ R Ii = ∅ for some( A Ii , R Ii ).Let RT be the DRTA obtained from the given NSA S .Theorem 6 is formalized and proved. Theorem 6. L ( RT ) = L ( S ) .Proof. This proof is similar to the one in [5]. ⇐ : This part of proof is almost identical to the one in [3].We ought to show that if Π = H H · · · is a run of RT overan infinite word α = α α · · · ∈ L ( S ), then (1) a node τ existsin every state in Π from some point on, (2) τ turns acceptinginfinitely often, and (3) τ has a fixed name Ii . The argumentin [3] guarantees the existence of such a node τ with the firsttwo properties. The only complication comes from renaming.We have the situation that τ with name Ii exists in H m , but itis renamed to Ii ′ in the succeeding state H m + . This happenswhen the left sibling τ ′ , whose index label h ( τ ′ ) = h ( τ ), of τ in H m is removed from H m + . However, it can only happento τ finitely many times, as the left siblings with the sameindex labels of τ are finite and the new created siblings whoseindex labels are the same as τ will be placed to the right of τ . Therefore, τ is eventually assigned a fixed name Ii , whichprovide us the third property. ⇒ : Given an ω -word α = α α · · · ∈ L ( RT ), there exists anaccepting run Π = H H · · · of RT over α . We ought to showthat there is also an accepting run of S over α . Π is acceptingmeans that there exists an Ii ∈ I such that Π eventually nevervisits R Ii , but visits A Ii infinitely often. Since renamed nodesor deleted nodes are rejecting, all nodes named by Ii have tobe the same node. It follows that a node τ eventually staysin every state in a suffix of Π and τ turns accepting infinitelyoften. The rest of the proof is the same as the one in [3]. (cid:3) Theorem 7.
Given a NSA S with n states and k Streett pairs,we can construct a DRTA with n n ( n !) n states, O ( n n ) Rabinpairs for k = ω ( n ) and n n k nk states, O ( k nk ) Rabin pairs fork = O ( n ) that recognizes the language L ( S ) .Proof. For the state complexity, by Lemma 5, we can calculatethe number of structural ordered trees with state and indexlabels (i.e. µ -Safra trees for Streett determinization withoutnames, E and F ). According to the result in [5], [6], there are atmost n n structural ordered trees. For every structural orderedtree, there are at most n n possibilities of state labeling. Besides,the number of possibilities of index labeling is bounded by( n !) n for k = ω ( n ), and k nk for k = O ( n ). Thus, the state complexity is n n · n n · ( n !) n = n n ( n !) n for k = ω ( n ), and n n · n n · k nk = n n k nk for k = O ( n ).For the index complexity, we have that for any branch fromthe root to a leaf of a H-Safra tree, there are at most µ nodes,say τ , such that j ( τ ) ,
0. Moreover, a H-Safra tree containsat most n nodes, say τ , with j ( τ ) = n + µ nodes in a branch. The name of a node isdenoted by x y . x y . · · · . x y n + µ n + µ , where x i ∈ { , j , j , . . . , j µ } ( j m is obtained by Mini for 1 ≤ m ≤ µ ) and y i ∈ { , , . . . , n } . Thenumber of i such that x i = n . Thus, the numberof names is nn + µ ! · n n · ( µ !) n = O ( µ n µ ) . Since µ = min( n , k ), for k = ω ( n ), by replacing µ with n ,the index complexity O ( n n ) is obtained; for k = O ( n ), byreplacing µ with k , O ( k nk ) is obtained. (cid:3) B. Construction of LIR-H-Safra Trees for Streett Determiniza-tion
Fix a NSA S = ( Σ , Q , Q , δ, h G , B i k ). The initial LIR-H-Safra tree for Streett determinization LH I of S is H I with aLIR. The order of all nodes in the LIR follows the order anode is generated.Given a LIR-H-Safra tree LH of S and a σ ∈ Σ , we constructa new LIR-H-Safra tree ˆ LH , called the σ -successor of LH ,and the signature sig of the transition, also in six steps similarto the transformation from NSA to DRTA. The differencesare: (1) For a node τ in LH , if p ( τ ) changes during thetransformation, τ is rejecting; otherwise, τ is stable. (2) Thesignature is defined by sig = ( st , p ). If there is no acceptingor rejecting node, sig = ∅ . Otherwise, in the case ˆ τ is thenode with the minimal position in the LIR among acceptingor rejecting nodes in the transformation, it has p = p (ˆ τ ), st : = acc if ˆ τ is accepting, and st : = rej if ˆ τ is rejecting. Asa result, an equivalent DPTA PT = ( Σ , Q PT , Q PT , δ PT , λ PT )can be obtained. Here Q PT is the set of LIR-H-Safra trees forStreett determinization w.r.t S ; Q PT is the initial LIR-H-Safratree for Streett determinization; δ PT is a transition relation thatis established during the construction of LIR-H-Safra trees forStreett determinization, consisting of transitions (typically δ )which are quintuples of the form LH σ −−→ δ sig ˆ LH where δ sig isthe signature of the transition δ , with σ ranging over Σ , and LH ranging over Q PT ; λ RT = { λ , λ , · · · , λ n ( µ + , λ n ( µ + + } isthe parity acceptance condition. Notice that for each 1 ≤ i ≤ n ( µ + λ i : = { δ ∈ δ PT | δ sig = ( acc , i ) } λ i − : = { δ ∈ δ PT | δ sig = ( rej , i ) } λ n ( µ + + : = { δ ∈ δ PT | δ sig = ∅ or δ sig = ( rej , } Given an input ω -word α : ω → Σ , we call the sequence Π = ( LH , α (0) , LH ) ( LH , α (1) , LH )( LH , α (2) , LH ) . . . oftransitions such that LH = LH I , and for all i ∈ ω , LH i + isthe α ( i )-successor of LH i , the LIR-H-Safra Streett trace of theNSA S over α . We view the LIR-H-Safra Streett trace of S over α as the run of the DPTA PT over α . Then we say that8 is accepted by the DPTA if the minimal index k for which Inf ( Π ) ∩ λ k , ∅ is even.Let PT be the DPTA obtained from the given NSA S .Theorem 8 is formalized. Theorem 8. L ( PT ) = L ( S ) .Proof. As it has been proved that S is equivalent to the DRTA RT in Section V-A, we further prove this theorem by showing L ( PT ) = L ( RT ). ⇐ : Given an ω -word α ∈ L ( RT ), there is a node τ thatis accepting infinitely often and its name keeps unchangedeventually in the H-Safra Streett trace about α . It indicates thatthe position of τ in the LIR is non-increasing. Note that theposition of τ in the LIR decreases when a node ˆ τ at a smallerposition with h (ˆ τ ) , h ( τ ) is removed. However, this can onlyhappen for finitely many times. The node τ will eventuallyremain in the same position p in the LIR and every node τ ′ with p ( τ ′ ) ≤ p will be stable. Hence, no odd priority < p occurs infinitely often. And from that time onward, the node τ is accepting infinitely many times. Therefore, the smallestpriority occurring infinitely often is even. It indicates that α ∈ L ( PT ). ⇒ : Let α be an ω -word in L ( PT ). There is a LIR-H-SafraStreett trace Π and an index 2 i such that Inf ( Π ) ∩ λ i , ∅ and Inf ( Π ) ∩ λ k = ∅ for any k < i . It indicates that each node τ with p ( τ ) ≤ i remains stable in the LIR from a time onward.That is τ is not rejecting. Meanwhile, the node on position i is accepting infinitely often from that time onward. Thus α ∈ L ( RT ). (cid:3) Theorem 9.
Given a NSA S with n states and k Streett pairs,we can construct a DPTA with n ( n + − n !) n + = O ( n log n ) states, n ( n + priorities for k = ω ( n ) and n ( k + − n ! k nk = O ( nk log nk ) states, n ( k + prioritiesfor k = O ( n ) that recognizes the language L ( S ) .Proof. The number of nodes in a LIR-H-Safra tree is also atmost n ( µ + n ( µ + − t ( n , m ) denote thenumber of LIR-H-Safra trees without index labels, say ˜ LH ,such that there are m nodes in ˜ LH and n states in the statelabel of the root of ˜ LH . Fist, we have t ( n , n ( µ + = ( n ( µ + − n !. A conclusion has been proved in [10] that for every m ≤ n ( µ + t ( n , m − ≤ t ( n , m ). Hence, P n ( µ + i = t ( n , i ) ≤ n ( µ + − n !. If there are n ′ ( n ′ < n ) states labelled inthe root, the number of the LIR-H-Safra trees without indexlabels is 2( n ′ ( µ ′ + − n ′ ! (cid:16) n ′ n (cid:17) ≤ n ′ ( µ ′ + − n !, where µ ′ = min( n ′ , k ). Thus, the number of LIR-H-Safra trees withoutindex label is P nn ′ = n ′ ( µ ′ + − n ! ≤ n ( µ + − n !.By the result in [5], [6], the number of possibilities of indexlabeling is bounded by ( n !) n for k = ω ( n ), and k nk for k = O ( n ).It follows that the number of LIR-H-Safra trees is at most3( n ( n + − n !) n + = O ( n log n ) for k = ω ( n ) by replacing µ with n and 3( n ( k + − n ! k nk = O ( nk log nk ) for k = O ( n ) byreplacing µ with k . (cid:3) VI. L
OWER B OUND C OMPLEXITY
As for the state lower bound, it means the minimum statesrequired by the equivalent deterministic automata, regard-less of whether the acceptance condition is state-based ortransition-based. In this section, we prove a lower boundstate complexity for determinization construction from NSAto DR(T)A, which exactly matches the state complexity ofthe proposed determinization construction. Further, we putforward a lower bound state complexity for determinizationconstruction from NSA to DP(T)A, which is the same as thestate complexity of the proposed determinization constructionin the exponent.
