pbl/galop2016/ribaslides.pdf · Introduction Introduction Fix D non-empty and ﬁnite. Full...

A Dialectica-Like Approach to Tree Automata

Colin RIBA

LIP - ENS de Lyon

Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 1 / 26

Introduction

IntroductionFix D non-empty and finite. Full Σ-labelled D-ary trees are maps

t : D∗ −→ Σ

Monadic-Second Order Logic (MSO) (over D-ary trees)Logic with

I existential quantification over arbitrary sets of tree positions(labellings D∗ → 2)

I boolean operations(here ⊗,(, and units)

Decidable (Rabin 1969).I Effective but non-elementary translation to tree automata.I Acceptance can be formulated using games (Gurevitch & Harrington).

Alternating tree automata (Muller & Schupp, Emmerson & Jutla):I “Easy” (= linear) negation (by determinacy of games).I Interpretation of existential quantification: exponential.

Here:I Curry-Howard like approach, categorical formulation:

Automata as objects, Executions as morphisms

Introduction

t : D∗ −→ Σ

Introduction

t : D∗ −→ Σ

Introduction

t : D∗ −→ Σ

Introduction

t : D∗ −→ Σ

Automata as objects, Executions as morphismsColin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 2 / 26

Introduction

Categorical model for deduction rules on automata

Objects A(t)I Automaton A instantiated with an input tree t

(t can have an input as well)

Morphisms A(t) −→ B(u)

I (winning) P-strategies in a simple linear arrow between (a generalization)of the acceptance games for A(t) and B(u).

Realizability semantics:I Soundness: If σ is a realizer of A → B then L(A) ⊆ L(B).I Weak converse (weak completeness).

Example of rules:1 ; A , B ` C

1 ; B ` A( B

I Symmetric monoidal closed structure.I Fibred structure (substitution) + existential quantification.I Deduction rules for the !(−) modality.

Introduction

Categorical model for deduction rules on automataObjects A(t)

I Automaton A instantiated with an input tree t(t can have an input as well)

1 ; B ` A( B

Introduction

1 ; B ` A( B

Introduction

Realizability semantics:I Soundness: If σ is a realizer of A → B then L(A) ⊆ L(B).

I Weak converse (weak completeness).Example of rules:

1 ; A , B ` C1 ; B ` A( B

Introduction

1 ; B ` A( B

Introduction

1 ; B ` A( B

Introduction

1 ; B ` A( B

Zigzag Games for Tree Automata

Outline

Introduction

Indexed Structure

Non-Deterministic Automata

Conclusion

Tree AutomataAlternating tree automata A = (QA,qıA, δA,ΩA)where ΩA ⊆ Qω

δA : QA × Σ −→ The free distr. lattice over (QA × D)

If A is “total” we can assumeδA : (q,a) 7−→

∨u∈U

∧x∈X (q′u,x ,du,x )

↑ ↑P O

Here: Monoidal closed structure.I Sets U and X dealt-with along the lines of the Dialectica interpretation.

We takeA = (QA,qıA,U,X , δA,ΩA)

with δA : QA × Σ −→ U × X −→ (D −→ QA)

and games of the form

(q,a) 7−→∨

u∈U∧

x∈X∧

d∈D δA(qA,a,u, x ,d)↑ ↑ ↑P O O

∨u∈U

↑ ↑P O

with δA : QA × Σ −→ U × X −→ (D −→ QA)

(q,a) 7−→∨

u∈U∧

x∈X∧

∨u∈U

↑ ↑P O

with δA : QA × Σ −→ U × X −→ (D −→ QA)

(q,a) 7−→∨

u∈U∧

x∈X∧

∨u∈U

↑ ↑P O

Here: Monoidal closed structure.

