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

The Index and Wadge Problems For Tree Languages

Alessandro Facchini (IDSIA, Lugano) Based on joint works with:

• J. Duparc (U. Lausanne)
• H. Michalewski, F. Murlak, M. Skrzypczak (U. Warsaw)

Workshop on Wadge Theory and Automata, 28.01.2015, Torino

The problem

Understand a formalism L, used to specify properties of a class C of structures

The problem

FO:

Understand a formalism L, used to specify properties of a class C of structures

MSO: ∃X

• ∀x(first(x) → Xx)

∧ ∀x(last(x) → ¬Xx) ∧ ∀x∀y(Rxy → (Xx ↔ ¬Xy)

• ∀x∃y Rxy

The problem

FO FO quantifier alternation hierarchy ∀ ∃ ∃∀ ∃∀∃ ∃∀∃∀ ∀∃∀∃ ∀∃∀ ∀∃ At

...

Understand a formalism L, used to specify properties of a class C of structures

The problem

MSO MSO quantifier alternation hierarchy ∀ ∃ ∃∀ ∃∀∃ ∃∀∃∀ ∀∃∀∃ ∀∃∀ ∀∃

...

Understand a formalism L, used to specify properties of a class C of structures

FO

The problem

• is the f.a.h. strict over C?
• are its levels decidable?

To understand FO / MSO over C means:

Why to bother about this problem (I)

The Definability Problem for L

(L fragment of MSO) Why to bother about this problem (I)

given an MSO property P (of words or trees), decide whether P is (definable) in L

a solution to the definability problem for L provides a deep insight into the structure of the class L given an MSO property P (of words or trees), decide whether P is (definable) in L The Definability Problem for L

(L fragment of MSO) Why to bother about this problem (I)

Definability Problem for FO over finite words

MSO MSO quantifier alternation hierarchy ∀ ∃ ∃∀ ∃∀∃ ∃∀∃∀ ∀∃∀∃ ∀∃∀ ∀∃

...

FO

a finite monoid M is aperiodic if there exists n ∈ ω such that xn = xn+1, for all x ∈ M.

Theorem 1. [McNaughton-Paper (1971)] P is FO definable iff P is star free Theorem 2. [Sch¨ utzenberger (1965)] P is star free iff its syntactic monoid is aperiodic.

Definability Problem for FO over finite words

• Corollary. The definability of FO over finite words

is solvable.

Definability Problem for FO over finite words

Theorem [Sch¨ utzenberger-McNaughton-Papert]. An MSO property P is FO definable iff the syntactic monoid of P is aperiodic.

Definability Problem for FO over finite words

Theorem [Sch¨ utzenberger-McNaughton-Papert]. An MSO property P is FO definable iff the syntactic monoid of P is aperiodic.

logic algebra lang.

Definability Problem for FO over infinite words

logic algebra lang.

Theorem [Ladner-Thomas]. An MSO property P over infinite words is FO definable iff the syn- tactic monoid of P is aperiodic.

The MSO-hierarchy problem over (infinite) words

MSO MSO quantifier alternation hierarchy ∀ ∃ ∃∀ ∃∀∃ ∃∀∃∀ ∀∃∀∃ ∀∃∀ ∀∃

...

FO

The MSO-hierarchy problem over (infinite) words

MSO MSO quantifier alternation hierarchy ∃ FO

Theorem [B¨ uchi (1962)]. MSO = B¨ uchi automata.

• ver infinite words.

The problem revisited (I)

Understand a formalism L, used to specify properties of a class C of structures L = (parity) automata C = infinite words / infinite trees

nondeterministic

Automata over infinite words

1 2

• a finite alphabet Σ,
• a finite set of states Q,
• an initial state qI ∈ Q,
• a transition function δ : Q × Σ → ℘(Q),
• a rank function rank : Q → ω.

a,b b a,b

1 2

a a a a a b a b a

ρ :

...

Automata over infinite words

a,b b a,b

1 2

a a a a a b a b a 1

ρ :

1 2 ...

