Denotations for parity automata

S. Salvati INRIA, I. Walukiewicz CNRSUniversité de Bordeaux

Shonan Meeting: Higher-Order Model Checking

Verification and Models

Schematology

Programs

Schemes

Evaluation treesTyping/Logic/… Semantics

verifi

catio

n execution

adequacy

syn.abst.

syn.exec.

evaluationsyn. prop.

Schematology

Programs

Schemes

Evaluation treesTyping/Logic/…

Semantics

verifi

catio

n

execution

adequacy

syn.abst.

syn.exec.

evaluationsyn. prop.

Schematology

Programs

Schemes

Evaluation trees

Typing/Logic/… Semantics

verifi

catio

n execution

adequacy

syn.abst.

syn.exec.

evaluationsyn. prop.

Schematology

Programs

Schemes

Evaluation trees

Typing/Logic/… Semantics

verifi

catio

n execution

adequacy

syn.abst.

syn.exec.

evaluationsyn. prop.

Schematology

Programs

Schemes

Evaluation trees

Typing/Logic/… Semantics

verifi

catio

n execution

adequacy

syn.abst.

syn.exec.

evaluationsyn. prop.

Schematology

Programs

Schemes

Evaluation treesTyping/Logic/… Semantics

verifi

catio

n execution

adequacy

syn.abst.

syn.exec.

evaluationsyn. prop.

Schematology

Programs

Schemes

Evaluation treesTyping/Logic/… Semantics

verifi

catio

n execution

adequacy

syn.abst.

syn.exec.

evaluation

syn. prop.

Schematology

Programs

Schemes

Evaluation treesTyping/Logic/… Semantics

verifi

catio

n execution

adequacy

syn.abst.

syn.exec.

evaluationsyn. prop.

Schematology

Programs

Schemes

Evaluation treesTyping/Logic/… Semantics

verifi

catio

n execution

adequacy

syn.abst.

syn.exec.

evaluation

abst.

syn. prop.

Schematology

Programs

Schemes

Evaluation treesSemantic Domains Semantics

verifi

catio

n execution

adequacy

syn.abst.

syn.exec.

evaluation

abst.

syn. prop.

Programs and recognizability

Specification

Higher-OrderPrograms

CorrectPrograms

InterpretationDomain

Recognizer [[·]]

[[·]]−1

Programs and recognizability

Specification

Higher-OrderPrograms

CorrectPrograms

InterpretationDomain

Recognizer [[·]]

[[·]]−1

Programs and recognizability

Specification

Higher-OrderPrograms

CorrectPrograms

InterpretationDomain

Recognizer [[·]]

[[·]]−1

Motivations

I Relating finite state methods with denotational methods

I Reveal the invariants behind behavioral propertiesI Obtain decidability results by finiteness propertiesI Compositional and Higher-Order by construction

Motivations

I Relating finite state methods with denotational methodsI Reveal the invariants behind behavioral properties

I Obtain decidability results by finiteness propertiesI Compositional and Higher-Order by construction

Motivations

I Relating finite state methods with denotational methodsI Reveal the invariants behind behavioral propertiesI Obtain decidability results by finiteness properties

I Compositional and Higher-Order by construction

Motivations

I Relating finite state methods with denotational methodsI Reveal the invariants behind behavioral propertiesI Obtain decidability results by finiteness propertiesI Compositional and Higher-Order by construction

Types: 0 is a type and (A → B) is a type if A and ̲ are types.

Tree signature Σ = {a, b, . . . } all constants of type 0 → 0 → 0 orof type 0.

λY -calculus

ΛY : MA,NB ::= xA | cA | (λxA.MB)A→B | (MA→BNA)B| (YMA→A)A

(β) (λx .M)N = M[N/x ]

(η) λx .Mx = M when x /∈ fv(M)

(δ) YM = M(YM)

Böhm tree for ΛY

Böhm trees are a sort of infinite normal form for ΛY -terms

If M reduces to λx1 . . . xn.hM1 . . .Mn:

BT (M) = λx1 . . . xn.h

BT (M1) BT (Mn)

otherwise:BT (M) = Ω

When M is closed and of type 0, BT (M) is an infinite tree: ahigher-order tree.

Böhm tree for ΛY

Böhm trees are a sort of infinite normal form for ΛY -terms

If M reduces to λx1 . . . xn.hM1 . . .Mn:

BT (M) = λx1 . . . xn.h

BT (M1) BT (Mn)

otherwise:BT (M) = Ω

When M is closed and of type 0, BT (M) is an infinite tree: ahigher-order tree.

Böhm tree for ΛY

Böhm trees are a sort of infinite normal form for ΛY -terms

If M reduces to λx1 . . . xn.hM1 . . .Mn:

BT (M) = λx1 . . . xn.h

BT (M1) BT (Mn)

otherwise:BT (M) = Ω

When M is closed and of type 0, BT (M) is an infinite tree: ahigher-order tree.

Böhm tree for ΛY

Böhm trees are a sort of infinite normal form for ΛY -terms

If M reduces to λx1 . . . xn.hM1 . . .Mn:

BT (M) = λx1 . . . xn.h

BT (M1) BT (Mn)

otherwise:BT (M) = Ω

When M is closed and of type 0, BT (M) is an infinite tree: ahigher-order tree.