I Sets U and X dealt-with along the lines of the Dialectica interpretation.We take

A = (QA,qıA,U,X , δA,ΩA)

with δA : QA × Σ −→ U × X −→ (D −→ QA)

(q,a) 7−→∨

u∈U∧

x∈X∧

∨u∈U

↑ ↑P O

with δA : QA × Σ −→ U × X −→ (D −→ QA)

(q,a) 7−→∨

u∈U∧

x∈X∧

∨u∈U

↑ ↑P O

with δA : QA × Σ −→ U × X −→ (D −→ QA)

(q,a) 7−→∨

u∈U∧

x∈X∧

∨u∈U

↑ ↑P O

with δA : QA × Σ −→ U × X −→ (D −→ QA)

(q,a) 7−→∨

u∈U∧

x∈X∧

Zigzag Games(Full, Positive) Simple Games:

A = (U,X ) where U,X non-empty (finite) sets

The Category DZ:I Objects: A = (U,X ), B = (V ,Y ), etcI Morphisms σ : A→ B are zigzag strategies:

Aσ−( B

O uv Py O

Monoidal Structure:

A⊗ B := (U × V , X × Y ) with unit I := (1,1)

Aσ−( B

O uv Py O

Monoidal Structure:

Aσ−( B

O uv Py O

Monoidal Structure:

D-Synchronicity

In DZ, the objectD := (1,D)

can be equipped with a monoid structure:

I −→ DO •

• Pd O

D ⊗ D −→ DO (•, •)

• Pd O

P (d ,d)

Hence there is a monad (−)⊗ D on DZ.

The Category DZD of D-Synchronous Strategies: DZD := Kl(D)

DZD A −→ BO u

v P(y ,d) O

P x (DZD[A,B] = DZ[A,B ⊗ D])

D-SynchronicityIn DZ, the object

D := (1,D)

I −→ DO •

• Pd O

D ⊗ D −→ DO (•, •)

• Pd O

P (d ,d)

DZD A −→ BO u

v P(y ,d) O

D-SynchronicityIn DZ, the object

D := (1,D)

I −→ DO •

• Pd O

D ⊗ D −→ DO (•, •)

• Pd O

P (d ,d)

DZD A −→ BO u

v P(y ,d) O

Linear Synchronous Arrow of Acceptance GamesGiven automata A = (QA , qıA , U , X , δA , ΩA) over Σ

and B = (QB , qıB , V , Y , δB , ΩB) over Γ

and trees t : D∗ → Σ and u : D∗ → Γ let

DZD A(t) −→ B(u)

(p,qA) (p,qB)O u

v P(y ,d) O

P x(p.d ,q′A) (p.d ,q′B)

whereq′A = δA(qA , t(p) , u , x , d)q′B = δB(qB , u(p) , v , y , d)

I A play is winning (for P) iff

qıA · . . . · qA · q′A · . . . ∈ ΩA =⇒ qıB · . . . · qB · q′B · . . . ∈ ΩB

I t ∈ L(A) iff there is a winning σ : I −→ A(t)with I := (1 , • , 1 , 1 , 1 , 1)

DZD A(t) −→ B(u)

(p,qA) (p,qB)

O uv P

(y ,d) OP x

(p.d ,q′A) (p.d ,q′B)

DZD A(t) −→ B(u)(p,qA) (p,qB)

O uv P

(y ,d) OP x

(p.d ,q′A) (p.d ,q′B)

O uv P

(y ,d) OP x

(p.d ,q′A) (p.d ,q′B)

O uv P

(y ,d) OP x

(p.d ,q′A) (p.d ,q′B)

O uv P

(y ,d) OP x

(p.d ,q′A) (p.d ,q′B)

Monoidal ClosureConsider

A = (U,X ) B = (V ,Y )

C = (W ,Z )

Dialectica-Like arrow in DZ:

A(DZ B := (V U × X U×Y , U × Y )

A⊗ B σ−→ C

BΛ(σ)−→ (A(DZ C)

(u, z) OP y

Monoidal Closure in DZD:

DZD[A,B] = DZ[A,B ⊗ D](A(DZD B)⊗ D = A(DZ (B ⊗ D)

