The Index and Wadge Problems For Tree Languages Alessandro Facchini - PowerPoint PPT Presentation

Pattern method

Pattern method

Pattern method

Pattern method

Sketch of the proof of Wagner’s theorem

Sketch of the proof of Wagner’s theorem d W ( L ) < ω ω Σ 1 1 n Π 1 . n . . • for every deterministic B¨ uchi automaton A , L ( A ) ∈ Π 0 2 • by McNaughton’s theorem, every regular language is in BC( Π 0 2 ). Σ 1 1 Π 1 1 BC ( Π 0 2 )

Sketch of the proof of Wagner’s theorem 1. For each ordinal α = ω ` k 1 n k + · · · + ω ` 0 1 n 0 < ω ! 1 construct a canonical automata of Wadge degree α . Do it by providing canonical patterns for • the degrees ω n 1 , with n ≥ 0, • operations on automata corresponding to ordinal sum 2. prove that the class of canonical automata is closed under the defined operations 3. prove that canonical automata represent the ≡ W -classes of all deterministically recognizable languages

Sketch of the proof of Wagner’s theorem L ( i,k ) For every index ( i, k ) let L ( i,k ) := { w ∈ { i, . . . , k } ω | lim sup w ( i ) is even. } i →∞ Fact . There is a deterministic automaton of index ( i, k ) recognizing L ( i,k ) . A paradigmatic class of languages

Sketch of the proof of Wagner’s theorem L ( i,k ) Proposition . The followings hold: 1. the languages L ( i,k ) form a strict hierarchy w.r.t. ≤ W L (0 , 0) L (0 , 2) L (0 , 1) L (0 , 3) ... L (1 , 1) L (1 , 2) L (1 , 3) L (1 , 4) 2. for every k ≥ 0, d W ( L (0 ,k ) ) = d W ( L (1 ,k +1) ) = ω k 1 3. If A has index ( i, k ), then L ( A ) ≤ W L ( i,k ) . A paradigmatic class of languages

Sketch of the proof of Wagner’s theorem L ( i,k ) Corollary . Let L be a regular language. If L ( i,k ) ≤ W L , then L is not in ( i, k ). Proof . Assume L ∈ ( i, k ). Then L ≤ W L ( i,k ) . Since L ( i,k ) ≤ W L , it holds that L ( i,k ) ≤ W L ( i,k ) . Contradiction. A paradigmatic class of languages

Sketch of the proof of Wagner’s theorem F ( i,k ) ( ? if i = k = 1 ι = > else. j a ` a j ι i a i i < j a ` 6 = a j i ≤ k j ≤ k (i,k)-flower

Sketch of the proof of Wagner’s theorem F ( i,k ) Proposition . For every ( i, k ), F ( i,k ) ≡ W L ( i,k ) . Thence, F ( i,k ) has index ( i, k ). (i,k)-flower

Sketch of the proof of Wagner’s theorem A ∨ B q I τ / ∈ { a, b } b a q A q B I I ( ? if L ( A ) = L ( B ) = ; ι = > else. sum

Sketch of the proof of Wagner’s theorem A ⊕ B s τ 6 = b b q B I n A := A ⊕ · · · ⊕ A | {z } n times sequential composition