A. L-Game
Definition 6 ( L -game [14]) . An L-game for two players, Adamand Eva, is a tuple G = ( V , V E , V A , p I , Σ , Move , L ), where • V is a set of positions which is partitioned into the positions for Eva V E and the positions for Adam V A , • p I ∈ V is the initial position of G , • Σ is the labelling alphabet , • Move ⊆ V × Σ × V is the set of possible moves , and • L ⊆ Σ ω is the winning condition .A tuple ( p , σ, p ′ ) ∈ Move indicates that there is a movefrom p to p ′ , which produces a letter σ . A play is a maximalsequence π = ( p , σ , p , σ , p , σ , . . . ) such that p = p I ,and for each i , ( p i , σ i , p i + ) ∈ Move. The player who be-longs to the current position will choose the next move. Let π Σ = ( σ , σ , σ , . . . ). If π Σ ∈ L , Eva wins the play. Otherwise,Adam wins the play.A strategy for the player X is a function which tells theplayer what move he should choose depending on the finitehistory of moves played so far. A strategy is called a winning strategy for Eva (resp. Adam), if Eva (resp. Adam) wins everyplay with this strategy. A strategy with memory m for Eva isdescribed as ( M , update, choice, init), in which M is a setof memory with the size being m , update is a mapping from M × Move to M , choice is a mapping from V E × M to Move,and init ∈ M . A player X wins a game with memory m if ithas a winning strategy with memory m .The following Lemma proved in [14] provides an argumentfor proving lower bounds on determinization problems. Lemma 10.
If Eva wins an L -game, and requires memory m for that, then every deterministic Rabin automaton for L hasstates at least m [14]. B. Lower Bound State Complexity for NSA to DR(T)A
Inspired by the approach in [14], in order to prove the lowerbound state complexity for the determinization constructionfrom NSA to DR(T)A, the essence is to define full Streettautomata and the relevant game.For convenience, we first introduce some notations. For atree T , every node τ ∈ T can be expressed by a sequence se ( τ ) = se ( τ )(0) se ( τ )(1) se ( τ )(2) · · · , where se ( τ )( i ) ( i ≥
0) is apositive integer. For the root τ r of T , we have se ( τ r ) =
1. Asfor any other node τ , se ( τ ) = se ( τ p ) i , where τ p is the parent of9 and i = + |{ τ ′ ∈ T | τ ′ is the left sibling of τ }| . For any twonodes τ and τ ′ , we define τ < lex τ ′ if se ( τ ) is the proper prefixof se ( τ ′ ); or there exists i such that se ( τ )( i ) < se ( τ ′ )( i ) and forall j < i , se ( τ )( j ) = se ( τ ′ )( j ). Further, τ ≤ lex τ ′ if τ < lex τ ′ or se ( τ ) = se ( τ ′ ). Definition 7 (Full Streett Automata) . A full Streett automaton is a quintuple ( Q , Σ , Q , δ, h G , B i [ k ] ) where Q is a finite setof states, Q ⊆ Q is a set of initial states, Σ = P ( Q ×{∅ , G , . . . , G k , B , . . . , B k } × Q ) is the alphabet, and the transi-tion relation is defined by δ ⊆ Q × Σ × Q . h G , B i [ k ] are Streettpairs, where k is a positive integer, and G i and B i are setsof transitions for 1 ≤ i ≤ k . For a Streett pair h G i , B i i anda letter σ ∈ Σ , a transition δ = ( p , σ, q ) ∈ G i (or B i ) iff( p , G i (or B i ) , q ) ∈ σ .For the full Streett automaton with n states S n = ( Q , Σ , Q , δ, h G , B i [ k ] ), where Q is also the set of initial states, and L ( S n ) = L n . A DRTA RT = ( Q RT , Σ , Q RT , δ RT , λ RT ) can be constructedvia H-Safra trees for Streett determinization .We introduce some useful notations. For a set of states S ⊆ Q , let Σ S be the set of letters σ ∈ Σ such that S q ∈ S δ ( q , σ ) = S .We also let L Sn = L n ∩ Σ ω S and Q SRT = { H ∈ Q RT : l ( τ r ) = S where τ r is the root of H } . Thus, for all words u ∈ Σ ∗ S andall H ∈ Q SRT , we have δ RT ( H , u ) ∈ Q SRT .Given a set of states S ⊆ Q , we define a L Sn - S -game G S suchthat Eva wins G S but she cannot win with memory less than | Q SRT | . This indicates that any determinization Rabin automatonaccepting L Sn has at least | Q SRT | states. Definition 8 ( L Sn - S -game) . The L Sn - S -game is a tuple G S = ( V , V E , V A , p I , Σ + S , Move , L Sn ), where V E is a singleton set { p E } and V A consists of the initial position p I and one position p H for each H-Safra tree H ∈ Q SRT . The Move of G S includes: • ( p I , u , p E ), u is a non- ǫ word in Σ + S . • ( p E , ǫ, p H ), for each H-Safra tree H in Q SRT . • ( p H , u , p E ), if there exists a node ˆ τ in ˆ H = δ RT ( H , u ) thatsatisfies one of the three following conditions during thetransformation from H to ˆ H :1) ˆ τ is accepting, and for all ˆ τ ′ ≤ lex ˆ τ in ˆ H , ˆ τ ′ is notrejecting, h (ˆ τ ′ ) = h ( τ ′ ) and l (ˆ τ ′ ) = l ( τ ′ ),2) j (ˆ τ ) < j ( τ ), and for all ˆ τ ′ < lex ˆ τ in ˆ H , ˆ τ ′ is notrejecting, h (ˆ τ ′ ) = h ( τ ′ ), and l (ˆ τ ′ ) = l ( τ ′ ),3) j (ˆ τ ) = j ( τ ), l (ˆ τ ) ⊃ l ( τ ), and for all ˆ τ ′ < lex ˆ τ in ˆ H , ˆ τ ′ is not rejecting, h (ˆ τ ′ ) = h ( τ ′ ) and l (ˆ τ ′ ) = l ( τ ′ ),for each H-Safra tree H in Q SRT and a word u ∈ Σ + S . Notethat τ and τ ′ are the nodes in H with se ( τ ) = se (ˆ τ ) and se ( τ ′ ) = se (ˆ τ ′ ), respectively.The L Sn - S -game has a flower shape, which is intuitivelyillustrated in Fig. 13. The central position is controlled byEva and the petals belong to Adam. Moreover, each petalcorresponds to a H-Safra tree . Lemma 11.
Eva has a winning strategy in G S . Proof.
There is a winning strategy for Eva: if a word u wasproduced after a finite play and Eva is to make a move from p E p H p I p H p H p H p H ... uεu Fig. 13. The L Sn - S -game G S p E , then she chooses to go to a position indexed by δ RT ( H , u )where H = Q RT .To see that Eva wins the L Sn - S -game G S with this strategy,we consider the run ρ RT of RT on the word defined by theplay ( p I , u , p E )( p E , ε, p H )( p H , u , p E )( p E , ε, p H ) · · · , whichrefers to the word u u u · · · . Each segment ρ RT ( H i , u i , H i + )( i ≥
1) of the run ρ RT and a corresponding node τ i ∈ H i + satisfie one of the conditions 1, 2 and 3 in Definition 8. Wedenote τ = ≤ lex -min { τ i | τ i occurs infinitely often } , then each τ ′ , such that τ ′ ≤ lex τ , is not rejecting in each segment of ρ RT .Obviously, if τ is infinitely often accepting, then Eva wins.Assume that there is a position in ρ RT such that τ is notaccepting, but the value of j ( τ ) becomes smaller infinitelyoften from the position onwards. However, this can onlyhappen finitely often since j ( τ ) has the minimal value 0, whichis a contradiction.Also, assume that from some position in ρ RT onwards, τ is not accepting and the index label remains constant.Nevertheless, the state label l ( τ ) would grow monotonouslyand would infinitely often grow strictly. It can only happenfinitely many times since l ( τ ) ⊆ l ( τ p ), which is a contradiction.Therefore, Eva wins G S with this strategy. (cid:3) Next, for each H-Safra tree H ∈ Q SRT , a game G SH is defined,which is a modification of G S by removing the position p H of Adam and the corresponding moves. For this game, thefollowing Lemma holds. Lemma 12.