A(DZD B := (V U × X U×Y×D , U × Y )

A = (U,X ) B = (V ,Y ) C = (W ,Z )

A(DZ C := (W U × X U×Z , U × Z )

A⊗ B σ−→ C

(u, z) OP y

A(DZD B := (V U × X U×Y×D , U × Y )

A = (U,X ) B = (V ,Y ) C = (W ,Z )

A(DZ C := (W U × X U×Z , U × Z )

A⊗ B σ−→ C

O v(?, ?) P

(u, z) OP y

A(DZD B := (V U × X U×Y×D , U × Y )

A = (U,X ) B = (V ,Y ) C = (W ,Z )

A(DZ C := (W U × X U×Z , U × Z )

A⊗ B σ−→ CO ( , v)

O v(?, ) P

(u, z) OP y

A(DZD B := (V U × X U×Y×D , U × Y )

A = (U,X ) B = (V ,Y ) C = (W ,Z )

A(DZ C := (W U × X U×Z , U × Z )

A⊗ B σ−→ CO (u, v)

O v(?, ) P

(u, z) OP y

A(DZD B := (V U × X U×Y×D , U × Y )

A = (U,X ) B = (V ,Y ) C = (W ,Z )

A(DZ C := (W U × X U×Z , U × Z )

A⊗ B σ−→ CO (u, v)

O v(?, ) P

(u, z) OP y

A(DZD B := (V U × X U×Y×D , U × Y )

A = (U,X ) B = (V ,Y ) C = (W ,Z )

A(DZ C := (W U × X U×Z , U × Z )

A⊗ B σ−→ CO (u, v)

O v(f , ) P

(u, z) OP y

A(DZD B := (V U × X U×Y×D , U × Y )

A = (U,X ) B = (V ,Y ) C = (W ,Z )

A(DZ C := (W U × X U×Z , U × Z )

A⊗ B σ−→ CO ( , v)

O v(f , ?) P

(u, z) OP y

A(DZD B := (V U × X U×Y×D , U × Y )

A = (U,X ) B = (V ,Y ) C = (W ,Z )

A(DZ C := (W U × X U×Z , U × Z )

A⊗ B σ−→ CO (u, v)

f (u) Pz O

O v(f , ?) P

(u, z) OP y

A(DZD B := (V U × X U×Y×D , U × Y )

A = (U,X ) B = (V ,Y ) C = (W ,Z )

A(DZ C := (W U × X U×Z , U × Z )

A⊗ B σ−→ CO (u, v)

f (u) Pz O

P (x , y)

O v(f , ?) P

(u, z) OP y

A(DZD B := (V U × X U×Y×D , U × Y )

A = (U,X ) B = (V ,Y ) C = (W ,Z )

A(DZ C := (W U × X U×Z , U × Z )

A⊗ B σ−→ CO (u, v)

f (u) Pz O

P (x , y)

O v(f ,F ) P

(u, z) OP y

A(DZD B := (V U × X U×Y×D , U × Y )

A = (U,X ) B = (V ,Y ) C = (W ,Z )

A(DZ C := (W U × X U×Z , U × Z )

A⊗ B σ−→ C

O v(f ,F ) P

(u, z) OP y

A(DZD B := (V U × X U×Y×D , U × Y )

A = (U,X ) B = (V ,Y ) C = (W ,Z )

A(DZ C := (W U × X U×Z , U × Z )

A⊗ B σ−→ C

O v(f ,F ) P(u, z) O

A(DZD B := (V U × X U×Y×D , U × Y )

A = (U,X ) B = (V ,Y ) C = (W ,Z )

A(DZ C := (W U × X U×Z , U × Z )

A⊗ B σ−→ CO (u, v)

f (u) Pz O

A(DZD B := (V U × X U×Y×D , U × Y )

A = (U,X ) B = (V ,Y ) C = (W ,Z )

A(DZ C := (W U × X U×Z , U × Z )