Automata over infinite words

a,b b a,b

1 2

a a a a a b a b a 1 1

ρ :

...

Automata over infinite words

1 2

a,b b a,b

1 2

a a a a a b a b a 1 1 1

ρ :

...

Automata over infinite words

1 2

a,b b a,b

1 2

a a a a a b a b a 1 1 1 1

ρ :

...

Automata over infinite words

1 2

a,b b a,b

1 2

a a a a a b a b a 1 1 1 1 1

ρ :

...

Automata over infinite words

1 2

a,b b a,b

1 2

a a a a a b a b a 1 1 1 1 1 1

ρ :

...

Automata over infinite words

1 2

a,b b a,b

1 2

a a a a a b a b a 1 1 1 1 1 1 1

ρ :

...

Automata over infinite words

1 2

a,b b a,b

1 2

a a a a a b a b a 1 1 1 1 1 1 1 1

ρ :

...

Automata over infinite words

1 2

a,b b a,b

1 2

a a a a a b a b a 1 1 1 1 1 1 1 1 2

ρ :

... ...

Automata over infinite words

1 2

a,b b a,b

1 2

Automata over infinite words

• a run ρ of A over w is accepting iff the highest

priority occurring infinitely often in ρ is even

• an infinite word w is accepted by A iff A has an

accepting run over w,

• a language L of infinite words is accepted by A

iff A accepts all words in L.

Automata over infinite words

• a finite alphabet Σ,
• a finite set of states Q,
• an initial state qI ∈ Q,
• a transition function δ : Q × Σ → ℘(Q),
• a rank function rank : Q → ω.

index(A) := (min rank(Q), max rank(Q))

1 2

a,b b a,b

1 2

(0,0) (1,1) (0,1) (1,2) (0,3) (0,2) (1,3) (1,4)

...

Problems : (i) Strictness of the hierarchy ? (ii) Compute the minimal index needed to recognize a given language (Nondeterministic index problem)

Automata over infinite words

The nondeterministic index hierarchy

Automata over infinite words

1 2

a,b b a,b

1 2

a b a,b a,b b a

1 2 2 3

Automata over infinite words

(0,0) (1,1) (1,2)

The nondeterministic index hierarchy

Fact [B¨ uchi (1962)]. The nondeterministic hierarchy collapses to (1, 2) over infinite words.

Problems : (i) Strictness of the hierarchy ? (ii) Compute the minimal index needed to recognize a given language

Automata over infinite words

1 2

a,b b a,b

1 2

a b a,b

deterministic

Automata over infinite words

1 2

• a finite alphabet Σ,
• a finite set of states Q,
• an initial state qI ∈ Q,
• a transition function δ : Q × Σ → Q,
• a rank function rank : Q → ω.

Automata over infinite words Theorem [McNaugthon (1966)]. Nondeterministic B¨ uchi automata and deterministic M¨ uller automata are equivalent

• ver infinite words.

Corollary . Nondeterministic B¨ uchi automata and deterministic parity automata are equivalent over infinite words.

(0,0) (1,1) (0,1) (1,2) (0,3) (0,2) (1,3) (1,4)

...

Problems : (i) Strictness of the hierarchy ? (ii) Compute the minimal index needed to recognize a given language

Automata over infinite words

The deterministic index hierarchy

Automata over infinite words Theorem [essentialy in Wagner (1979)]. The deterministic index problem over infinite words is decidable.

automata Wadge

Automata over infinite words Theorem [Wagner (1979)]. Regular ω-languages have exactly the Wadge degrees of the form ωk

1nk + . . . ω1 1n1 + n0

for k < ω and n1, . . . , nk < ω. Moreover the following facts hold:

• 1. the Wadge degree of a regular ω-language is decidable
• 2. The deterministic index of a regular ω-language L is (0, k)
• r (1, k + 1) iff ωk−1

1

< dW (L) ≤ ωk

1.