For any two H-Safra trees H , H ′ in Q SRT ,there exists a word u such that ( p H ′ , u , p E ) is a move in G SH , δ RT ( H ′ , u ) = δ RT ( H , u ) = H , and for any node τ in H , τ is notaccepting. Proof.
This lemma requires an analysis of the differencesbetween the two H-Safra trees H and H ′ . For the ≤ lex -minimalnodes τ in H and τ ′ in H ′ where se ( τ ) = se ( τ ′ ), but l ( τ ) , l ( τ ′ )or h ( τ ) , h ( τ ′ ), a letter σ is defined first which has thefollowing two cases, denoted as σ ′ and σ ′′ , respectively.(i) If τ and τ ′ are the left most child of their parents τ p and τ ′ p , respectively, σ ′ is produced such that { ( s , ∅ , s p ) | s ∈ l ( τ ) ∪ l ( τ ′ ) and s p ∈ l ( τ p ) } ⊆ σ ′ .10ii) If τ and τ ′ have left siblings τ l and τ ′ l , respectively, it isapparent that l ( τ l ) = l ( τ ′ l ) and h ( τ l ) = h ( τ ′ l ). Then we construct σ ′′ such that { ( s , ∅ , s p ) | s ∈ l ( τ ) ∪ l ( τ ′ ) and s p ∈ l ( τ p ) \ l ( τ l ) } ⊆ σ ′′ .For these two cases, after reading σ at H and H ′ , we have l ( τ ) = l ( τ ′ ). Every node ˆ τ < lex τ in H and ˆ τ ′ < lex τ ′ in H ′ remain unchanged. Meanwhile, for each node τ m > lex τ in H and τ ′ m > lex τ ′ in H ′ , we have l ( τ m ) = ∅ and l ( τ ′ m ) = ∅ .Next, for two different nodes τ and τ ′ , there are four casesto be considered:(1) j ( τ ) > j ( τ ′ ). In the case that τ and τ ′ are theleft most child of their parents τ p and τ ′ p , respectively, let w = σ j ( τ ′ ) σ j ( τ ′ ) − · · · σ . Here, for each 1 ≤ k ≤ j ( τ ′ ), σ k = { ( s , G k , s ) | s ∈ l ( τ ′ ) } . By reading σ ′ w , H and H ′ canreach ˆ H and ˆ H ′ , respectively. The parent of τ ′ is acceptingand τ stays unchanged. In the case that τ and τ ′ haveleft siblings τ l and τ ′ l , respectively, it has l ( τ l ) = l ( τ ′ l ) and h ( τ l ) = h ( τ ′ l ). Let s be a state in l ( τ l ). We construct a word w = σ j ( τ l ) σ j ( τ l ) − · · · σ j ( τ ) + . Here, for each j ( τ ) + ≤ k ≤ j ( τ l ),( s , G k , s ) ∈ σ k . By reading σ ′′ w , a new node τ s is createdas the sibling of τ l with l ( τ s ) = { s } , h ( τ s ) = h ( τ ), and τ ′ s iscreated as the sibling of τ ′ l with l ( τ ′ s ) = { s } , h ( τ ′ s ) = h ( τ ). Then τ s and τ ′ s are accepting in the next transformation. Later, let l ( τ ) = l ( τ ′ ) = ∅ and remove τ and τ ′ , which makes τ s renamed(rejected), and τ ′ s not rejected. After the above operations, ˆ H and ˆ H ′ are obtained, respectively.(2) j ( τ ) < j ( τ ′ ). Construct a word w = σ j ( τ ′ ) σ j ( τ ′ ) − · · · σ j ( τ ) + . Here, for each j ( τ ) + ≤ k ≤ j ( τ ′ ), it has σ k = { ( s , G k , s ) | s ∈ l ( τ ′ ) } . By reading σ w , H and H ′ can reachˆ H and ˆ H ′ , respectively.(3) j ( τ ) = j ( τ ′ ) and l ( τ ) ⊃ l ( τ ′ ). After reading σ at H and H ′ , ˆ H and ˆ H ′ are obtained, respectively.(4) j ( τ ) = j ( τ ′ ) and l ( τ ′ ) \ l ( τ ) , ∅ . We first construct aword w , which makes τ ′ being accepting after reading w at H ′ . Then construct a letter ˆ σ such that ( s , B j ( τ ) , s ) ∈ ˆ σ for eachstate s ∈ l ( τ ). As a consequence, by reading w ˆ σ at H and H ′ , τ becomes rejected and τ ′ is accepting. Furthermore, ˆ H andˆ H ′ are obtained, respectively.For the four cases, the next transformation makes both ˆ H and ˆ H ′ move to H .Therefore, in the transformation from H ′ to H , (1) and (4)satisfy condition 1) of Definition 8. What is more, (2) and(3) satisfy condition 2) and 3), respectively. Meanwhile, thereexists no accepting node during the transformation from H to H . (cid:3) Further, by Lemma 12, the following lemma is obtained.