A⊗ B σ−→ CO (u, v)

f (u) Pz O

P (F (u, z), y)

A(DZD B := (V U × X U×Y×D , U × Y )

A = (U,X ) B = (V ,Y ) C = (W ,Z )

A(DZ C := (W U × X U×Z , U × Z )

A⊗ B σ−→ CO (u, v)

f (u) Pz O

P (F (u, z), y)

A(DZD B := (V U × X U×Y×D , U × Y )

A = (U,X ) B = (V ,Y ) C = (W ,Z )

A(DZ B := (V U × X U×Y , U × Y )

A⊗ B σ−→ CO (u, v)

f (u) Pz O

P (F (u, z), y)

A(DZD B := (V U × X U×Y×D , U × Y )

A = (U,X ) B = (V ,Y ) C = (W ,Z )

A(DZ B := (V U × X U×Y , U × Y )

A⊗ B σ−→ CO (u, v)

f (u) Pz O

P (F (u, z), y)

A(DZD B := (V U × X U×Y×D , U × Y )

Some Operations on AutomataConsider automata over Σ

A = (QA , qıA , U , X , δA , ΩA) B = (QB , qıB , V , Y , δB , ΩB)

LetA⊗ B := (QA ×QB , (qıA,q

ıB) , U × V , X × Y , δA⊗B , ΩA⊗B)

A( B := (QA ×QB , (qıA,qıB) , V U × X U×Y×D , U × Y , δA(B , ΩA(B)

Some Properties:I L(A⊗ B) = L(A) ∩ L(B)

I There is an automaton

⊥ := (B , ff , 1 , 1 , δ⊥ , B∗ · ttω)

such thatI L(⊥) = ∅I If ΩA is Borel, then t ∈ L(A( ⊥) iff t /∈ L(A)

ıB) , U × V , X × Y , δA⊗B , ΩA⊗B)

⊥ := (B , ff , 1 , 1 , δ⊥ , B∗ · ttω)

ıB) , U × V , X × Y , δA⊗B , ΩA⊗B)

⊥ := (B , ff , 1 , 1 , δ⊥ , B∗ · ttω)

ıB) , U × V , X × Y , δA⊗B , ΩA⊗B)

withδA⊗B((qA,qB) , a , (u, v) , (x , y) , d) := (q′A,q

q′A := δA(qA , a , u , x , d) and q′B := δB(qB , a , v , y , d)

⊥ := (B , ff , 1 , 1 , δ⊥ , B∗ · ttω)

ıB) , U × V , X × Y , δA⊗B , ΩA⊗B)

withδA(B((qA,qB) , a , (f ,F ) , (u, y) , d) := (q′A,q

q′A := δA(qA , a , u , F (u, y ,d) , d) and q′B := δB(qB , a , f (u) , y , d)

⊥ := (B , ff , 1 , 1 , δ⊥ , B∗ · ttω)

ıB) , U × V , X × Y , δA⊗B , ΩA⊗B)

⊥ := (B , ff , 1 , 1 , δ⊥ , B∗ · ttω)

ıB) , U × V , X × Y , δA⊗B , ΩA⊗B)

⊥ := (B , ff , 1 , 1 , δ⊥ , B∗ · ttω)

Indexed Structure

Outline

Introduction

Indexed Structure

Conclusion

Indexed Structure

DZD-games A(t)→ B(u)

interpret judgments of the form

1;A(t) ` B(u)

Generalization to “open” games on arbitrary input alphabet:

Σ;A ` B

Indexed StructureI Over a category T (to be defined soon)

I Objects: alphabets Σ, Γ, etc (non-empty finite sets)I Morphisms: T[Σ, Γ] such that

D∗ → (Σ→ Γ) ⊆ T[Σ, Γ] and T[1,Σ] ' D∗ → Σ

I with substitution:Σ;A(u) ` B(v) t ∈ T[Γ,Σ]

Γ;A(u t) ` B(v t)I and existential quantification (on automata):