Higher-order control flowfold f a l = if l=[] then a else f (hd l) (fold f a (tl l))

M = Yλfold f a l.ite (=l []) a (f (hd l) (fold f a (tl l)))

ite

=

l []

[] ::

f

hd

l

ite

=

tl

l

[]

[] ite

=

tl

tl

l

[]

[] ::

f

hd

hd

l

n

n-1

n

λfal .

Finite models

[[M,ν ]]

Axioms[[MN, ν]] = [[M, ν]] • [[N, ν]]

[[λx .M, ν]] • f = [[M, ν[f /x ]]][[Y , ν]] • f = f • ([[Y , ν]] • f )

Lemma (Correctness)If M =βδ N, then for every ν,[[M, ν]] = [[N, ν]].

A B C

A → B

(A → B) → C

f

g

f • g

Finite models

[[M,ν ]]

Axioms[[MN, ν]] = [[M, ν]] • [[N, ν]]

[[λx .M, ν]] • f = [[M, ν[f /x ]]][[Y , ν]] • f = f • ([[Y , ν]] • f )

Lemma (Correctness)If M =βδ N, then for every ν,[[M, ν]] = [[N, ν]].

A B C

A → B

(A → B) → C

f

g

f • g

Finite models

[[M,ν ]]

Axioms[[MN, ν]] = [[M, ν]] • [[N, ν]]

[[λx .M, ν]] • f = [[M, ν[f /x ]]][[Y , ν]] • f = f • ([[Y , ν]] • f )

Lemma (Correctness)If M =βδ N, then for every ν,[[M, ν]] = [[N, ν]].

A B C

A → B

(A → B) → C

f

g

f • g

Finite models

[[M,ν ]]

Axioms[[MN, ν]] = [[M, ν]] • [[N, ν]]

[[λx .M, ν]] • f = [[M, ν[f /x ]]][[Y , ν]] • f = f • ([[Y , ν]] • f )

Lemma (Correctness)If M =βδ N, then for every ν,[[M, ν]] = [[N, ν]].

A B C

A → B

(A → B) → C

f

g

f • g

Finite models

[[M,ν ]]

Axioms[[MN, ν]] = [[M, ν]] • [[N, ν]]

[[λx .M, ν]] • f = [[M, ν[f /x ]]][[Y , ν]] • f = f • ([[Y , ν]] • f )

Lemma (Correctness)If M =βδ N, then for every ν,[[M, ν]] = [[N, ν]].

A B C

A → B

(A → B) → C

f

g

f • g

Finite models

[[M,ν ]]

Axioms[[MN, ν]] = [[M, ν]] • [[N, ν]][[λx .M, ν]] • f = [[M, ν[f /x ]]]

[[Y , ν]] • f = f • ([[Y , ν]] • f )

Lemma (Correctness)If M =βδ N, then for every ν,[[M, ν]] = [[N, ν]].

A B C

A → B

(A → B) → C

f

g

f • g

Finite models

[[M,ν ]]

Axioms[[MN, ν]] = [[M, ν]] • [[N, ν]][[λx .M, ν]] • f = [[M, ν[f /x ]]][[Y , ν]] • f = f • ([[Y , ν]] • f )

Lemma (Correctness)If M =βδ N, then for every ν,[[M, ν]] = [[N, ν]].

A B C

A → B

(A → B) → C

f

g

f • g

Finite models

[[M,ν ]]

Axioms[[MN, ν]] = [[M, ν]] • [[N, ν]][[λx .M, ν]] • f = [[M, ν[f /x ]]][[Y , ν]] • f = f • ([[Y , ν]] • f )

Lemma (Correctness)If M =βδ N, then for every ν,[[M, ν]] = [[N, ν]].

A B C

A → B

(A → B) → C

f

g

f • g

Recognizability in the simply typed λ-calculus

L is recognizable iff:

L = {M | [[M, ∅]] ∈ R}

A

RL

[[_, ∅]]−1

Recognizability in the simply typed λ-calculus

L is recognizable iff:

L = {M | [[M, ∅]] ∈ R}

A

R

L[[_, ∅]]−1

Recognizability in the simply typed λ-calculus

L is recognizable iff:

L = {M | [[M, ∅]] ∈ R}

A

RL

[[_, ∅]]−1

Basic properties

Recognizable languages of λ-terms are:I conservative extensions of recognizable languages of strings

and trees,

I closed under boolean operations,I closed under inverse higher-order homomorphism,I not closed under relabeling.

I Singleton languages are recognizable [Statman 82]I Emptiness is undecidable [Loader 01]I Membership is non-elementary

Basic properties

Recognizable languages of λ-terms are:I conservative extensions of recognizable languages of strings

and trees,I closed under boolean operations,

I closed under inverse higher-order homomorphism,I not closed under relabeling.

I Singleton languages are recognizable [Statman 82]I Emptiness is undecidable [Loader 01]I Membership is non-elementary

Basic properties

Recognizable languages of λ-terms are:I conservative extensions of recognizable languages of strings

and trees,I closed under boolean operations,I closed under inverse higher-order homomorphism,

I not closed under relabeling.

I Singleton languages are recognizable [Statman 82]I Emptiness is undecidable [Loader 01]I Membership is non-elementary

Basic properties

Recognizable languages of λ-terms are:I conservative extensions of recognizable languages of strings

and trees,I closed under boolean operations,I closed under inverse higher-order homomorphism,I not closed under relabeling.