Sketch of the proof of Wagner’s theorem Canonical automata • for 1 < α < ω , • for 0 < k < ω , • C 1 = F (0 , 0) , C α = C 1 ⊕ ( α − 1) E 1 , C ω k = F (0 ,k ) , D 1 = F (1 , 1) , D α = D 1 ⊕ ( α − 1) E 1 , D ω k = F (1 ,k +1) , E 1 = C 1 ∨ D 1 , E α = α E 1 E k = C k ∨ D k , • given ↵ = ! ` k n k + · · · + ! ` 0 n 0 < ! ! , with ! > ` k > 0, ` k > · · · > ` 0 , and 0 < n i < ! : – if ` 0 = 0: C ↵ = C n 0 ⊕ n 1 E ! ` 1 ⊕ · · · ⊕ n k E ! ` k , D ↵ = D n 0 ⊕ n 1 E ! ` 1 ⊕ · · · ⊕ n k E ! ` k , E ↵ = E n 0 ⊕ n 1 E ! ` 1 ⊕ · · · ⊕ n k E ! ` k – else: C ↵ = C ! ` 0 ⊕ ( n 0 − 1) E ! ` 0 ⊕ n 1 E ! ` 1 ⊕ · · · ⊕ n k E ! ` k , D ↵ = D ! ` 0 ⊕ ( n 0 − 1) E ! ` 0 ⊕ n 1 E ! ` 1 ⊕ · · · ⊕ n k E ! ` k , E ↵ = E ! ` 0 ⊕ ( n 0 − 1) E ! ` 0 ⊕ n 1 E ! ` 1 ⊕ · · · ⊕ n k E ! ` k

Sketch of the proof of Wagner’s theorem Theorem . Given ↵ = ! ` k n k + · · · + ! ` 0 n 0 < ! ! , with ! > ` k > 0, ` k > · · · > ` 0 , and 0 < n i < ! : d W ( C ↵ ) = d W ( D ↵ ) = d W ( E ↵ ) = ω ` k 1 n k + · · · + ω ` 0 1 n 0 and C α C α +1 E α D α +1 D α

Sketch of the proof of Wagner’s theorem Theorem . The following holds: 1. the class of canonical automata is closed under sum and sequential composition, 2. for every automata A , (a) one can e ff ectively construct a Wadge equivalent ca- nonical one, (b) L ( A ) ≤ W L ( i,k ) i ff L ( A ) ∈ ( i, k )

Sketch of the proof of Wagner’s theorem Proof . 1. Calculate. 2. (a) define an algorithm, first normalizing priorities, then checking for pattern (flower and sum) (b) Clearly if L ( A ) ∈ ( i, k ) then L ( A ) ≤ W L ( i,k ) . For the other direction, as a corollary of the pre- vious point it holds that Fact . L ( A ) ≤ W L ( i,k ) i ff A does not contain an ( i, k )-flower (i,k)-flower

The landscape of infinite words The case of regular languages of infinite words Nondeterministic Deterministic Borel hierarchy Wadge hierarchy index hierarchy index hierarchy BC( Π 0 (1,2) Strict 2 ) ω ω Decidable Decidable Decidable Decidable

What about regular languages of infinite trees?

What about regular languages of infinite trees? In general: - we just know that hierarchies are strict, - (almost) no decidability result (except for very low levels)

Parity tree automata ( Σ , Q, q I ∈ Q, δ , rank : Q → N ) deterministic: δ : Q × Σ → Q × Q nondeterministic: δ : Q × Σ → ℘ ( Q × Q ) alternating: δ : Q × Σ → B + ( { ε , 0 , 1 } × Q ) strong parity condition: no restriction weak parity condition: restriction if q reachable from r , then rank ( q ) ≤ rank ( r )

Parity tree automata deterministic automaton

Parity tree automata deterministic automaton

Parity tree automata nondeterministic automaton

Parity tree automata nondeterministic automaton

Parity tree automata alternating automaton

Parity tree automata alternating automaton

Parity games 1 3 0 5 6 2

Parity games 3 1 3 0 5 6 2

Parity games 30 1 3 0 5 6 2

Parity games 305 1 3 0 5 6 2

Parity games 3055.....5 1 3 0 5 6 2

Parity games 3055.....56 1 3 0 5 6 2

Parity games 3055.....56.... ∈ { 0 , . . . , 6 } ω 1 3 0 5 6 2

Parity games Player ♦ wins i ff the greatest priority occurring infinitely often is even 1 3 0 5 6 2

Parity games Determinacy: For every node, one of the two players has a winning strategy Positional determinacy: For every node, one of the two players has a positional winning strategy

Parity games 1 3 0 5 6 2

Parity games Theorem [Emerson-Jutla / Mostowski]. Parity games are positional determined

Deterministic languages K = { t ∈ T { 0 , 1 } | on each branch there are finitely many nodes labelled by 1 } • { 0 , 1 } K is Π 1 1 -complete. • q I := 0 • ( i, j ) 7! ( j, j ) • rank (i)=i

Deterministic languages K { is Σ 1 1 -complete. Deterministic ?