Σ;∃ΓA ` BΣ× Γ;A ` B(π)

andΣ× Γ;A ` B(π)

Σ;∃ΓA ` B

(where π := λ_.λ(a, b).a ∈ T[Σ× Γ,Σ])

Indexed Structure

DZD-games A(t)→ B(u) interpret judgments of the form

1;A(t) ` B(u)

Σ;A ` B

D∗ → (Σ→ Γ) ⊆ T[Σ, Γ] and T[1,Σ] ' D∗ → Σ

Σ;∃ΓA ` B

Indexed Structure

1;A(t) ` B(u)

Σ;A ` B

D∗ → (Σ→ Γ) ⊆ T[Σ, Γ] and T[1,Σ] ' D∗ → Σ

Σ;∃ΓA ` B

Indexed Structure

1;A(t) ` B(u)

Σ;A ` B

D∗ → (Σ→ Γ) ⊆ T[Σ, Γ] and T[1,Σ] ' D∗ → Σ

Σ;∃ΓA ` B

Indexed Structure

1;A(t) ` B(u)

Σ;A ` B

D∗ → (Σ→ Γ) ⊆ T[Σ, Γ] and T[1,Σ] ' D∗ → Σ

Γ;A(u t) ` B(v t)

I and existential quantification (on automata):

Σ;∃ΓA ` B

Indexed Structure

1;A(t) ` B(u)

Σ;A ` B

D∗ → (Σ→ Γ) ⊆ T[Σ, Γ] and T[1,Σ] ' D∗ → Σ

Σ;∃ΓA ` B

(where π := λ_.λ(a, b).a ∈ T[Σ× Γ,Σ])Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 12 / 26

Indexed Structure

The Base Category TGiven an alphabet Σ, write

Σ := (Σ,1) (DZD-object)

The Base Category T:I Objects: alphabets Σ, Γ, etcI Morphisms: T[Σ, Γ] := DZD[Σ, Γ] that is

DZD Σt−→ Γ

O ab Pd O

(⋃n∈N

Dn × Σn+1

)−→ Γ

Remarks.I D∗ → (Σ→ Γ) ⊆ T[Σ, Γ]

I D∗ → Σ ' T[1,Σ]

Indexed Structure

DZD Σt−→ Γ

O ab Pd O

(⋃n∈N

Dn × Σn+1

)−→ Γ

I D∗ → Σ ' T[1,Σ]

Indexed Structure

DZD Σt−→ Γ

O ab Pd O

(⋃n∈N

Dn × Σn+1

)−→ Γ

I D∗ → Σ ' T[1,Σ]

Indexed Structure

DZD Σt−→ Γ

O ab Pd O

(⋃n∈N

Dn × Σn+1

)−→ Γ

I D∗ → Σ ' T[1,Σ]

Indexed Structure

Comonoid IndexingIn DZD, each object

Σ = (Σ,1)

can be equipped with a comonoid structure:

DZD Σ −→ IO a

• Pd O

DZD Σ −→ Σ⊗ ΣO a

(a,a) Pd O

Hence for each Σ, there is a comonad Σ⊗ (−) on DZD.In the coKleisli category Kl(Σ):

Kl(Σ) A −→ BO (a,u)

v P(y ,d) O

P x (Kl(Σ)[A,B] = DZD[Σ⊗ A,B])

Indexed Structure

Σ = (Σ,1)

DZD Σ −→ IO a

• Pd O

(a,a) Pd O

Hence for each Σ, there is a comonad Σ⊗ (−) on DZD.