I Singleton languages are recognizable [Statman 82]I Emptiness is undecidable [Loader 01]I Membership is non-elementary

Basic properties

Recognizable languages of λ-terms are:I conservative extensions of recognizable languages of strings

and trees,I closed under boolean operations,I closed under inverse higher-order homomorphism,I not closed under relabeling.

I Singleton languages are recognizable [Statman 82]

I Emptiness is undecidable [Loader 01]I Membership is non-elementary

Basic properties

Recognizable languages of λ-terms are:I conservative extensions of recognizable languages of strings

and trees,I closed under boolean operations,I closed under inverse higher-order homomorphism,I not closed under relabeling.

I Singleton languages are recognizable [Statman 82]I Emptiness is undecidable [Loader 01]

I Membership is non-elementary

Basic properties

Recognizable languages of λ-terms are:I conservative extensions of recognizable languages of strings

and trees,I closed under boolean operations,I closed under inverse higher-order homomorphism,I not closed under relabeling.

I Singleton languages are recognizable [Statman 82]I Emptiness is undecidable [Loader 01]I Membership is non-elementary

Theory of Böhm trees

ΛY -theoriesBöhm theories Init.

Finite models

MSOL

Böhm theories: BT (M) = BT (N) implies for all ν, [[M, ν]] = [[N, ν]]

I expressiveness of finite Böhm models? (See Pawel’s talk)I axiomatization of finite Böhm models?

Theory of Böhm trees

ΛY -theoriesBöhm theories Init.

Finite models

MSOL

Böhm theories: BT (M) = BT (N) implies for all ν, [[M, ν]] = [[N, ν]]

I expressiveness of finite Böhm models? (See Pawel’s talk)I axiomatization of finite Böhm models?

Theory of Böhm trees

ΛY -theoriesBöhm theories Init.

Finite models

MSOL

Böhm theories: BT (M) = BT (N) implies for all ν, [[M, ν]] = [[N, ν]]

I expressiveness of finite Böhm models? (See Pawel’s talk)I axiomatization of finite Böhm models?

Theory of Böhm trees

ΛY -theoriesBöhm theories Init.

Finite models

MSOL

Böhm theories: BT (M) = BT (N) implies for all ν, [[M, ν]] = [[N, ν]]

I expressiveness of finite Böhm models? (See Pawel’s talk)I axiomatization of finite Böhm models?

Theory of Böhm trees

ΛY -theoriesBöhm theories Init.

Finite models

MSOL

Böhm theories: BT (M) = BT (N) implies for all ν, [[M, ν]] = [[N, ν]]

I expressiveness of finite Böhm models? (See Pawel’s talk)

I axiomatization of finite Böhm models?

Theory of Böhm trees

ΛY -theoriesBöhm theories Init.

Finite models

MSOL

Böhm theories: BT (M) = BT (N) implies for all ν, [[M, ν]] = [[N, ν]]

I expressiveness of finite Böhm models? (See Pawel’s talk)I axiomatization of finite Böhm models?

Theory of Böhm trees

ΛY -theoriesBöhm theories Init.

Finite models

MSOL

Böhm theories: BT (M) = BT (N) implies for all ν, [[M, ν]] = [[N, ν]]

I expressiveness of finite Böhm models? (See Pawel’s talk)I axiomatization of finite Böhm models?

Monotone (Scott) models

((MA,≤A)A, [[_,_]])I (M0,≤0) is a complete lattice,

I (MA→B ,≤A→B) is the complete lattice of monotone functionsf from (MA,≤A) to (MB ,≤B), i.e. a ≤A b impliesf (a) ≤B f (b), ordered pointwise.

I [[Y , ν]](f ) =∧{f n(>) | n ∈ N}.

I given f ∈ MA, g ∈ MB , (f 7→ g)(h) ={

g when g ≤ h⊥ otherwise

Monotone (Scott) models

((MA,≤A)A, [[_,_]])I (M0,≤0) is a complete lattice,I (MA→B ,≤A→B) is the complete lattice of monotone functions

f from (MA,≤A) to (MB ,≤B), i.e. a ≤A b impliesf (a) ≤B f (b), ordered pointwise.

I [[Y , ν]](f ) =∧{f n(>) | n ∈ N}.

I given f ∈ MA, g ∈ MB , (f 7→ g)(h) ={

g when g ≤ h⊥ otherwise

Monotone (Scott) models

((MA,≤A)A, [[_,_]])I (M0,≤0) is a complete lattice,I (MA→B ,≤A→B) is the complete lattice of monotone functions

f from (MA,≤A) to (MB ,≤B), i.e. a ≤A b impliesf (a) ≤B f (b), ordered pointwise.

I [[Y , ν]](f ) =∧{f n(>) | n ∈ N}.

I given f ∈ MA, g ∈ MB , (f 7→ g)(h) ={

g when g ≤ h⊥ otherwise

Monotone (Scott) models

((MA,≤A)A, [[_,_]])I (M0,≤0) is a complete lattice,I (MA→B ,≤A→B) is the complete lattice of monotone functions

f from (MA,≤A) to (MB ,≤B), i.e. a ≤A b impliesf (a) ≤B f (b), ordered pointwise.

I [[Y , ν]](f ) =∧{f n(>) | n ∈ N}.

I given f ∈ MA, g ∈ MB , (f 7→ g)(h) ={

g when g ≤ h⊥ otherwise

Digretion: a extensional non-Böhm model

Take (M0,≤) = (P({q0, q1}),⊆), we let M be the model so that(MA,≤A) is generated as in the monotone model. We then let:

I S ↓0= S ∩ {q0} for S ⊆ Q,I f ↓0 (g) = f (g) ↓0 for f of type A → B and g of type A.