Properties of deterministic automata

• A = hΣ, QA, qA

I , δA, rankAi

• qs 2 QA,
• (q0, a) 7! {(q1, q2), (q2, qs)}
• B = hΣ, QB, qB

I , δB, rankBi

The substitution AB is obtained by taking

• δ(q0, a) = {(q1, q2), (q2, qB

I )}

Substitution Properties of deterministic automata

A: B:

Properties of deterministic automata

A: B: AB:

Properties of deterministic automata

Properties of deterministic automata

• Fact. Identity of languages is preserved by

substitution.

• L(B) = L(C)
• ⇒
• L(AB) = L(AC)

What about coarser equivalence relations ?

≡W ≡ind

L(A) ≡ind L(B) iff min{index(A0) : L(A) = L(A0)} = min{index(B0) : L(B) = L(B0)} L(A) ≡W L(B) iff there are continuous f, g s.t. f −1L(A) = L(B) and g−1L(B) = L(A)

Properties of deterministic automata

• L(B) ≡W L(C)
• ? ⇒ ?
• L(AB) ≡W L(AC)

NPA do not satisfy this property in both cases because (set-theoretic) union does not satisfy it Properties of deterministic automata

• Fact. In the case of deterministic parity automata,

substitution preserves both Wadge and index equivalences. This is not the case for non-deterministic automata.

Σ = {0, 1, 2}

• (0∗(1 + 2))ω
• (0∗2)ω
• Σ∗0ω

Failing with Wadge equivalence

Σ = {0, 1, 2}

• (0∗(1 + 2))ω
• (0∗2)ω
• Σ∗0ω

(0∗2)ω ∪ Σ∗0ω >W Σω (0∗(1 + 2))ω ∪ Σ∗0ω = Σω (0∗(1 + 2))ω ≡W (0∗2)ω

Failing with Wadge/Index equivalence [1] [2] [3] [2] [1] [1 v 3] [2 v 3]

Possibility to obtain decidable results by applying a «forbidden pattern» technique Why bother with this ?

Possibility to obtain decidable results by applying a «forbidden pattern» technique Idea: to each Wadge/index equivalence class, we associate a canonical automaton Why bother with this ?

Pattern method

Pattern method

Pattern method

Pattern method

Pattern method

Sketch of the proof of Wagner’s theorem

Sketch of the proof of Wagner’s theorem

. . .

Σ1

n

Σ1

1

Π1

n

Π1

1

BC(Π0

2)

dW (L) < ωω

1

• for every deterministic B¨

uchi automaton A, L(A) ∈ Π0

2

• by McNaughton’s theorem,

every regular language is in BC(Π0

2).

Sketch of the proof of Wagner’s theorem

• 1. For each ordinal α = ω`k

1 nk + · · · + ω`0 1 n0 < ω! 1

construct a canonical automata of Wadge degree α. Do it by providing canonical patterns for

• the degrees ωn

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
• f all deterministically recognizable languages

Sketch of the proof of Wagner’s theorem For every index (i, k) let

A paradigmatic class of languages

L(i,k) L(i,k) := {w ∈ {i, . . . , k}ω | lim sup

i→∞

w(i) is even.}

• Fact. There is a deterministic automaton of index (i, k)

recognizing L(i,k).

Sketch of the proof of Wagner’s theorem

...

A paradigmatic class of languages

L(i,k)

• Proposition. The followings hold:
• 1. the languages L(i,k) form a strict hierarchy

w.r.t. ≤W

• 2. for every k ≥ 0, dW (L(0,k)) = dW (L(1,k+1)) = ωk

1

• 3. If A has index (i, k), then L(A) ≤W L(i,k).

L(0,0) L(1,1) L(0,1) L(0,2) L(0,3) L(1,3) L(1,4) L(1,2)

Sketch of the proof of Wagner’s theorem

A paradigmatic class of languages

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.