In the coKleisli category Kl(Σ):

v P(y ,d) O

Indexed Structure

Σ = (Σ,1)

DZD Σ −→ IO a

• Pd O

(a,a) Pd O

Hence for each Σ, there is a comonad Σ⊗ (−) on DZD.In the coKleisli category Kl(Σ):

v P(y ,d) O

Indexed Structure

The Category DialZTree morphisms t ∈ T[Σ, Γ] give functors

t∗ : Kl(Γ)→ Kl(Σ) with id∗Σ = idKl(Σ) and (t u)∗ = u∗ t∗

The Category DialZ: (by Grothendieck construction on (−)∗ : T→ Cat)I Objects: (Σ,U,X ), (Γ,V ,Y ), etcI Morphisms: (t , σ) : (Σ,U,X ) −→ (Γ,V ,Y ) where

t ∈ T[Σ, Γ] and σ : (Σ× U,X ) −→DZD (V ,Y )

(t∗(Γ,V ,Y ) = (Σ,V ,Y ) and Kl(Σ)[A,B] = DZD[Σ⊗ A,B])

Properties.I DialZ is fibred over T (via first projection).I The fibre over Σ is Kl(Σ).I The fibres are monoidal closed, with

A(Kl(Σ) B := A(DZD B

since Kl(Σ)[A⊗ B,C] = DZD[Σ⊗ A⊗ B,C]' DZD[Σ⊗ B,A(DZD C]= Kl(Σ)[B,A(Kl(Σ) C]

Indexed Structure

Properties.I DialZ is fibred over T (via first projection).I The fibre over Σ is Kl(Σ).

I The fibres are monoidal closed, with

Indexed Structure

Fibrewise Linear ArrowGiven automata A = (QA , qıA , U , X , δA , ΩA) over ∆

and tree morphisms t ∈ T[Σ,∆] and u ∈ T[Σ, Γ] let

Kl(Σ) A(t) −→ B(u)

(p,a,qA) (p,a,qB)O (a,u)

v P(y ,d) O

P x(p.d ,a.a,q′A) (p.d ,a.a,q′B)

whereq′A = δA(qA , t(p,a.a) , u , x , d)q′B = δB(qB , u(p,a.a) , v , y , d)

Indexed Structure

Kl(Σ) A(t) −→ B(u)

(p,a,qA) (p,a,qB)

O (a,u)v P

(y ,d) OP x

(p.d ,a.a,q′A) (p.d ,a.a,q′B)

Indexed Structure

Kl(Σ) A(t) −→ B(u)(p,a,qA) (p,a,qB)

O (a,u)v P

(y ,d) OP x

Indexed Structure

O (a,u)v P

(y ,d) OP x

Indexed Structure

O (a,u)v P

(y ,d) OP x

Indexed Structure

Categories of Tree AutomataThe Category DA.

I Objects over Σ: A(t) where A over Γ and t ∈ T[Σ, Γ].I Morphisms from A(u) (over Σ) to B(v) (over Γ) are pairs (t , σ) with

t ∈ T[Σ, Γ] and σ : A(u) −→Kl(Σ) B(v t)

Properties.I Fibre morphisms Σ;A(u) ` B(v) are strategies

σ : A(u) −→Kl(Σ) B(v)

I Substitution (over T) and fibred monoidal closure inherited from DialZ.

The Category Aut.I Objects over Σ: A over Σ. (A = A(id))I Morphisms (t , σ) where t is induced by a FinSet function ψ : Σ→ Γ.

Properties.I Fibred over FinSet:

δA[ψ](q,a,u, x ,d) := δA(q, ψ(a),u, x ,d) (ψ : Σ→ Γ)

I Fibrewise monoidal closed.I Soundness: If σ : A → B is winning then L(A) ⊆ L(B).

Indexed Structure

σ : A(u) −→Kl(Σ) B(v)

Indexed Structure

σ : A(u) −→Kl(Σ) B(v)

Indexed Structure

σ : A(u) −→Kl(Σ) B(v)

Indexed Structure

Existential Quantifications in AutFix a projection π : Σ× Γ→ Σ (in FinSet).Sound interpretation of

Σ;∃ΓA ` B

(actually adjunction ∃Σ,Γ ` π∗) + compatibility with substitution.