We then define:fix(f ) = νx .f (x ↓0)))

I for f1 = q1 7→ q0 ∨ q0 7→ q1, we have fix(f1) = ∅,I and for f2 = q0 7→ q0 ∨ q1 7→ q1, we have fix(f2) = {q0}

But f2 = f1 ◦ f1.So if a is a constant so that [[a]] = f1, interpreting Y as fix gives[[Y (λx .ax)]] = ∅ and [[Y (λx .a(ax))]] = {q0}.

Digretion: a extensional non-Böhm model

Take (M0,≤) = (P({q0, q1}),⊆), we let M be the model so that(MA,≤A) is generated as in the monotone model. We then let:

I S ↓0= S ∩ {q0} for S ⊆ Q,I f ↓0 (g) = f (g) ↓0 for f of type A → B and g of type A.

We then define:fix(f ) = νx .f (x ↓0)))

I for f1 = q1 7→ q0 ∨ q0 7→ q1, we have fix(f1) = ∅,I and for f2 = q0 7→ q0 ∨ q1 7→ q1, we have fix(f2) = {q0}

But f2 = f1 ◦ f1.So if a is a constant so that [[a]] = f1, interpreting Y as fix gives[[Y (λx .ax)]] = ∅ and [[Y (λx .a(ax))]] = {q0}.

Digretion: a extensional non-Böhm model

Take (M0,≤) = (P({q0, q1}),⊆), we let M be the model so that(MA,≤A) is generated as in the monotone model. We then let:

I S ↓0= S ∩ {q0} for S ⊆ Q,I f ↓0 (g) = f (g) ↓0 for f of type A → B and g of type A.

We then define:fix(f ) = νx .f (x ↓0)))

I for f1 = q1 7→ q0 ∨ q0 7→ q1, we have fix(f1) = ∅,I and for f2 = q0 7→ q0 ∨ q1 7→ q1, we have fix(f2) = {q0}

But f2 = f1 ◦ f1.So if a is a constant so that [[a]] = f1, interpreting Y as fix gives[[Y (λx .ax)]] = ∅ and [[Y (λx .a(ax))]] = {q0}.

Digretion: a extensional non-Böhm model

Take (M0,≤) = (P({q0, q1}),⊆), we let M be the model so that(MA,≤A) is generated as in the monotone model. We then let:

I S ↓0= S ∩ {q0} for S ⊆ Q,I f ↓0 (g) = f (g) ↓0 for f of type A → B and g of type A.

We then define:fix(f ) = νx .f (x ↓0)))

I for f1 = q1 7→ q0 ∨ q0 7→ q1, we have fix(f1) = ∅,I and for f2 = q0 7→ q0 ∨ q1 7→ q1, we have fix(f2) = {q0}

But f2 = f1 ◦ f1.So if a is a constant so that [[a]] = f1, interpreting Y as fix gives[[Y (λx .ax)]] = ∅ and [[Y (λx .a(ax))]] = {q0}.

Ω-blind trivial propertiesTheorem (S. Waluckiewicz 13)Scott models recognize boolean combinations of Ω-blind trivial properties.

a

b

a

Ω b

a

b b

b

a

b b

a

b b

q1

q2 q2

q1 q1 q1 q1

q2 q2 q2 q2q2 q2 q2 q2

δ(a, q1) = (q2, q2) δ(b, q2) = (q1, q1)

The recognizing model is defined: (M0,≤0) = (P({q1, q2},⊆))In [S. Walukiewicz 13], it is showed how to build insightful models.

Ω-blind trivial propertiesTheorem (S. Waluckiewicz 13)Scott models recognize boolean combinations of Ω-blind trivial properties.

a

b

a

Ω b

a

b b

b

a

b b

a

b b

q1

q2 q2

q1 q1 q1 q1

q2 q2 q2 q2q2 q2 q2 q2

δ(a, q1) = (q2, q2) δ(b, q2) = (q1, q1)

The recognizing model is defined: (M0,≤0) = (P({q1, q2},⊆))In [S. Walukiewicz 13], it is showed how to build insightful models.

Ω-blind trivial propertiesTheorem (S. Waluckiewicz 13)Scott models recognize boolean combinations of Ω-blind trivial properties.

a

b

a

Ω b

a

b b

b

a

b b

a

b b

q1

q2 q2

q1 q1 q1 q1

q2 q2 q2 q2q2 q2 q2 q2

δ(a, q1) = (q2, q2) δ(b, q2) = (q1, q1)

The recognizing model is defined: (M0,≤0) = (P({q1, q2},⊆))In [S. Walukiewicz 13], it is showed how to build insightful models.

Ω-blind trivial propertiesTheorem (S. Waluckiewicz 13)Scott models recognize boolean combinations of Ω-blind trivial properties.

a

b

a

Ω b

a

b b

b

a

b b

a

b b

q1

q2 q2

q1 q1 q1 q1

q2 q2 q2 q2q2 q2 q2 q2

δ(a, q1) = (q2, q2) δ(b, q2) = (q1, q1)

The recognizing model is defined: (M0,≤0) = (P({q1, q2},⊆))In [S. Walukiewicz 13], it is showed how to build insightful models.