Sketch of the proof of Wagner’s theorem

(i,k)-flower F(i,k)

ι = ( ? if i = k = 1 > else.

aj a`

j i < j i ≤ k j ≤ k i a` 6= aj

ai

ι

Sketch of the proof of Wagner’s theorem

(i,k)-flower F(i,k)

• Proposition. For every (i, k), F(i,k) ≡W L(i,k).

Thence, F(i,k) has index (i, k).

Sketch of the proof of Wagner’s theorem

sum A ∨ B

ι = ( ? if L(A) = L(B) = ; > else. a b τ / ∈ {a, b} qI qA

I

qB

I

Sketch of the proof of Wagner’s theorem

sequential composition A ⊕ B

b qB

I

s τ 6= b nA := A ⊕ · · · ⊕ A | {z }

n times

Sketch of the proof of Wagner’s theorem

Canonical automata

• for 0 < k < ω,

Cωk = F(0,k), Dωk = F(1,k+1), Ek = Ck ∨ Dk,

• for 1 < α < ω,

Cα = C1 ⊕ (α − 1)E1, Dα = D1 ⊕ (α − 1)E1, Eα = αE1

• C1 = F(0,0),

D1 = F(1,1), E1 = C1 ∨ D1,

• given ↵ = !`knk + · · · + !`0n0 < !!, with ! > `k > 0,

`k > · · · > `0, and 0 < ni < !: – if `0 = 0: C↵ = Cn0 ⊕ n1E!`1 ⊕ · · · ⊕ nkE!`k , D↵ = Dn0 ⊕ n1E!`1 ⊕ · · · ⊕ nkE!`k , E↵ = En0 ⊕ n1E!`1 ⊕ · · · ⊕ nkE!`k – else: C↵ = C!`0 ⊕ (n0 − 1)E!`0 ⊕ n1E!`1 ⊕ · · · ⊕ nkE!`k , D↵ = D!`0 ⊕ (n0 − 1)E!`0 ⊕ n1E!`1 ⊕ · · · ⊕ nkE!`k , E↵ = E!`0 ⊕ (n0 − 1)E!`0 ⊕ n1E!`1 ⊕ · · · ⊕ nkE!`k

Sketch of the proof of Wagner’s theorem dW (C↵) = dW (D↵) = dW (E↵) = ω`k

1 nk + · · · + ω`0 1 n0

Cα

Dα Eα Cα+1 Dα+1

• Theorem. Given ↵ = !`knk + · · · + !`0n0 < !!,

with ! > `k > 0, `k > · · · > `0, and 0 < ni < !: and

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 effectively construct a Wadge equivalent ca- nonical one, (b) L(A) ≤W L(i,k) iff L(A) ∈ (i, k)

Sketch of the proof of Wagner’s theorem

(i,k)-flower

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) iff

A does not contain an (i, k)-flower

The case of regular languages of infinite words

Nondeterministic index hierarchy Deterministic index hierarchy Borel hierarchy Wadge hierarchy (1,2) Decidable Strict Decidable Decidable Decidable ωω BC(Π0

2)

The landscape of infinite words

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

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

1 3 6 2 5

Parity games

1 3 6 2 5

3

Parity games

1 3 6 2 5

30

Parity games

1 3 6 2 5

305

Parity games

1 3 6 2 5

3055.....5

Parity games

1 3 6 2 5

3055.....56

Parity games

1 3 6 2 5

3055.....56.... ∈ {0, . . . , 6}ω

Parity games

1 3 6 2 5

Player ♦ wins iff the greatest priority occurring infinitely often is even

Parity games

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

Parity games

Determinacy: For every node, one of the two players has a winning strategy

1 3 6 2 5

Parity games

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} K is Π1

1-complete.

• {0, 1}
• qI := 0
• (i, j) 7! (j, j)
• rank(i)=i

Deterministic languages

K{ is Σ1

1-complete.

Deterministic ?

t ∈ W(1,3) iff ♦ has a w.s. in the induced parity game

Game languages t ∈ TΣ, where Σ = {⌃, ⇤} × {1, 2, 3}

2 1 3 1 2 3 2 . . . . . . . . . . . . . . . .