Existential quantifications in DialZ: (following the simple fibration)

∃Σ,Γ(Σ× Γ , U , X ) := (Σ , Γ× U , X ) ((Γ× U,X ) = Γ⊗ (U,X ))

Existential quantifications in Aut:Given

A = (QA,qıA,U,X , δA,ΩA) over Σ× Γ

let∃Σ,ΓA := (QA,qıA, Γ× U,X , δ∃ΓA,ΩA) over Σ

In particular:1;A(t) ` ∃ΣA

Indexed Structure

Σ;∃ΓA ` B

(actually adjunction ∃Σ,Γ ` π∗) + compatibility with substitution.Existential quantifications in DialZ: (following the simple fibration)

Indexed Structure

Σ;∃ΓA ` B

Indexed Structure

Σ;∃ΓA ` B

Indexed Structure

A Sound Deduction System – Examples of rulesAxioms, Cuts, and Substitution.

Σ ; A1(t1) , . . . , An(tn) , A(t) ` A(t)etc

Symmetric Monoidal Structure.

Σ ; A1(t1) , . . . , An(tn) , A(t) , B(u) ` C(v)

Σ ; A1(t1) , . . . , An(tn) , A(t)⊗ B(u) ` C(v)etc

(where A(t)⊗ B(u) = (A[π1]⊗ B[π2])〈t , u〉)Linear Arrow.

Σ ; A1(t1) , . . . , An(tn) , A(t) ` B(u)

Σ ; A1(t1) , . . . , An(tn) ` A(t)( B(u)etc

(where A(t)( B(u) = (A[π1]( B[π2])〈t , u〉)Falsity.

Σ ; A1(t1) , . . . , An(tn) , ⊥ ` A(t)

(+ Existential Quantifications)Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 19 / 26

Outline

Introduction

Indexed Structure

Conclusion

δA : QA × Σ −→ U × 1 −→ (D −→ QA)

Deduction rules for ExponentialsΣ; !A1, . . . , !An ` AΣ; !A1, . . . , !An ` !A

Σ;A1, . . . ,An,A ` BΣ;A1, . . . ,An, !A ` B

Σ;A1, . . . ,An ` BΣ;A1, . . . ,An, !A ` B

Σ;A1, . . . ,An, !A, !A ` BΣ;A1, . . . ,An, !A ` B

I Can be realized for!A = non-deterministic automaton equivalent to A

I Requires a form of positionality of strategies !(−) is not a functor in our setting.

Extraction from realizable existential statments: (well-known fact)I The rule

1; I ` ∃ΣA =⇒ there is t ∈ T[1,Σ] s.t. 1; I ` A(t)

in general only holds for non-deterministic automata.

δA : QA × Σ −→ U × 1 −→ (D −→ QA)

Σ;A1, . . . ,An,A ` BΣ;A1, . . . ,An, !A ` B

Σ;A1, . . . ,An ` BΣ;A1, . . . ,An, !A ` B

Σ;A1, . . . ,An, !A, !A ` BΣ;A1, . . . ,An, !A ` B

δA : QA × Σ −→ U × 1 −→ (D −→ QA)

Σ;A1, . . . ,An,A ` BΣ;A1, . . . ,An, !A ` B

Σ;A1, . . . ,An ` BΣ;A1, . . . ,An, !A ` B

Σ;A1, . . . ,An, !A, !A ` BΣ;A1, . . . ,An, !A ` B

δA : QA × Σ −→ U × 1 −→ (D −→ QA)

Σ;A1, . . . ,An,A ` BΣ;A1, . . . ,An, !A ` B

Σ;A1, . . . ,An ` BΣ;A1, . . . ,An, !A ` B

Σ;A1, . . . ,An, !A, !A ` BΣ;A1, . . . ,An, !A ` B

in general only holds for non-deterministic automata.Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 21 / 26

Non-Determinization (Simulation)Simulation Theorem (Emerson & Jutla, Muller & Schupp).