Ω-blind trivial propertiesTheorem (S. Waluckiewicz 13)Scott models recognize boolean combinations of Ω-blind trivial properties.

a

b

a

Ω b

a

b b

b

a

b b

a

b b

q1

q2 q2

q1 q1 q1 q1

q2 q2 q2 q2q2 q2 q2 q2

δ(a, q1) = (q2, q2) δ(b, q2) = (q1, q1)

The recognizing model is defined: (M0,≤0) = (P({q1, q2},⊆))In [S. Walukiewicz 13], it is showed how to build insightful models.

Ω-blind trivial propertiesTheorem (S. Waluckiewicz 13)Scott models recognize boolean combinations of Ω-blind trivial properties.

a

b

a

Ω b

a

b b

b

a

b b

a

b b

q1

q2 q2

q1 q1 q1 q1

q2 q2 q2 q2q2 q2 q2 q2

δ(a, q1) = (q2, q2) δ(b, q2) = (q1, q1)

The recognizing model is defined: (M0,≤0) = (P({q1, q2},⊆))In [S. Walukiewicz 13], it is showed how to build insightful models.

Ω-blind trivial propertiesTheorem (S. Waluckiewicz 13)Scott models recognize boolean combinations of Ω-blind trivial properties.

a

b

a

Ω b

a

b b

b

a

b b

a

b b

q1

q2 q2

q1 q1 q1 q1

q2 q2 q2 q2q2 q2 q2 q2

δ(a, q1) = (q2, q2) δ(b, q2) = (q1, q1)

The recognizing model is defined: (M0,≤0) = (P({q1, q2},⊆))In [S. Walukiewicz 13], it is showed how to build insightful models.

Ω-blind trivial propertiesTheorem (S. Waluckiewicz 13)Scott models recognize boolean combinations of Ω-blind trivial properties.

a

b

a

Ω b

a

b b

b

a

b b

a

b b

q1

q2 q2

q1 q1 q1 q1

q2 q2 q2 q2q2 q2 q2 q2

δ(a, q1) = (q2, q2) δ(b, q2) = (q1, q1)

The recognizing model is defined: (M0,≤0) = (P({q1, q2},⊆))

In [S. Walukiewicz 13], it is showed how to build insightful models.

Ω-blind trivial propertiesTheorem (S. Waluckiewicz 13)Scott models recognize boolean combinations of Ω-blind trivial properties.

a

b

a

Ω b

a

b b

b

a

b b

a

b b

q1

q2 q2

q1 q1 q1 q1

q2 q2 q2 q2q2 q2 q2 q2

δ(a, q1) = (q2, q2) δ(b, q2) = (q1, q1)

The recognizing model is defined: (M0,≤0) = (P({q1, q2},⊆))In [S. Walukiewicz 13], it is showed how to build insightful models.