W(1,3)

Game languages t ∈ TΣ, where Σ = {⌃, ⇤} × {1, 2, 3}

1 1 3 1 3 2 2 . . . . . . . . . . . . . . . .

W ]

(1,3)

t ∈ W ]

(1,3)

iff t(w)|2 ≤ min{t(w0)|2, t(w1)|2}, for each w ∈ dom(t), and ♦ has a w.s. in the induced parity game

Game languages

...

W(1,1) W(0,0) W(0,1) W(0,2) W(0,3) W(1,3) W(1,4) W(1,2)

Theorem [Arnold, Niwinski (2008)]. The game languages form a hierarchy with respect to Wadge reducibility, i.e.

...

W ]

(0,0)

W ]

(1,1)

W ]

(1,2)

W ]

(1,3)

W ]

(1,4)

W ]

(0,3)

W ]

(0,2)

W ]

(0,1)

Moreover, W(0,1) is Π1

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 t ∈ TΣ, where Σ = {⌃, ⇤} × {1, 2, 3}

• {1, 2, 3}
• qI := 1
• i →(⌃,j) (0, j) ∨ (1, j)

i →(⇤,j) (0, j) ∧ (1, j)

• rank(i)=i

‘game languages are alternating’

2 1 3 1 2 3 2 . . . . . . . . . . . . . . . .

W(1,3)

Game languages t ∈ TΣ, where Σ = {⌃, ⇤} × {1, 2, 3} ‘game languages are nondeterministic’

2 1 3 1 2 3 2 . . . . . . . . . . . . . . . .

W(1,3)

• {1, 2, 3, >}
• qI := 1
• i !(⌃,j) {(>, j), (j, >)}

i !(⇤,j) (j, j)

• rank(i)=i

Game languages t ∈ TΣ, where Σ = {⌃, ⇤} × {1, 2, 3} ‘weak game languages are weak alternating’

1 1 3 1 3 2 2 . . . . . . . . . . . . . . . .

W ]

(1,3)

• {1, 2, 3, ⊥}
• qI := 1
• i →(⌃,j)

( (0, j) ∨ (1, j) i ≤ j ⊥ i > j i →(⇤,j) ( (0, j) ∧ (1, j) i ≤ j ⊥ i > j

• rank(i)=i

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.

Parity tree automata

∆0

ω

∆0

3

∆1

2

Σ1

1

Π1

1

deterministic weak regular

?

Deterministic tree languages

• Theorem. For deterministic languages of infinite trees, the

following holds:

• [Murlak (2006)] the Wadge hierarchy is decidable,
• [Niwinski-Walukiewicz (1996)] the deterministic index is

decidable,

• [Niwinski-Walukiewicz (2005)] the nondeterministic index is

decidable,

• [Murlak (2008)] the weak index is decidable,
• [Murlak (2008)] the weak index and the Borel rank coincide.

Parity tree automata

(table by S. Hummel)

Why the understanding regular languages of infinite trees is so difficult?

• 1. there is no canonical representation of a

regular tree language: determinism non determinism

• 2. regular tree languages are (topologically)

very complicated ( ) 6= ∆1

2

• (ρ, t) 7! t is continuous,
• the set of successful runs of a nondeterministic

parity automaton is in Π1

1,

• regular languages are closed under complementation.

Is it possible to determine the largest class of parity automata extending the deterministic one for which substitution is compatible with the Wadge equivalence and the index equivalence?

Towards game automata

(we want to use a pattern technique!)

Towards game automata

We call a transition (q, a) ambiguous if it con- tains either an ✏ move or two occurrences of some direction d ∈ {0, 1}.

(q, a) → (1, p) ∨ (1, r) (q, a) → (1, p) ∨ (ε, r)

Towards game automata

• Proposition. Let C be a class of alternating automata
• over a fixed alphabet with at least two letters,
• closed under substitution and
• containing the one-state all-rejecting and all-

accepting automata,

• containing an automaton of non trivial index,

Substitution preserves the Wadge equivalence / index equivalence in C iff no automaton of C has an ambiguous transition.