Lemma 13.
For every H-Safra tree H in Q SRT , Adam has awinning strategy in the correspongding G SH . Proof.
There is a winning strategy for Adam as follows. Whenhe plays a word u from p I such that δ RT ( H , u ) where H = Q RT , the best choice for Eva is to move to p H on the basisof the proof of Lemma 11. However, this position has beenremoved, she is forced to move to another position p H ′ ( H ′ , H ). Then Adam moves according to Lemma 12, and he can always answer to the proposal of Eva similarly in the play.Meanwhile, an infinite word α is produced. It is obvious that RT does not accept α because of Lemma 12. Therefore, Adamhas a winning strategy in G SH . (cid:3) Then, it is easy to infer the following lemma.
Lemma 14.
Eva has no winning strategy with memory lessthan | Q SRT | in G S . Proof.
For a contradiction, we suppose that Eva has a winningstrategy with memory | Q SRT |−
1. Then there would be a position p H which is never visited by this strategy. It is a contradictionwith Lemma 13. (cid:3) Similar to the approach in [14], the main theorem is readyto be proved.
Theorem 15.
Every DR(T)A accepting L ( S n ) has states atleast | Q RT | = n n ( n !) n for k = ω ( n ) and n n k nk for k = O ( n ) . This theorem means that the proved lower bound statecomplexity for the determinization construction from NSA toDR(T)A exactly matches the state complexity of the proposeddeterminization construction by H-Safra trees.
C. Lower Bound State Complexity for NSA to DP(T)A
To prove the lower bound state complexity for determiniza-tion construction from NSA to DP(T)A, an appropriate L -game, for recognizing the complement language of the NSA,is constructed first.For the full Streett automaton S n = ( Q , Σ , Q , δ, h G , B i [ k ] ), aDPTA PT = ( Q PT , Σ , Q PT , δ PT , λ PT ) can be constructed via LIT-H-Safra trees . Let L cn be the complement of L ( S n ), Σ ω S denote the infinite words over Σ S , and L cSn = L cn ∩ Σ ω S . For any S ⊆ Q , let Q SPT = { LH ∈ Q PT : l ( ǫ ) = S where ǫ is the root of LH } be the set of LIR-H-Safra trees in which state label ofthe root is S . We choose a subset Q S hPT of Q SPT , which satisfies:For any two LIR-H-Safra trees LH , LH ′ ∈ Q S hPT and any nodes τ in LH , τ ′ in LH ′ , if se ( τ ) = se ( τ ′ ), then h ( τ ) = h ( τ ′ ).Given a set of states S ⊆ Q , we define a L cSn - S -game G cS such that Eva wins G cS but she cannot win with memory lessthan | Q S hPT | . Definition 9 ( L cSn - S -game) . The L cSn - S -game is a tuple G cS = ( V , V E , V A , p I , Σ + S , Move , L cSn ), where V E is a singleton set { p E } and V A consists of the initial position p I and one position p LH for each LIR-H-Safra tree LH ∈ Q S hPT . The Move of G cS includes: • ( p I , u , p E ), u is a non- ǫ word in Σ + S . • ( p E , ǫ, p LH ), for each LIR-H-Safra tree LH in Q S hPT . • ( p LH , u , p E ), if there exists a node τ in LH with p ( τ ) = i and τ satisfies one of the two following conditions in thetransition from LH to ˆ LH = δ PT ( LH , u ):1) τ is rejecting and the priority of the transition is 2 i − τ ′ in LH such that p ( τ ′ ) < p ( τ ), it requiresthat l (ˆ τ ′ ) = l ( τ ′ ) and h (ˆ τ ′ ) = h ( τ ′ ),2) h (ˆ τ ) = h ( τ ), l (ˆ τ ) ⊂ l ( τ ), and the priority of thetransition is larger than 2 i , and for each τ ′ in LH p ( τ ′ ) < p ( τ ), it requires that l (ˆ τ ′ ) = l ( τ ′ )and h (ˆ τ ′ ) = h ( τ ′ ),for each LIR-H-Safra tree LH in Q S hPT and a word u ∈ Σ + S .Note that ˆ τ and ˆ τ ′ are nodes in ˆ LH with p (ˆ τ ) = p ( τ ) and p (ˆ τ ′ ) = p ( τ ′ ), respectively. Lemma 16.