First step towards ParityConditions: weak MSOL

weak MSOLa

b

a

Ω b

a

b b

b

a

b b

a

b b

q2

q2 q2

q1q2 q1q2 q1q2 q1q2

q0q2 q0q2 q0q2 q0q2q0q2 q0q2 q0q2 q0q2

δ(a, q1) ={{(q0, q0)}}

rk(qi) = i

δ(b, q2) ={{(q1, q1), (q2, q2)}}δ(a, q0) = δ(b, q0){{(q0, q0)}}

δ(a, q2) ={{(q2, q2)}}δ(b, q1) ={{(q1, q1)}}

weak MSOLa

b

a

Ω b

a

b b

b

a

b b

a

b b

q2

q2 q2

q1q2 q1q2 q1q2 q1q2

q0q2 q0q2 q0q2 q0q2q0q2 q0q2 q0q2 q0q2

δ(a, q1) ={{(q0, q0)}} rk(qi) = iδ(b, q2) ={{(q1, q1), (q2, q2)}}

δ(a, q0) = δ(b, q0){{(q0, q0)}}δ(a, q2) ={{(q2, q2)}}δ(b, q1) ={{(q1, q1)}}

weak MSOLa

b

a

Ω b

a

b b

b

a

b b

a

b b

q2

q2 q2

q1q2 q1q2 q1q2 q1q2

q0q2 q0q2 q0q2 q0q2q0q2 q0q2 q0q2 q0q2

δ(a, q1) ={{(q0, q0)}} rk(qi) = iδ(b, q2) ={{(q1, q1), (q2, q2)}}

δ(a, q0) = δ(b, q0){{(q0, q0)}}δ(a, q2) ={{(q2, q2)}}δ(b, q1) ={{(q1, q1)}}

weak MSOLa

b

a

Ω b

a

b b

b

a

b b

a

b b

q2

q2 q2

q1q2 q1q2 q1q2 q1q2

q0q2 q0q2 q0q2 q0q2q0q2 q0q2 q0q2 q0q2

δ(a, q1) ={{(q0, q0)}} rk(qi) = iδ(b, q2) ={{(q1, q1), (q2, q2)}}

δ(a, q0) = δ(b, q0){{(q0, q0)}}δ(a, q2) ={{(q2, q2)}}δ(b, q1) ={{(q1, q1)}}

weak MSOLa

b

a

Ω b

a

b b

b

a

b b

a

b b

q2

q2 q2

q1q2 q1q2 q1q2 q1q2

q0q2 q0q2 q0q2 q0q2q0q2 q0q2 q0q2 q0q2

δ(a, q1) ={{(q0, q0)}} rk(qi) = iδ(b, q2) ={{(q1, q1), (q2, q2)}}

δ(a, q0) = δ(b, q0){{(q0, q0)}}δ(a, q2) ={{(q2, q2)}}δ(b, q1) ={{(q1, q1)}}

weak MSOLa

b

a

Ω b

a

b b

b

a

b b

a

b b

q2

q2 q2

q1q2 q1q2 q1q2 q1q2

q0q2 q0q2 q0q2 q0q2q0q2 q0q2 q0q2 q0q2

δ(a, q1) ={{(q0, q0)}} rk(qi) = iδ(b, q2) ={{(q1, q1), (q2, q2)}}

δ(a, q0) = δ(b, q0){{(q0, q0)}}δ(a, q2) ={{(q2, q2)}}δ(b, q1) ={{(q1, q1)}}

weak MSOLa

b

a

Ω b

a

b b

b

a

b b

a

b b

q2

q2 q2

q1q2 q1q2 q1q2 q1q2

q0q2 q0q2 q0q2 q0q2q0q2 q0q2 q0q2 q0q2

δ(a, q1) ={{(q0, q0)}} rk(qi) = iδ(b, q2) ={{(q1, q1), (q2, q2)}}

δ(a, q0) = δ(b, q0){{(q0, q0)}}δ(a, q2) ={{(q2, q2)}}δ(b, q1) ={{(q1, q1)}}

Structure of weak Parity automata accepting runs

4

3

2

1

0

Layered monotone modelsThe layered monotone model over the finite latticesL0 = (Q0,≤0), …, Lk = (Qk ,≤k):

D = ({(DA,vA)}A∈T t, ρ) ρ : Cst → D

whereI D0 = L0 × · · · × Lk and f v0 g is the product order,I e = (a1, . . . , ak), e|i = (a1, . . . , ai),

I DB→C = [DB →l DC ] = {f ∈ [DB →m DC ] | ∀g , g ′ ∈DB ,∀i ≤ k, g|i = g ′|i ⇒ (f (g))|i = (f (g

′))|i}vB→C= pointwise ordering.

I DB→C |i = [DB|i →l DC |i ]I for f ∈ DB→C , f|i(g|i) = (f (g))|i ,I DB→C ,i = {f ∈ DB→B | ∀g ∈ DB ,∀j 6= i , πj(f (g)) = ⊥DC,j}.

LemmaFor all A, DA is isomorphic to DA,0 × · · · × DA,k .

Layered monotone modelsThe layered monotone model over the finite latticesL0 = (Q0,≤0), …, Lk = (Qk ,≤k):

D = ({(DA,vA)}A∈T t, ρ) ρ : Cst → D

whereI D0 = L0 × · · · × Lk and f v0 g is the product order,I e = (a1, . . . , ak), e|i = (a1, . . . , ai),

I DB→C = [DB →l DC ] = {f ∈ [DB →m DC ] | ∀g , g ′ ∈DB ,∀i ≤ k, g|i = g ′|i ⇒ (f (g))|i = (f (g

′))|i}vB→C= pointwise ordering.

I DB→C |i = [DB|i →l DC |i ]I for f ∈ DB→C , f|i(g|i) = (f (g))|i ,I DB→C ,i = {f ∈ DB→B | ∀g ∈ DB ,∀j 6= i , πj(f (g)) = ⊥DC,j}.

LemmaFor all A, DA is isomorphic to DA,0 × · · · × DA,k .

Layered monotone modelsThe layered monotone model over the finite latticesL0 = (Q0,≤0), …, Lk = (Qk ,≤k):

D = ({(DA,vA)}A∈T t, ρ) ρ : Cst → D

whereI D0 = L0 × · · · × Lk and f v0 g is the product order,I e = (a1, . . . , ak), e|i = (a1, . . . , ai),I DB→C = [DB →l DC ] = {f ∈ [DB →m DC ] | ∀g , g ′ ∈

DB , ∀i ≤ k, g|i = g ′|i ⇒ (f (g))|i = (f (g′))|i}

vB→C= pointwise ordering.

I DB→C |i = [DB|i →l DC |i ]I for f ∈ DB→C , f|i(g|i) = (f (g))|i ,I DB→C ,i = {f ∈ DB→B | ∀g ∈ DB ,∀j 6= i , πj(f (g)) = ⊥DC,j}.

LemmaFor all A, DA is isomorphic to DA,0 × · · · × DA,k .

Layered monotone modelsThe layered monotone model over the finite latticesL0 = (Q0,≤0), …, Lk = (Qk ,≤k):

D = ({(DA,vA)}A∈T t, ρ) ρ : Cst → D

whereI D0 = L0 × · · · × Lk and f v0 g is the product order,I e = (a1, . . . , ak), e|i = (a1, . . . , ai),I DB→C = [DB →l DC ] = {f ∈ [DB →m DC ] | ∀g , g ′ ∈

DB , ∀i ≤ k, g|i = g ′|i ⇒ (f (g))|i = (f (g′))|i}

vB→C= pointwise ordering.I DB→C |i = [DB|i →l DC |i ]

I for f ∈ DB→C , f|i(g|i) = (f (g))|i ,I DB→C ,i = {f ∈ DB→B | ∀g ∈ DB ,∀j 6= i , πj(f (g)) = ⊥DC,j}.

LemmaFor all A, DA is isomorphic to DA,0 × · · · × DA,k .