Each non-ambiguous transition has one of the four forms:

• (0, p),
• (1, p),
• (0, p) ∨ (1, q), or
• (0, p) ∧ (1, q).

Towards game automata

Towards game automata Definition [DFM (2011)] : A game automaton (GA) is an alternating automaton without ambiguous transitions.

Towards game automata t ∈ TΣ, where Σ = {⌃, ⇤} × {1, 2, 3}

• {1, 2, 3}
• qI := 1
• i →(⌃,j) (0, j) ∨ (1, j)

i →(⇤,j) (0, j) ∧ (1, j)

• rank(i)=i

‘game languages are game’

2 1 3 1 2 3 2 . . . . . . . . . . . . . . . .

W(1,3) Definition [DFM (2011)] : A game automaton (GA) is an alternating automaton without ambiguous transitions.

The landscape of automata

ATA GA DPA WGA

Parity tree automata

∆0

ω

∆0

3

∆1

2

Σ1

1

Π1

1

deterministic weak regular

?

game

?

Weak game automata weak parity condition: restriction if q reachable from r, then rank(q) ≤ rank(r)

Weak game automata

• we isolate a finite class of patterns
• we check that they corresponds to operations on Wadge degrees
• based on those patterns, we define canonical automata
• we check that canonical automata are closed under such
• perations
• we verify that for each WGA there is a Wadge-equivalent

canonical one Solving the Wadge problem for WGA

i A B

a

i A B

a b

+ A C

a c

• A

C

a c

+ L M

a b c

• +

a, b a, b, c c

• L

M

a b c

+

• a, b

a, b, c c

Weak game automata canonical patterns

[γ1]✏1 ♦ [γ2]✏2 = complicated . . .

Weak game automata

• perations on Wadge degrees

exp1(α) = ωα

1 , and expk+1(α) = exp(expk(α))

Weak game automata Let [Φ] = {[↵]✏ ↵ ∈ Φ, ✏ ∈ {+, −, ±}}, with Φ denoting the set

• f ordinals of the form P0

n=N n + ↵ where

• ↵ < ! and
• each n is of the form expn(!)⌘ + P1

p=P expn(p)kp, for

some ⌘ < !! and kp < !. Wadge degrees for canonical WGA (I) Proposition [DFM (2009)]. [Φ] is closed under t, ⇧, loop+, 9 (and their duals) and the result of the operation can be computed effectively.

Weak game automata Let Ω be the set of ordinals of the form Σ0

i=K exp(αi)ηi where

• αK, αK−1, . . . , α0 is a strictly decreasing sequence of ordi-

nals from Φ, and

• ηi < ω for cofαi = ω1 or cofαi < ω, and
• ηi < ωω for cofαi = ω.

Proposition [DFM (2011)]. [Ω] is closed under t, loop+, loop reset+, 9 (and their duals) and the result of the opera- tion can be computed effectively. Wadge degrees for canonical WGA (II)

Weak game automata Theorem [DFM (2011)] : There is an effective procedure that takes as input a WGA and returns as output a Wadge equivalent canonical one.

Weak Index Borel Hierarchy Wadge Hierarchy WGA ? Decidable Decidable

Weak game automata

Weak Index Borel Hierarchy Wadge Hierarchy WGA ? Decidable Decidable Weak Index = Borel Rank

Weak game automata

Weak Index Borel Hierarchy Wadge Hierarchy WGA Decidable Decidable Decidable Weak Index = Borel Rank

Weak game automata

What about Game Automata ?