Game languages W (1 , 3) 2 t ∈ T Σ , where Σ = { ⌃ , ⇤ } × { 1 , 2 , 3 } 3 1 2 1 2 3 . . . . . . . . . . . . . . . . t ∈ W (1 , 3) i ff ♦ has a w.s. in the induced parity game

Game languages W ] (1 , 3) 1 t ∈ T Σ , where Σ = { ⌃ , ⇤ } × { 1 , 2 , 3 } 2 1 3 1 2 3 . . . . . . . . . . . . . . . . t ∈ W ] (1 , 3) i ff t ( w ) | 2 ≤ min { t ( w 0) | 2 , t ( w 1) | 2 } , for each w ∈ dom( t ), and ♦ has a w.s. in the induced parity game

Game languages Theorem [Arnold, Niwinski (2008)]. The game languages form a hierarchy with respect to Wadge reducibility, i.e. W (0 , 0) W (0 , 1) W (0 , 2) W (0 , 3) ... W (1 , 1) W (1 , 4) W (1 , 2) W (1 , 3) W ] W ] W ] W ] (0 , 1) (0 , 2) (0 , 0) (0 , 3) ... W ] W ] W ] W ] (1 , 1) (1 , 2) (1 , 4) (1 , 3) Moreover, W (0 , 1) is Π 1 1 -complete.

Game languages Theorem . For every index ( i, k ), W ] ( i,k ) , W ( i,k ) ∈ ( i, k ) and • if A is weak alternating of index ( i, k ), then L ( A ) ≤ W ] ( i,k ) , • if A is strong alternating of index ( i, k ), then L ( A ) ≤ W ( i,k ) Proof . Reductions are given by encoding the arena (run).

Game languages W (1 , 3) 2 t ∈ T Σ , where Σ = { ⌃ , ⇤ } × { 1 , 2 , 3 } 3 1 2 1 2 3 . . . . . . . . . . • { 1 , 2 , 3 } . . . . . . • q I := 1 • i → ( ⌃ ,j ) (0 , j ) ∨ (1 , j ) i → ( ⇤ ,j ) (0 , j ) ∧ (1 , j ) • rank (i)=i ‘game languages are alternating’

Game languages W (1 , 3) 2 t ∈ T Σ , where Σ = { ⌃ , ⇤ } × { 1 , 2 , 3 } 3 1 2 1 2 3 . . . . . . . . • { 1 , 2 , 3 , > } . . . . . . . . • q I := 1 • i ! ( ⌃ ,j ) { ( > , j ) , ( j, > ) } i ! ( ⇤ ,j ) ( j, j ) • rank (i)=i ‘game languages are nondeterministic’

Game languages W ] (1 , 3) 1 t ∈ T Σ , where Σ = { ⌃ , ⇤ } × { 1 , 2 , 3 } 2 1 • { 1 , 2 , 3 , ⊥ } • q I := 1 3 1 2 3 ( (0 , j ) ∨ (1 , j ) i ≤ j • i → ( ⌃ ,j ) . . . . . . . ⊥ i > j . . . . . . . . . ( (0 , j ) ∧ (1 , j ) i ≤ j i → ( ⇤ ,j ) ⊥ i > j • rank (i)=i ‘weak game languages are weak alternating’

Game languages Theorem [Bradfield (1998), Arnold (2002)]. The index and weak index hierarchies are strict over infinite trees. Proof . Assume it collapses to ( i, k ). Then there is A ∈ ( i, k + 1) such that L ( A ) = W ( i,k +1) . Thus W ( i,k +1) ≤ W W ( i,k ) . Contradiction.