Eva has a winning strategy in G cS . Proof.
There is a winning strategy for Eva: if a word u wasproduced after a finite play and Eva is to make a movefrom p E , then she chooses to go to a position indexed by δ PT ( LH , u ) where LH = Q PT .To see that Eva wins the L cSn - S -game G cS with this strategy,we consider the run ρ PT of PT on the word defined by theplay ( p I , u , p E )( p E , ǫ, p LH )( p LH , u , p E )( p E , ǫ, p LH ) · · · ,which refers to the word u u u · · · . Each segment ρ PT ( LH k , u k , LH k + ) ( k ≥
1) of the run ρ PT satisfies one of the conditions1), 2) and 3) in Definition 9, and there exists a node τ k ∈ LH k with p ( τ k ) = i k . Let i min be the minimal one that occursinfinitely often among these i k and τ min be the node on position i min in the LIR. Hence, no priority smaller than 2 i min − ρ PT . It is obvious that if τ min isinfinitely often rejecting, then the minimal priority occurringinfinitely often is 2 i min − ρ PT , and Eva wins.Next, we assume that there is a position po of ρ PT such that τ min is not rejecting, but satisfies condition 2) in Definition 9infinitely often from the position po onwards. Consequently,the state label of τ min would reduce monotonously from theposition po onwards, and would infinitely often reduce strictly.It is a contradiction.Therefore, Eva wins G cS with this strategy. (cid:3) Similar to the lower bound state complexity from NSAto DR(T)A in Section VI-B, for each LIR-H-Safra tree LH ∈ Q S hPT , a game G cSLH can be defined by removing thecorresponding position p LH and the relevant moves from G cS .The following lemma shows that Adam has a winning strategyin G cSLH . Lemma 17.
For any two LIR-H-Safra trees LH , LH ′ in Q S hPT ,there exists a word u such that ( p LH ′ , u , p E ) is a move in G cSLH , δ PT ( LH ′ , u ) = δ PT ( LH , u ) = LH , and the minimal priority inthe transitions from LH to LH after reading u is even. Proof.
We first identify the position-minimal nodes τ in LH and τ ′ in LH ′ such that p ( τ ) = p ( τ ′ ) = i , and se ( τ ) , se ( τ ′ )or l ( τ ) , l ( τ ′ ). We use W to denote a set of words such thatfor each w ∈ W , τ is accepting and after reading w at LH , thepriority is 2 i . Then two cases are considered:(1) se ( τ ) = se ( τ ′ ). It has h ( τ ) = h ( τ ′ ). The only differencebetween τ and τ ′ is the state labels. In the case that l ( τ ) \ l ( τ ′ ) , ∅ , a word w ∈ W is read at LH and LH ′ . Let σ bea letter such that ( s , B j ( τ ′ ) , s ) ∈ σ , where s ∈ l ( τ ′ ). By reading w σ , LH and LH ′ can reach ˆ LH and ˆ LH ′ , respectively. In thetransformation from LH to ˆ LH , τ is accepting and the priorityis 2 i . Meanwhile, τ ′ is rejecting and the priority is 2 i − LH ′ to ˆ LH ′ . The next transformationmakes both ˆ LH and ˆ LH ′ move to LH . In the case that l ( τ ) ⊂ l ( τ ′ ), let w be a word in W such that τ ′ is not acceptingor rejecting after reading w at LH ′ . As a result, ˆ LH andˆ LH ′ are obtained. The next transformation makes both ˆ LH and ˆ LH ′ move to LH and the priority is larger than 2 i in thetransformation from ˆ LH ′ to LH .(2) se ( τ ) , se ( τ ′ ). Let w be a word in W such that τ ′ is rejecting and after reading w at LH ′ , the priority is 2 i −
1. Then, ˆ LH and ˆ LH ′ are obtained, respectively. The nexttransformation makes both LH and LH ′ move to LH .As a result, in the transformation from LH ′ to LH , thefirst case of (1) and (2) satisfy condition 1) of Definition9. The second case of (1) satisfies condition 2). Meanwhile,the minimal priority is 2 i in the transformation from LH to LH . (cid:3) Thus, Eva has no winning strategy with memory less than | Q S hPT | in G cS . Based on the approach in [14] and Lemma 10,we can obtain the following result. Lemma 18.
