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
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 2 / 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
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 2 / 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
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 2 / 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
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 2 / 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 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.
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 3 / 26
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)
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.
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 3 / 26
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)
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.
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 3 / 26
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)
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 ` C1 ; B ` A( B
I Symmetric monoidal closed structure.I Fibred structure (substitution) + existential quantification.I Deduction rules for the !(−) modality.
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 3 / 26
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)
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.
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 3 / 26
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)
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.
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 3 / 26
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)
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.
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 3 / 26
Zigzag Games for Tree Automata
Outline
Introduction
Zigzag Games for Tree Automata
Indexed Structure
Non-Deterministic Automata
Conclusion
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 4 / 26
Zigzag Games for Tree Automata
Tree AutomataAlternating tree automata A = (QA,qıA, δA,ΩA)where ΩA ⊆ Qω
A and
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
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 5 / 26
Zigzag Games for Tree Automata
Tree AutomataAlternating tree automata A = (QA,qıA, δA,ΩA)where ΩA ⊆ Qω
A and
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
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 5 / 26
Zigzag Games for Tree Automata
Tree AutomataAlternating tree automata A = (QA,qıA, δA,ΩA)where ΩA ⊆ Qω
A and
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
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 5 / 26
Zigzag Games for Tree Automata
Tree AutomataAlternating tree automata A = (QA,qıA, δA,ΩA)where ΩA ⊆ Qω
A and
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 take
A = (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
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 5 / 26
Zigzag Games for Tree Automata
Tree AutomataAlternating tree automata A = (QA,qıA, δA,ΩA)where ΩA ⊆ Qω
A and
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
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 5 / 26
Zigzag Games for Tree Automata
Tree AutomataAlternating tree automata A = (QA,qıA, δA,ΩA)where ΩA ⊆ Qω
A and
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
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 5 / 26
Zigzag Games for Tree Automata
Tree AutomataAlternating tree automata A = (QA,qıA, δA,ΩA)where ΩA ⊆ Qω
A and
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
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 5 / 26
Zigzag Games for Tree Automata
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
P x
Monoidal Structure:
A⊗ B := (U × V , X × Y ) with unit I := (1,1)
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 6 / 26
Zigzag Games for Tree Automata
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
P x
Monoidal Structure:
A⊗ B := (U × V , X × Y ) with unit I := (1,1)
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 6 / 26
Zigzag Games for Tree Automata
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
P x
Monoidal Structure:
A⊗ B := (U × V , X × Y ) with unit I := (1,1)
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 6 / 26
Zigzag Games for Tree Automata
D-Synchronicity
In DZ, the objectD := (1,D)
can be equipped with a monoid structure:
I −→ DO •
• Pd O
P •
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])
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 7 / 26
Zigzag Games for Tree Automata
D-SynchronicityIn DZ, the object
D := (1,D)
can be equipped with a monoid structure:
I −→ DO •
• Pd O
P •
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])
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 7 / 26
Zigzag Games for Tree Automata
D-SynchronicityIn DZ, the object
D := (1,D)
can be equipped with a monoid structure:
I −→ DO •
• Pd O
P •
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])
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 7 / 26
Zigzag Games for Tree Automata
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)
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 8 / 26
Zigzag Games for Tree Automata
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 uv P
(y ,d) OP 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)
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 8 / 26
Zigzag Games for Tree Automata
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 uv P
(y ,d) OP 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)
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 8 / 26
Zigzag Games for Tree Automata
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 uv P
(y ,d) OP 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)
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 8 / 26
Zigzag Games for Tree Automata
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 uv P
(y ,d) OP 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)
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 8 / 26
Zigzag Games for Tree Automata
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 uv P
(y ,d) OP 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)
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 8 / 26
Zigzag Games for Tree Automata
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
O
BΛ(σ)−→ (A(DZ C)
O vP
(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 )
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 9 / 26
Zigzag Games for Tree Automata
Monoidal ClosureConsider
A = (U,X ) B = (V ,Y ) C = (W ,Z )
Dialectica-Like arrow in DZ:
A(DZ C := (W U × X U×Z , U × Z )
A⊗ B σ−→ C
O
BΛ(σ)−→ (A(DZ C)
O vP
(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 )
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 9 / 26
Zigzag Games for Tree Automata
Monoidal ClosureConsider
A = (U,X ) B = (V ,Y ) C = (W ,Z )
Dialectica-Like arrow in DZ:
A(DZ C := (W U × X U×Z , U × Z )
A⊗ B σ−→ C
O
BΛ(σ)−→ (A(DZ C)
O v(?, ?) P
(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 )
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 9 / 26
Zigzag Games for Tree Automata
Monoidal ClosureConsider
A = (U,X ) B = (V ,Y ) C = (W ,Z )
Dialectica-Like arrow in DZ:
A(DZ C := (W U × X U×Z , U × Z )
A⊗ B σ−→ CO ( , v)
BΛ(σ)−→ (A(DZ C)
O v(?, ) P
(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 )
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 9 / 26
Zigzag Games for Tree Automata
Monoidal ClosureConsider
A = (U,X ) B = (V ,Y ) C = (W ,Z )
Dialectica-Like arrow in DZ:
A(DZ C := (W U × X U×Z , U × Z )
A⊗ B σ−→ CO (u, v)
BΛ(σ)−→ (A(DZ C)
O v(?, ) P
(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 )
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 9 / 26
Zigzag Games for Tree Automata
Monoidal ClosureConsider
A = (U,X ) B = (V ,Y ) C = (W ,Z )
Dialectica-Like arrow in DZ:
A(DZ C := (W U × X U×Z , U × Z )
A⊗ B σ−→ CO (u, v)
w P
BΛ(σ)−→ (A(DZ C)
O v(?, ) P
(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 )
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 9 / 26
Zigzag Games for Tree Automata
Monoidal ClosureConsider
A = (U,X ) B = (V ,Y ) C = (W ,Z )
Dialectica-Like arrow in DZ:
A(DZ C := (W U × X U×Z , U × Z )
A⊗ B σ−→ CO (u, v)
w P
BΛ(σ)−→ (A(DZ C)
O v(f , ) P
(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 )
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 9 / 26
Zigzag Games for Tree Automata
Monoidal ClosureConsider
A = (U,X ) B = (V ,Y ) C = (W ,Z )
Dialectica-Like arrow in DZ:
A(DZ C := (W U × X U×Z , U × Z )
A⊗ B σ−→ CO ( , v)
BΛ(σ)−→ (A(DZ C)
O v(f , ?) P
(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 )
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 9 / 26
Zigzag Games for Tree Automata
Monoidal ClosureConsider
A = (U,X ) B = (V ,Y ) C = (W ,Z )
Dialectica-Like arrow in DZ:
A(DZ C := (W U × X U×Z , U × Z )
A⊗ B σ−→ CO (u, v)
f (u) Pz O
BΛ(σ)−→ (A(DZ C)
O v(f , ?) P
(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 )
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 9 / 26
Zigzag Games for Tree Automata
Monoidal ClosureConsider
A = (U,X ) B = (V ,Y ) C = (W ,Z )
Dialectica-Like arrow in DZ:
A(DZ C := (W U × X U×Z , U × Z )
A⊗ B σ−→ CO (u, v)
f (u) Pz O
P (x , y)
BΛ(σ)−→ (A(DZ C)
O v(f , ?) P
(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 )
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 9 / 26
Zigzag Games for Tree Automata
Monoidal ClosureConsider
A = (U,X ) B = (V ,Y ) C = (W ,Z )
Dialectica-Like arrow in DZ:
A(DZ C := (W U × X U×Z , U × Z )
A⊗ B σ−→ CO (u, v)
f (u) Pz O
P (x , y)
BΛ(σ)−→ (A(DZ C)
O v(f ,F ) P
(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 )
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 9 / 26
Zigzag Games for Tree Automata
Monoidal ClosureConsider
A = (U,X ) B = (V ,Y ) C = (W ,Z )
Dialectica-Like arrow in DZ:
A(DZ C := (W U × X U×Z , U × Z )
A⊗ B σ−→ C
O
BΛ(σ)−→ (A(DZ C)
O v(f ,F ) P
(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 )
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 9 / 26
Zigzag Games for Tree Automata
Monoidal ClosureConsider
A = (U,X ) B = (V ,Y ) C = (W ,Z )
Dialectica-Like arrow in DZ:
A(DZ C := (W U × X U×Z , U × Z )
A⊗ B σ−→ C
O
BΛ(σ)−→ (A(DZ C)
O v(f ,F ) P(u, z) O
P 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 )
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 9 / 26
Zigzag Games for Tree Automata
Monoidal ClosureConsider
A = (U,X ) B = (V ,Y ) C = (W ,Z )
Dialectica-Like arrow in DZ:
A(DZ C := (W U × X U×Z , U × Z )
A⊗ B σ−→ CO (u, v)
f (u) Pz O
BΛ(σ)−→ (A(DZ C)
O v(f ,F ) P(u, z) O
P 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 )
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 9 / 26
Zigzag Games for Tree Automata
Monoidal ClosureConsider
A = (U,X ) B = (V ,Y ) C = (W ,Z )
Dialectica-Like arrow in DZ:
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)
BΛ(σ)−→ (A(DZ C)
O v(f ,F ) P(u, z) O
P 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 )
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 9 / 26
Zigzag Games for Tree Automata
Monoidal ClosureConsider
A = (U,X ) B = (V ,Y ) C = (W ,Z )
Dialectica-Like arrow in DZ:
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)
BΛ(σ)−→ (A(DZ C)
O v(f ,F ) P(u, z) O
P 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 )
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 9 / 26
Zigzag Games for Tree Automata
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−→ CO (u, v)
f (u) Pz O
P (F (u, z), y)
BΛ(σ)−→ (A(DZ C)
O v(f ,F ) P(u, z) O