I If A is an alternating (and regular) automaton then there is anon-deterministic (and regular) !A with

L(!A) = L(A)

I Requires finite memory strategies.In our setting:

I The operation !A is an easy adaptation of known constructions.I Units I and ⊥ are non-deterministic.

Non-determinism preserved by _⊗ _ and ∃(_).The “co-unit law” Σ; !A ` A is easy to realize.

I The difficult point is the ruleΣ;N ` AΣ;N ` !A

requires positional strategies.Known fact:

I If N and A have parity conditions then Σ;N ` A has a Rabin winningcondition (disjunction of parity conditions).If P wins a Rabin game, then P has a positional winning strategy.

L(!A) = L(A)

I Requires finite memory strategies.

In our setting:I The operation !A is an easy adaptation of known constructions.I Units I and ⊥ are non-deterministic.

L(!A) = L(A)

I The operation !A is an easy adaptation of known constructions.

I Units I and ⊥ are non-deterministic.Non-determinism preserved by _⊗ _ and ∃(_).The “co-unit law” Σ; !A ` A is easy to realize.

L(!A) = L(A)

requires positional strategies.

Known fact:I If N and A have parity conditions then Σ;N ` A has a Rabin winning

condition (disjunction of parity conditions).If P wins a Rabin game, then P has a positional winning strategy.

L(!A) = L(A)

Weak CompletenessWeak Completeness.Assume

L(A) ⊆ L(B) (over Σ)

ThenL(∃Σ(!A⊗ !B⊥)) = ∅

and from the (regular) P-strategy witnessing

L(∃Σ(!A⊗ !B⊥) ( ⊥

)= T[1,1]

the categorical combinators give a strategy

σ : !A −→ ?B over Σ

(where ?B = (!B⊥)⊥)

Relation to classical logic:

Σ;A1 , . . . , An ` ((?A ⇒ ?B) ⇒ ?A) ⇒ ?A is derivable

(where A ⇒ B := !A( B)

Weak CompletenessWeak Completeness.Assume

L(A) ⊆ L(B) (over Σ)

ThenL(∃Σ(!A⊗ !B⊥)) = ∅

and from the (regular) P-strategy witnessing

L(∃Σ(!A⊗ !B⊥) ( ⊥

)= T[1,1]

the categorical combinators give a strategy

σ : !A −→ ?B over Σ

(where ?B = (!B⊥)⊥)Relation to classical logic:

Σ;A1 , . . . , An ` ((?A ⇒ ?B) ⇒ ?A) ⇒ ?A is derivable

(where A ⇒ B := !A( B)Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 23 / 26

Conclusion

Outline

Introduction

Indexed Structure

Conclusion

ConclusionA Dialectica-Like Approach to Tree Automata

I Categories with morphisms based on simple (linear) games.

I Symmetric Monoidal Closed Structure.I Fibred structure based on the simple fibration.

I Substitution and Existential quantification.I Non-determinization satisfies the deduction rules of the !(−) modality.I Not presented here:

I Zigzag games via a distributive law on “simple self-dualization” in the toposof trees.

Further works

I Application to the proof theory of MSO:I Intuitionistic and linear versions of MSO.I Weak completeness gives realizers for true negative formulas.I For ω-words: Application to extraction of synchronous realizers for∀∃-formulas (“Church synthesis”).

I !(−) as a proper LL exponential. Non-deterministic strategies ?

Conclusion

I Categories with morphisms based on simple (linear) games.I Symmetric Monoidal Closed Structure.

I Fibred structure based on the simple fibration.I Substitution and Existential quantification.

I Non-determinization satisfies the deduction rules of the !(−) modality.I Not presented here:

Further works

Conclusion

Further works

Conclusion

I Non-determinization satisfies the deduction rules of the !(−) modality.

I Not presented here:I Zigzag games via a distributive law on “simple self-dualization” in the topos

of trees.

Further works

Conclusion

Further works

Conclusion

Further works

Conclusion

Thanks for your attention !

http://perso.ens-lyon.fr/colin.riba/

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