Layered monotone modelsThe layered monotone model over the finite latticesL0 = (Q0,≤0), …, Lk = (Qk ,≤k):

D = ({(DA,vA)}A∈T t, ρ) ρ : Cst → D

whereI D0 = L0 × · · · × Lk and f v0 g is the product order,I e = (a1, . . . , ak), e|i = (a1, . . . , ai),I DB→C = [DB →l DC ] = {f ∈ [DB →m DC ] | ∀g , g ′ ∈

DB , ∀i ≤ k, g|i = g ′|i ⇒ (f (g))|i = (f (g′))|i}

vB→C= pointwise ordering.I DB→C |i = [DB|i →l DC |i ]I for f ∈ DB→C , f|i(g|i) = (f (g))|i ,

I DB→C ,i = {f ∈ DB→B | ∀g ∈ DB ,∀j 6= i , πj(f (g)) = ⊥DC,j}.

LemmaFor all A, DA is isomorphic to DA,0 × · · · × DA,k .

Layered monotone modelsThe layered monotone model over the finite latticesL0 = (Q0,≤0), …, Lk = (Qk ,≤k):

D = ({(DA,vA)}A∈T t, ρ) ρ : Cst → D

whereI D0 = L0 × · · · × Lk and f v0 g is the product order,I e = (a1, . . . , ak), e|i = (a1, . . . , ai),I DB→C = [DB →l DC ] = {f ∈ [DB →m DC ] | ∀g , g ′ ∈

DB , ∀i ≤ k, g|i = g ′|i ⇒ (f (g))|i = (f (g′))|i}

vB→C= pointwise ordering.I DB→C |i = [DB|i →l DC |i ]I for f ∈ DB→C , f|i(g|i) = (f (g))|i ,I DB→C ,i = {f ∈ DB→B | ∀g ∈ DB ,∀j 6= i , πj(f (g)) = ⊥DC,j}.

LemmaFor all A, DA is isomorphic to DA,0 × · · · × DA,k .

Layered monotone modelsThe layered monotone model over the finite latticesL0 = (Q0,≤0), …, Lk = (Qk ,≤k):

D = ({(DA,vA)}A∈T t, ρ) ρ : Cst → D

whereI D0 = L0 × · · · × Lk and f v0 g is the product order,I e = (a1, . . . , ak), e|i = (a1, . . . , ai),I DB→C = [DB →l DC ] = {f ∈ [DB →m DC ] | ∀g , g ′ ∈

DB , ∀i ≤ k, g|i = g ′|i ⇒ (f (g))|i = (f (g′))|i}

vB→C= pointwise ordering.I DB→C |i = [DB|i →l DC |i ]I for f ∈ DB→C , f|i(g|i) = (f (g))|i ,I DB→C ,i = {f ∈ DB→B | ∀g ∈ DB ,∀j 6= i , πj(f (g)) = ⊥DC,j}.

LemmaFor all A, DA is isomorphic to DA,0 × · · · × DA,k .

Towards a Semantics of Y : Galois Connections

For f = (f1, . . . , fi) in DA|i we let:I f ↑ = (f1, . . . , fi ,>A,i),I f ↓ = (f1, . . . , fi ,⊥A,i)

For f = (f1, . . . , fi , fi+1) in DA|i+1 we let:

f = (f1, . . . , fi)

We have, for f ∈ DA|i and g ∈ DA|i+1:I g ≤ f iff g ≤ f ↑,I f ≤ g iff f ↓ ≤ g

Towards a Semantics of Y

We inductively define fixi as an element of DA→A|i :I fix0(f ) =

d{f n(>A,0) | n ∈ N}

I fix2i+1(f ) =⊔{f n((fix2i(f ))↓) | n ∈ N}

I fix2i+2(f ) =d{f n((fix2i+1(f ))↑) | n ∈ N}

DA|0

>A,0

fix0(f )

Towards a Semantics of Y

We inductively define fixi as an element of DA→A|i :I fix0(f ) =

d{f n(>A,0) | n ∈ N}

I fix2i+1(f ) =⊔{f n((fix2i(f ))↓) | n ∈ N}

I fix2i+2(f ) =d{f n((fix2i+1(f ))↑) | n ∈ N}

DA|2i

DA|2i+1

DA|2i+1

DA|2i+2

f

(fix2i+1(f ))↑

(fix2i(f ))↓

fix2i(f )

fix2i+1(f )

Towards a Semantics of Y

We inductively define fixi as an element of DA→A|i :I fix0(f ) =

d{f n(>A,0) | n ∈ N}

I fix2i+1(f ) =⊔{f n((fix2i(f ))↓) | n ∈ N}

I fix2i+2(f ) =d{f n((fix2i+1(f ))↑) | n ∈ N}

DA|2i

DA|2i+1

DA|2i+1

DA|2i+2f

(fix2i+1(f ))↑

(fix2i(f ))↓

fix2i(f )

fix2i+1(f )

Layered monotone models and weak automata

Theorem (S. Walukiewicz 15)Given D a layered monotone model and A ⊆ D0, M is recognizedby A iff BT (M) is accepted by a weak alternating parityautomaton.

I The model lives inside the monotone model where we haveremoved meaningless functions.

I Dualities in the model −→ means for reasonning for provingor refuting properties.

Layered monotone models and weak automata

Theorem (S. Walukiewicz 15)Given D a layered monotone model and A ⊆ D0, M is recognizedby A iff BT (M) is accepted by a weak alternating parityautomaton.

I The model lives inside the monotone model where we haveremoved meaningless functions.