P 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 )
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 9 / 26
Zigzag Games for Tree Automata
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−→ CO (u, v)
f (u) Pz O
P (F (u, z), y)
BΛ(σ)−→ (A(DZ C)
O v(f ,F ) P(u, z) O
P 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 )
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 9 / 26
Zigzag Games for Tree Automata
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)
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 10 / 26
Zigzag Games for Tree Automata
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)
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 10 / 26
Zigzag Games for Tree Automata
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)
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 10 / 26
Zigzag Games for Tree Automata
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)
withδA⊗B((qA,qB) , a , (u, v) , (x , y) , d) := (q′A,q
′B)
where
q′A := δA(qA , a , u , x , d) and q′B := δB(qB , a , v , y , d)
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)
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 10 / 26
Zigzag Games for Tree Automata
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)
withδA(B((qA,qB) , a , (f ,F ) , (u, y) , d) := (q′A,q
′B)
where
q′A := δA(qA , a , u , F (u, y ,d) , d) and q′B := δB(qB , a , f (u) , y , d)
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)
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 10 / 26
Zigzag Games for Tree Automata
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)
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 10 / 26
Zigzag Games for Tree Automata
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)
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 10 / 26
Indexed Structure
Outline
Introduction
Zigzag Games for Tree Automata
Indexed Structure
Non-Deterministic Automata
Conclusion
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 11 / 26
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])
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 12 / 26
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])
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 12 / 26
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])
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 12 / 26
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])
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 12 / 26
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])
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 12 / 26
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])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
P •
' t :
(⋃n∈N
Dn × Σn+1
)−→ Γ
Remarks.I D∗ → (Σ→ Γ) ⊆ T[Σ, Γ]
I D∗ → Σ ' T[1,Σ]
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 13 / 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
P •
' t :
(⋃n∈N
Dn × Σn+1
)−→ Γ
Remarks.I D∗ → (Σ→ Γ) ⊆ T[Σ, Γ]
I D∗ → Σ ' T[1,Σ]
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 13 / 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
P •
' t :
(⋃n∈N
Dn × Σn+1
)−→ Γ
Remarks.I D∗ → (Σ→ Γ) ⊆ T[Σ, Γ]
I D∗ → Σ ' T[1,Σ]
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 13 / 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
P •
' t :
(⋃n∈N
Dn × Σn+1
)−→ Γ
Remarks.I D∗ → (Σ→ Γ) ⊆ T[Σ, Γ]
I D∗ → Σ ' T[1,Σ]
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 13 / 26
Indexed Structure
Comonoid IndexingIn DZD, each object
Σ = (Σ,1)
can be equipped with a comonoid structure:
DZD Σ −→ IO a
• Pd O
P •
DZD Σ −→ Σ⊗ ΣO a
(a,a) Pd O
P •
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])
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 14 / 26
Indexed Structure
Comonoid IndexingIn DZD, each object
Σ = (Σ,1)
can be equipped with a comonoid structure:
DZD Σ −→ IO a
• Pd O
P •
DZD Σ −→ Σ⊗ ΣO a
(a,a) Pd O
P •
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])
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 14 / 26
Indexed Structure
Comonoid IndexingIn DZD, each object
Σ = (Σ,1)
can be equipped with a comonoid structure:
DZD Σ −→ IO a
• Pd O
P •
DZD Σ −→ Σ⊗ ΣO a
(a,a) Pd O
P •
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])
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 14 / 26
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]
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 15 / 26
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]
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 15 / 26
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]
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 15 / 26
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]
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 15 / 26
Indexed Structure
Fibrewise Linear ArrowGiven automata A = (QA , qıA , U , X , δA , ΩA) over ∆
and B = (QB , qıB , V , Y , δB , ΩB) over Γ
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)
I A play is winning (for P) iff
qıA · . . . · qA · q′A · . . . ∈ ΩA =⇒ qıB · . . . · qB · q′B · . . . ∈ ΩB
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 16 / 26
Indexed Structure
Fibrewise Linear ArrowGiven automata A = (QA , qıA , U , X , δA , ΩA) over ∆
and B = (QB , qıB , V , Y , δB , ΩB) over Γ
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) OP 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)
I A play is winning (for P) iff
qıA · . . . · qA · q′A · . . . ∈ ΩA =⇒ qıB · . . . · qB · q′B · . . . ∈ ΩB
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 16 / 26
Indexed Structure
Fibrewise Linear ArrowGiven automata A = (QA , qıA , U , X , δA , ΩA) over ∆
and B = (QB , qıB , V , Y , δB , ΩB) over Γ
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) OP 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)