Every DR(T)A that recognises the complementof L ( S n ) must contain at least | S S ⊆ Q Q S hPT | states.In [10], there is a result that the size of the smallest Rabinautomaton that recognises the complement of L ( S n ) is equalto the one of the smallest Streett automaton that recognises L ( S n ). Since parity automata are special Streett automata, themain theorem is inferred. Theorem 19.
Every DS(T)A or DP(T)A accepting L ( S n ) musthave states at least | S S ⊆ Q Q S hPT | = Ω ( n log n ) for k = ω ( n ) and Ω ( nk log nk ) for k = O ( n ) . Finally, we give the estimate for | S S ⊆ Q Q S hPT | . Since the indexlabel of each node is fixed, we can neglect the impact of theindex label. Therefore, by the proof of Theorem 9, we have | [ S ⊆ Q Q S hPT | = n ( µ + − n ! . Specifically, | S S ⊆ Q Q S hPT | = n ( n + − n ! = Ω ( n log n ) for k = ω ( n ) by replacing µ with n , and 3( n ( k + − n ! = Ω ( nk log nk ) for k = O ( n ) by replacing µ with k .By the result in Section V-B, the state complexity for theconstruction from NSA to DPTA is 3( n ( n + − n !) n + = O ( n log n ) for k = ω ( n ) and 3( n ( k + − n ! k nk = O ( nk log nk ) for k = O ( n ). So, the above lower bound is the same as the upperbound in the exponent. There is still a slight gap between thelower and upper bounds.VII. C ONCLUSION
In this paper, we present determinization transformationsfrom NSA with n states and k Streett pairs to DRTA with n n ( n !) n states, O ( n n ) Rabin pairs for k = ω ( n ) and n n k nk states, O ( k nk ) Rabin pairs for k = O ( n ); and to DPTAwith 3( n ( n + − n !) n + states, 2 n ( n +
1) priorities for k = ω ( n ) and 3( n ( k + − n ! k nk states, 2 n ( k +
1) priorities for k = O ( n ). Further, we prove a lower bound state complexity12or determinization construction from NSA to DR(T)A, whichmatches the state complexity of the proposed determinizationconstruction. Also, we put forward a lower bound state com-plexity for determinization construction from NSA to DP(T)Awhich is the same as the proposed determinization constructionin the exponent.In the near future, we will implement the proposed deter-minization constructions and evaluate efficiency of the algo-rithms in practice. R EFERENCES[1] R.S.Streett. Propositional dynamic logic of looping and converse.
Infor-mation and Control , 54:121-141, 1982.[2] J. R. B¨uchi. On a decision method in restricted second order arithmetic.In
Proceedings of the International Congress on Logic, Method, andPhilosophy of Science , pages 1-12. Stanford University Press, 1962.[3] Safra, S.: Exponential Determinization for omega-Automata with Strong-Fairness Acceptance Condition (Extended Abstract). STOC 1992: 275-282.[4] S. Safra, M. Y. Vardi, On ω -automata and temporal logic, in: Proceedingsof the 21st annual ACM symposium on Theory of computing (STOC’89),ACM, 1989, pp. 127-137.[5] Y. Cai, T. Zhang: Can nondeterminism help complementation? In Gan-dALF, pages 57-70, 2012.[6] Y. Cai, T. Zhang: Determinization complexities of ω automata, Technicalreport (2013), http://theory.stanford.edu/ ∼ tingz/tcs.pdf[7] Y. Cai, T. Zhang: Tight Upper Bounds for Streett and Parity Complemen-tation. Proceedings of the 20th Conference on Computer Science Logic(CSL 2011), Dagstuhl Publishing, 2011: 112-128.[8] Y. Cai, T. Zhang: A tight lower bound for Streett complementation.FSTTCS 2011: 339-350.[9] Piterman, N.: From nondeterministic B¨uchi and Streett automata todeterministic parity automata. Journal of Logical Methods in ComputerScience ω -automata. In Proc. 19th Conf. on Foundations of Software Technology and TheoreticalComputer Science , volume 1738 of
Lecture Notes in Computer Science ,pages 97-109, 1999.[14] Thomas Colcombet, Konrad Zdanowski: A Tight Lower Bound forDeterminization of Transition Labeled B¨uchi Automata. ICALP 2009:151-162[15] Qiqi Yan. Lower bounds for complementation of omega-automata viathe full automata technique. ICALP 2006: 589-600.,pages 97-109, 1999.[14] Thomas Colcombet, Konrad Zdanowski: A Tight Lower Bound forDeterminization of Transition Labeled B¨uchi Automata. ICALP 2009:151-162[15] Qiqi Yan. Lower bounds for complementation of omega-automata viathe full automata technique. ICALP 2006: 589-600.