Strong game automata

2 1 2 1 2 1 2 . . . . . . . . . . . . . . . .

t ∈ TΣ, where Σ = {⌃, ⇤} × {1, 2} W(1,2) t ∈ W(1,2) iff ♦ has a w.s. in the induced parity game

Strong game automata ˆ W(1,2) t ∈ ˆ W(1,2) iff ♦ has a w.s. in the induced parity game t ∈ TΣ, where Σ = {e, ⊥}

• if t(v) = e, then ⌃ can choose either to go left or right, and

rank(vR) = 2, and rank(vL) = 1,

• if t(v) = ⊥, then the whole subtree is labelled by ⊥,

⇤ can choose to go left or right and rank(vL) = rank(vR) = 1.

Strong game automata ˆ W(1,2) t ∈ ˆ W(1,2) iff ♦ has a w.s. in the induced parity game t ∈ TΣ, where Σ = {e, ⊥}

Strong game automata

• Proof. Let t ∈ T{⌃,⇤}×{1,2}.

By K¨

• nig’s lemma, Player ♦ has a winning strategy in Gt iff

she can produce a sequence of finite strategies σ0, σ1, σ2, . . . (viewed as subtrees of t) such that

• 1. σ0 consists of the root only;
• 2. for each n the strategy σn+1 extends σn in such a way that

below each leaf of σn a non-empty subtree is added, and all leaves of σn+1 have priority 2. Using this observation we can define the reduction.

• Proposition. W(1,2) ≤W

ˆ W(1,2)

Strong game automata

• Proposition. W(1,2) ≤W

ˆ W(1,2) Proof (cont). Let (τi)i∈ω be all finite unlabelled binary trees. Some of these trees naturally induce a strategy for ♦ in Gt. For those we define tτi ∈ T{e,⊥} co-inductively, as follows:

• tτi(Rj) = e for all j;
• if τj induces in Gt a strategy that is a legal extension of

the strategy induced by τi in the sense of item 2) above, then the subtree of tτi rooted at RjL is tτj;

• otherwise, all nodes in this subtree are labelled with ⊥.

Let f(t) = tσ0. By the initial observation, tσ0 ∈ ˆ W(1,2) iff ♦ has a winning strategy in Gt. The function f is continuous. Hence, W(1,2) ≤W ˆ W(1,2).

Strong game automata ˆ W(0,k) t ∈ ˆ W(0,k) iff ♦ has a w.s. in the induced parity game t ∈ TΣ(0,k), where Σ(0, k) = {e, a, 2, . . . , k} We associate a parity game with positions dom(t) such that

• if t(v) = a, then in v player ⇤ can choose to go to vL
• r to vR, and rank(vL) = 1, rank(vR) = 0,
• if t(v) = e, then in v player ⌃ can choose to go to vL
• r to vR, and rank(vL) = 1, rank(vR) = 2,
• if t(v) ∈ {3, . . . , k}, the only move from v is to vL and

rank(vL) = t(v).

Strong game automata ˆ W(0,k) t ∈ ˆ W(0,k) iff ♦ has a w.s. in the induced parity game t ∈ TΣ(0,k), where Σ(0, k) = {e, a, 2, . . . , k}

Strong game automata

• Proposition. W(i,k) ≤W

ˆ W(i,k)

Strong game automata: edelweiss

(0,k)-edelweiss (1,k+1)-edelweiss

Strong game automata: edelweiss

• Proposition. If A contains an (i, k)-edelweiss,

then ˆ W(i,k) ≤W L(A).

• Corollary. If A contains an (i, k)-edelweiss,

then W(i,k) ≤W L(A), thus L(A) is not in (i, k).

• Consider the graph of an automaton

A of index (i, k)

• Fix i ≤ n ≤ k, and consider the sub-

graph of nodes p with rank(p) ≤ n

• a SCC B is the subgraph is called n-

component

• it is called ∃-branching if it has the

pattern

p qL qR ∨ B

Strong game automata: computing the index

Strong game automata: computing the index

Let A of index (i, k). By induction from n = i. Let B be an n-component:

Strong game automata: computing the index Upper bound: Follow the recursive algorithm, pushing through an invariant guaranteeing that each n-component B can be re- placed with an “equivalent” class(B)-automaton.