I A play is winning (for P) iff
qıA · . . . · qA · q′A · . . . ∈ ΩA =⇒ qıB · . . . · qB · q′B · . . . ∈ ΩB
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 16 / 26
Indexed Structure
Fibrewise Linear ArrowGiven automata A = (QA , qıA , U , X , δA , ΩA) over ∆
and B = (QB , qıB , V , Y , δB , ΩB) over Γ
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) OP 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)
I A play is winning (for P) iff
qıA · . . . · qA · q′A · . . . ∈ ΩA =⇒ qıB · . . . · qB · q′B · . . . ∈ ΩB
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 16 / 26
Indexed Structure
Fibrewise Linear ArrowGiven automata A = (QA , qıA , U , X , δA , ΩA) over ∆
and B = (QB , qıB , V , Y , δB , ΩB) over Γ
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) OP 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)
I A play is winning (for P) iff
qıA · . . . · qA · q′A · . . . ∈ ΩA =⇒ qıB · . . . · qB · q′B · . . . ∈ ΩB
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 16 / 26
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).
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 17 / 26
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).
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 17 / 26
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).
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 17 / 26
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).
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 17 / 26
Indexed Structure
Existential Quantifications in AutFix a projection π : Σ× Γ→ Σ (in FinSet).Sound interpretation of
Σ;∃ΓA ` BΣ× Γ;A ` B(π)
andΣ× Γ;A ` B(π)
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
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 18 / 26
Indexed Structure
Existential Quantifications in AutFix a projection π : Σ× Γ→ Σ (in FinSet).Sound interpretation of
Σ;∃ΓA ` BΣ× Γ;A ` B(π)
andΣ× Γ;A ` B(π)
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
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 18 / 26
Indexed Structure
Existential Quantifications in AutFix a projection π : Σ× Γ→ Σ (in FinSet).Sound interpretation of
Σ;∃ΓA ` BΣ× Γ;A ` B(π)
andΣ× Γ;A ` B(π)
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
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 18 / 26
Indexed Structure
Existential Quantifications in AutFix a projection π : Σ× Γ→ Σ (in FinSet).Sound interpretation of
Σ;∃ΓA ` BΣ× Γ;A ` B(π)
andΣ× Γ;A ` B(π)
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
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 18 / 26
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
Non-Deterministic Automata
Outline
Introduction
Zigzag Games for Tree Automata
Indexed Structure
Non-Deterministic Automata
Conclusion
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 20 / 26
Non-Deterministic Automata
Non-Deterministic Automata
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.
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 21 / 26
Non-Deterministic Automata
Non-Deterministic Automata
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.
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 21 / 26
Non-Deterministic Automata
Non-Deterministic Automata
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.
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 21 / 26
Non-Deterministic Automata
Non-Deterministic Automata
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.Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 21 / 26
Non-Deterministic Automata
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.
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 22 / 26
Non-Deterministic Automata
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.
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 22 / 26
Non-Deterministic Automata
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.
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 22 / 26
Non-Deterministic Automata
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.
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 22 / 26
Non-Deterministic Automata
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 winning
condition (disjunction of parity conditions).If P wins a Rabin game, then P has a positional winning strategy.
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 22 / 26
Non-Deterministic Automata
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.
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 22 / 26
Non-Deterministic Automata
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
Non-Deterministic Automata
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
Zigzag Games for Tree Automata
Indexed Structure
Non-Deterministic Automata
Conclusion
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 24 / 26
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 ?
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 25 / 26
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 ?
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 25 / 26
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 ?
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 25 / 26
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 topos
of 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 ?
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 25 / 26
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 ?
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 25 / 26
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 ?
Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 25 / 26
Conclusion
Thanks for your attention !
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Colin RIBA (LIP - ENS de Lyon) Dialectica Tree Automata 26 / 26
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