I Dualities in the model −→ means for reasonning for provingor refuting properties.

Layered monotone models and weak automata

Theorem (S. Walukiewicz 15)Given D a layered monotone model and A ⊆ D0, M is recognizedby A iff BT (M) is accepted by a weak alternating parityautomaton.

I The model lives inside the monotone model where we haveremoved meaningless functions.

I Dualities in the model −→ means for reasonning for provingor refuting properties.

Models for MSOL

Color modalities (1)

Color modalities (2)

General principles

I Maintaining the information of the maximal color seen fromthe root of a Böhm tree to occurrences of variables.

I We use Scott domains enriched with this information.I As for weak MSOL we remove meaningless interpretations.

Enriched Scott domains

Fix a parity automaton A, rk(q) is the color associated to q.

Enriched domain

R0 = P({(q, r) : q ∈ Q and rk(q) ≤ r ≤ m})

h�r = {(q, i) ∈ h : r ≤ i} ∪ {(q, j) : (q, r) ∈ h, rk(q) ≤ j ≤ r}

LemmaFor h ∈ R0, q ∈ Q, and r , r1, r2 ∈ [m]:

I (h�r1)�r2 = h�max(r1,r2);I (q, rk(q)) ∈ h�r iff (q,max(r , rk(q)) ∈ h

h⇓q = {r | (q, r) ∈ h}

Result domain

Fix a parity automaton A, rk(q) is the color associated to q.Result domain

D0 = P(Q)

f · r = {(q, r) : q ∈ R0 and rk(q) ≤ r}

f ⇓q = f ∩ {q}

For h in R0, leth∂ = {q : (q, rk(q)) ∈ h}

Going higher-order

Enriched domain RA→B is the set of monotone functions from RAto RB so that:

∀g ∈ RA. ∀q ∈ Q. (f (g))⇓q = (f (g�rk(q)))⇓q

Where f ⇓q(g) = f (g)⇓q, f �rk(q)(g) = f (g)�rk(q) andf ∂(g) = f (g)∂

Result domainDA→B is the set of monotone functions from RA to DB so that:

∀g ∈ RA. ∀q ∈ Q. (f (g))⇓q = (f (g�rk(q)))⇓q

Where f ⇓q(g) = f (g)⇓q and f · r(g) = f (g) · r .

Interpretation of terms

[[x , ν]] = (ν(x))∂[[a, ν]]h1 . . . hk = {q : ∃(q1,...,qk)∈(q,a) qi ∈ (hi�rk(q))

∂ for all i}[[λx .M, ν]]h = [[M, ν[h/x ]]][[MN, ν]] = [[M, ν]]〈〈N, ν〉〉 where 〈〈N, ν〉〉 =

∨mr=0

([[N, ν�r ]] · r

)and ν�r (x) = ν(x)�r

[[Y , ν]]h = µfm.νfm−1 . . . µf1.νf0. (h�l)∂(∨l

i=0 fi · i)

Theorem (Soundness (S. Walukiewicz 15))If BT (M) = BT (N) then [[M, ν]] = [[N, ν]].

Theorem (Completeness (S. Walukiewicz 15))For M closed and of type 0, q ∈ [[M]] iff A has an accepting runstarting from q on BT (M).

Interpretation of terms

[[x , ν]] = (ν(x))∂[[a, ν]]h1 . . . hk = {q : ∃(q1,...,qk)∈(q,a) qi ∈ (hi�rk(q))

∂ for all i}[[λx .M, ν]]h = [[M, ν[h/x ]]][[MN, ν]] = [[M, ν]]〈〈N, ν〉〉 where 〈〈N, ν〉〉 =

∨mr=0

([[N, ν�r ]] · r

)and ν�r (x) = ν(x)�r

[[Y , ν]]h = µfm.νfm−1 . . . µf1.νf0. (h�l)∂(∨l

i=0 fi · i)

Theorem (Soundness (S. Walukiewicz 15))If BT (M) = BT (N) then [[M, ν]] = [[N, ν]].

Theorem (Completeness (S. Walukiewicz 15))For M closed and of type 0, q ∈ [[M]] iff A has an accepting runstarting from q on BT (M).

Interpretation of terms

[[x , ν]] = (ν(x))∂[[a, ν]]h1 . . . hk = {q : ∃(q1,...,qk)∈(q,a) qi ∈ (hi�rk(q))

∂ for all i}[[λx .M, ν]]h = [[M, ν[h/x ]]][[MN, ν]] = [[M, ν]]〈〈N, ν〉〉 where 〈〈N, ν〉〉 =

∨mr=0

([[N, ν�r ]] · r

)and ν�r (x) = ν(x)�r

[[Y , ν]]h = µfm.νfm−1 . . . µf1.νf0. (h�l)∂(∨l

i=0 fi · i)

Theorem (Soundness (S. Walukiewicz 15))If BT (M) = BT (N) then [[M, ν]] = [[N, ν]].

Theorem (Completeness (S. Walukiewicz 15))For M closed and of type 0, q ∈ [[M]] iff A has an accepting runstarting from q on BT (M).

Conclusion

Finite-State Automata

Finite Algebras Logic

Conclusion

Finite-State Automata

Finite Models Logic

Conclusion

Intersection Types

Finite Models Logic

Conclusion

Intersection Types

Finite Models ??

Announcement

Igor and I have a PhD fellowship starting this Autumn in Bordeaux.We will welcome any good student willing to work on this topic.