Lower bound. It holds that:

• if class(A) ≥ (i, k) then A contains a (i, k)-edelweiss,
• if A contains a (i, k)-edelweiss, then W(i,k) ≤W L(A).

Strong game automata: computing the index

Strong game automata: results

• Theorem. For languages of infinite trees recognized by game

automata, the following holds:

• [Niwinski-Walukiewicz (1996)] the deterministic index is

decidable,

• [FMS (2013)] the nondeterministic index is decidable,
• [FMS (2013)] the alternating index is decidable,
• [FMS (2014)] the weak index is decidable,
• [FMS (2014)] the weak index and the Borel rank coincide.

Strong game automata

What about the Wadge hierarchy for game automata? Idea: extend the WGA approach, i.e. use patterns in game automata to define effective operations on Wadge degrees.

Morality: Topological and index hierarchies are very closely related. Topology can help in understanding regular languages Use automata to understand the Wadge hierarchy within the second ambiguous projective class Future:

(0,0) (1,1) (0,1) (1,2) (0,3) (0,2) (1,3) (1,4)

...

Problems : (i) Strictness of the hierarchy ? OK (ii) Compute the minimal index needed to recognize a given language

Automata over infinite trees

The strong alternating index hierarchy

(0,0) (1,1) (0,1) (1,2) (0,3) (0,2) (1,3) (1,4)

...

Problems : (i) Strictness of the hierarchy ? OK (ii) Compute the minimal index needed to recognize a given language

Automata over infinite trees

The weak alternating index hierarchy

Deciding being open over infinite trees

Theorem: It is decidable whether a regular tree language is open. Proof: Given a tree language L,

• Lf = {s ∈ T (n) | n < !, s.T ⊆ L},
• Lint = {t ∈ T | ∃`, ∃s ∈ Lf : t|` = s}.

L is open iff L = Lint. Let A non-deterministic s.t. L(A) = L.

Deciding being open over infinite trees

Proof (cont.):

• s ∈ Lf iff there is a run of A over s such that the

leaves of s are labelled with q ∈ Q such that L(A@q) = T,

• Thence Lf is recognized by

Af = (Q, qI, δ, F = {q ∈ Q | L(A@q) = T}),

Deciding being open over infinite trees

Proof (cont.): Let Aint = (Q, qI, δint, rankint) where

• δint(q, c) =

( δ(q, c) if q / ∈ F (q, q) else,

• rankint(q) =

( 1 if q / ∈ F else. It holds that L(Aint) = Lint. To check if L is open, it is enough to check if L(Aint) = L(A).

Going further

Theorem [Bojanczyk-Place (2012)]: It is decidable whether a regular tree language is in BC(Σ0

1).

Theorem [F.-Michalewski (2014)]: It is decidable whether a regular tree language is in ∆0

2.

logic algebra topology games Going further

Theorem [Bojanczyk-Place (2012)]: It is decidable whether a regular tree language is in BC(Σ0

1).

Theorem [F.-Michalewski (2014)]: It is decidable whether a regular tree language is in ∆0

2.

Open problems

Alessandro Facchini (IDSIA, Lugano)

Workshop on Wadge Theory and Automata, 28.01.2015, Torino

• alternating and nondeterministic index problem
• weak (alternating) index vs Borel rank

Are there regular language of non finite Borel rank?

• Wadge degrees for game automata
• separation problem for regular languages

Two analytic sets can be separated by a Borel sets. This fails for co-analytic sets. Now, for Sigma - i.e. (i, 2k+1) - classes of trees (nondeterministic alternating hierarchies), separation fails [Arnold, Niwinski, Michalewski (2012)]. What about Pi - i.e. (i, 2k) - classes? Does the property holds? (for n=2 true, by a result of Rabin). Notice that this is true in case of deterministic parity word automata [Arnold, Niwinski, Michalewski (2012)].

• algebra vs topology
• Wadge hierarchy for quantitative extensions of regular languages

Future directions / Open problems