SOS and Modal Logic Revisited Bartek Klin University of Cambridge, - - PowerPoint PPT Presentation

SOS and Modal Logic

Revisited

Bartek Klin University of Cambridge, Warsaw University

CMCS, Paphos, 26/03/10

CMCS, Paphos, 26/03/10

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SOS and distributive laws

2

x

a

− → y x

a

− → y x ⊗ x

a

− → y ⊗ y t ::= nil | a | t ⊗ t a

a

− → nil

CMCS, Paphos, 26/03/10

/ 11

SOS and distributive laws

2

x

a

− → y x

a

− → y x ⊗ x

a

− → y ⊗ y ΣX = 1 + A + X2 t ::= nil | a | t ⊗ t a

a

− → nil

CMCS, Paphos, 26/03/10

/ 11

SOS and distributive laws

2

x

a

− → y x

a

− → y x ⊗ x

a

− → y ⊗ y ΣX = 1 + A + X2 BX = (PωX)A t ::= nil | a | t ⊗ t a

a

− → nil

CMCS, Paphos, 26/03/10

/ 11

SOS and distributive laws

2

x

a

− → y x

a

− → y x ⊗ x

a

− → y ⊗ y ΣX = 1 + A + X2 BX = (PωX)A λ : ΣB = ⇒ BΣ t ::= nil | a | t ⊗ t a

a

− → nil

CMCS, Paphos, 26/03/10

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SOS and distributive laws

2

x

a

− → y x

a

− → y x ⊗ x

a

− → y ⊗ y ΣX = 1 + A + X2 BX = (PωX)A λ : ΣB = ⇒ BΣ t ::= nil | a | t ⊗ t a

a

− → nil λX(nil)(a) = ∅ λX(a)(b) =

• {nil}

⇐ b = a ∅ ⇐ b = a λX(β ⊗ β′)(a) = {y ⊗ y′ | y ∈ β(a), y′ ∈ β′(a)}

CMCS, Paphos, 26/03/10

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SOS and distributive laws

2

x

a

− → y x

a

− → y x ⊗ x

a

− → y ⊗ y ΣX = 1 + A + X2 BX = (PωX)A λ : ΣB = ⇒ BΣ t ::= nil | a | t ⊗ t a

a

− → nil

CMCS, Paphos, 26/03/10

/ 11

SOS and distributive laws

2

x

a

− → y x

a

− → y x ⊗ x

a

− → y ⊗ y ΣX = 1 + A + X2 BX = (PωX)A λ : ΣB = ⇒ BΣ t ::= nil | a | t ⊗ t a

a

− → nil ΣA

a

• Σh
• A

h

• ΣBA

λA

BΣA

Ba

BA

• terms

Σ

CMCS, Paphos, 26/03/10

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SOS and distributive laws

2

x

a

− → y x

a

− → y x ⊗ x

a

− → y ⊗ y ΣX = 1 + A + X2 BX = (PωX)A λ : ΣB = ⇒ BΣ t ::= nil | a | t ⊗ t a

a

− → nil ΣA

a

• Σh
• A

h

• ΣBA

λA

BΣA

Ba

BA

• terms

Σ

Fact: Bisimilarity is a congruence (for any )

λ

CMCS, Paphos, 26/03/10

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Predicate liftings

3

Modal logic:

BX = (PωX)A φ ::= ⊤ | aφ

CMCS, Paphos, 26/03/10

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Predicate liftings

3

Modal logic:

BX = (PωX)A φ ::= ⊤ | aφ x | = ⊤ always x | = aφ ⇐ ⇒ ∃x

a

− → y. y | = φ

CMCS, Paphos, 26/03/10

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Predicate liftings

3

Modal logic:

BX = (PωX)A φ ::= ⊤ | aφ x | = ⊤ always x | = aφ ⇐ ⇒ ∃x

a

− → y. y | = φ | ∅

CMCS, Paphos, 26/03/10

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Predicate liftings

3

Modal logic:

BX = (PωX)A φ ::= ⊤ | aφ x | = ⊤ always x | = aφ ⇐ ⇒ ∃x

a

− → y. y | = φ | ∅ x | = ∅ ⇐ ⇒ x − →

CMCS, Paphos, 26/03/10

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Predicate liftings

3

Modal logic:

BX = (PωX)A φ ::= ⊤ | aφ x | = ⊤ always x | = aφ ⇐ ⇒ ∃x

a

− → y. y | = φ | ∅ x | = ∅ ⇐ ⇒ x − → | φ ∧ φ

CMCS, Paphos, 26/03/10

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Predicate liftings

3

Modal logic:

BX = (PωX)A φ ::= ⊤ | aφ x | = ⊤ always x | = aφ ⇐ ⇒ ∃x

a

− → y. y | = φ | ∅ x | = ∅ ⇐ ⇒ x − → | φ ∧ φ x | = φ ∧ ψ ⇐ ⇒ x | = φ, x | = ψ

CMCS, Paphos, 26/03/10

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Predicate liftings

3

Modal logic:

BX = (PωX)A φ ::= ⊤ | aφ x | = ⊤ always x | = aφ ⇐ ⇒ ∃x

a

− → y. y | = φ | ∅ x | = ∅ ⇐ ⇒ x − → | φ ∧ φ x | = φ ∧ ψ ⇐ ⇒ x | = φ, x | = ψ

Predicate liftings: β : B(2n) → 2

2 = {tt, ff}

CMCS, Paphos, 26/03/10

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Predicate liftings

3

Modal logic:

BX = (PωX)A φ ::= ⊤ | aφ x | = ⊤ always x | = aφ ⇐ ⇒ ∃x

a

− → y. y | = φ | ∅ x | = ∅ ⇐ ⇒ x − → | φ ∧ φ x | = φ ∧ ψ ⇐ ⇒ x | = φ, x | = ψ

Predicate liftings: β : B(2n) → 2

2 = {tt, ff} ⊤(β) = tt ⊤ : B1 → 2 a : B2 → 2 a(β) = tt ⇐ ⇒ tt ∈ β(a) ∅ : B1 → 2 ∅(β) = tt ⇐ ⇒ ∀a ∈ A. β(a) = ∅

CMCS, Paphos, 26/03/10

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Predicate liftings

3

Modal logic:

BX = (PωX)A φ ::= ⊤ | aφ x | = ⊤ always x | = aφ ⇐ ⇒ ∃x

a

− → y. y | = φ | ∅ x | = ∅ ⇐ ⇒ x − → | φ ∧ φ x | = φ ∧ ψ ⇐ ⇒ x | = φ, x | = ψ

Predicate liftings: β : B(2n) → 2

2 = {tt, ff} ⊤(β) = tt ⊤ : B1 → 2 a : B2 → 2 a(β) = tt ⇐ ⇒ tt ∈ β(a) ∅ : B1 → 2 ∅(β) = tt ⇐ ⇒ ∀a ∈ A. β(a) = ∅ a(−1 ∧ · · · ∧ −n) : B2n → 2 a(−1 ∧ · · · ∧ −n)(β) = tt ⇐ ⇒ (tt, . . . , tt) ∈ β(a)

CMCS, Paphos, 26/03/10

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Predicate liftings for syntax

4

ΣX = 1 + A + X2 t ::= nil | a | t ⊗ t

CMCS, Paphos, 26/03/10

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Predicate liftings for syntax

4

ΣX = 1 + A + X2 t ::= nil | a | t ⊗ t T : Σ1 → 2 T(t) = tt a : Σ1 → 2 a(t) = tt ⇐ ⇒ t = a [⊗](t) = tt ⇐ ⇒ t = tt ⊗ tt ⊗(t) = tt ⇐ ⇒ t = v ⊗ v′, tt ∈ {v, v′} [⊗] : Σ2 → 2 ⊗ : Σ2 → 2

CMCS, Paphos, 26/03/10

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Predicate liftings for syntax

4

ΣX = 1 + A + X2 t ::= nil | a | t ⊗ t T : Σ1 → 2 T(t) = tt a : Σ1 → 2 a(t) = tt ⇐ ⇒ t = a [⊗](t) = tt ⇐ ⇒ t = tt ⊗ tt ⊗(t) = tt ⇐ ⇒ t = v ⊗ v′, tt ∈ {v, v′} [⊗] : Σ2 → 2 ⊗ : Σ2 → 2 a ∨ [⊗] : Σ2 → 2

etc.

−1 ⊗ −2 : Σ22 → 2

CMCS, Paphos, 26/03/10

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Construction of liftings

5

Variable renaming:

β : B2n → 2 f : n → m β|f : B2m → 2

CMCS, Paphos, 26/03/10

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Construction of liftings

5

Variable renaming:

β : B2n → 2 f : n → m β|f : B2m → 2 x1 ⊗ x2 : Σ22 → 2 x ⊗ x : Σ2 → 2 [⊗]x =

CMCS, Paphos, 26/03/10

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Construction of liftings

5

β : B2n → 2

• σi : Σ2mi → 2
• i=1,...,n

β(σ1, . . . , σn) : BΣ2m → 2 m = m

i=1 mi

Composition: Variable renaming:

β : B2n → 2 f : n → m β|f : B2m → 2 x1 ⊗ x2 : Σ22 → 2 x ⊗ x : Σ2 → 2 [⊗]x =

CMCS, Paphos, 26/03/10

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Example

6

ΣX = 1 + A + X2 BX = (PωX)A [⊗] : Σ2 → 2 a : B2 → 2

CMCS, Paphos, 26/03/10

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Example

6

ΣX = 1 + A + X2 BX = (PωX)A [⊗] : Σ2 → 2 a : B2 → 2 [⊗]a : ΣB2 → 2 a[⊗] : BΣ2 → 2

CMCS, Paphos, 26/03/10

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Example

6

ΣX = 1 + A + X2 BX = (PωX)A [⊗] : Σ2 → 2 a : B2 → 2 [⊗]a : ΣB2 → 2 a[⊗] : BΣ2 → 2 x

a

− → y x

a

− → y x ⊗ x

a

− → y ⊗ y a

a

− → nil λ : ΣB = ⇒ BΣ

CMCS, Paphos, 26/03/10

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Example

6

ΣX = 1 + A + X2 BX = (PωX)A [⊗] : Σ2 → 2 a : B2 → 2 [⊗]a : ΣB2 → 2 a[⊗] : BΣ2 → 2 x

a

− → y x

a

− → y x ⊗ x

a

− → y ⊗ y a

a

− → nil λ : ΣB = ⇒ BΣ a[⊗] ◦ λ2 : ΣB2 → 2

CMCS, Paphos, 26/03/10

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Example

6

ΣX = 1 + A + X2 BX = (PωX)A [⊗] : Σ2 → 2 a : B2 → 2 [⊗]a : ΣB2 → 2 x

a

− → y x

a

− → y x ⊗ x

a

− → y ⊗ y a

a

− → nil λ : ΣB = ⇒ BΣ a[⊗] ◦ λ2 : ΣB2 → 2

CMCS, Paphos, 26/03/10

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Example

6

ΣX = 1 + A + X2 BX = (PωX)A [⊗] : Σ2 → 2 a : B2 → 2 [⊗]a : ΣB2 → 2 x

a

− → y x

a

− → y x ⊗ x

a

− → y ⊗ y a

a

− → nil λ : ΣB = ⇒ BΣ a[⊗] ◦ λ2 : ΣB2 → 2

=

CMCS, Paphos, 26/03/10

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Valid equations

7

β(σ1(x11, . . . , x1n1), . . . , σm(xm1, . . . , xmnm)) = σ(β1(y11, . . . , y1k1), . . . , βl(yl1, . . . , ylkl))

CMCS, Paphos, 26/03/10

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Valid equations

7

β(σ1(x11, . . . , x1n1), . . . , σm(xm1, . . . , xmnm)) = σ(β1(y11, . . . , y1k1), . . . , βl(yl1, . . . , ylkl))

CMCS, Paphos, 26/03/10

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Valid equations

7

β(σ1(x11, . . . , x1n1), . . . , σm(xm1, . . . , xmnm)) = σ(β1(y11, . . . , y1k1), . . . , βl(yl1, . . . , ylkl))

• LHS defines a -lifting

BΣ

• RHS+ define a -lifting of the same arity.

ΣB

then use to get a -lifting

λ

BΣ

CMCS, Paphos, 26/03/10

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Valid equations

7

β(σ1(x11, . . . , x1n1), . . . , σm(xm1, . . . , xmnm)) = σ(β1(y11, . . . , y1k1), . . . , βl(yl1, . . . , ylkl))

• LHS defines a -lifting

BΣ

• RHS+ define a -lifting of the same arity.

ΣB

then use to get a -lifting

λ

BΣ

The equation is valid if the two -liftings are equal.

BΣ

CMCS, Paphos, 26/03/10

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Compositionality

8

• Thm. For a family of -liftings,

(βi)i∈I B

• find a family of -liftings,

(σj)j∈J Σ

• for every possible LHS:

β(σ1(x11, . . . , x1n1), . . . , σm(xm1, . . . , xmnm))

• - find an RHS

σ(β1(y11, . . . , y1k1), . . . , βl(yl1, . . . , ylkl))

s.t. the equation is valid wrt .

λ

CMCS, Paphos, 26/03/10

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Compositionality

8

• Thm. For a family of -liftings,

(βi)i∈I B

• find a family of -liftings,

(σj)j∈J Σ

• for every possible LHS:

β(σ1(x11, . . . , x1n1), . . . , σm(xm1, . . . , xmnm))

• - find an RHS

σ(β1(y11, . . . , y1k1), . . . , βl(yl1, . . . , ylkl))

s.t. the equation is valid wrt .

λ

Then the logical equivalence defined by (βi)i∈I is a congruence on the coalgebra induced by .

λ

CMCS, Paphos, 26/03/10

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Example

9

a

a

− → nil x

a

− → y x

a

− → y x ⊗ x

a

− → y ⊗ y φ ::= ⊤ | aφ

• liftings:

B

Σ-liftings:

a : B2 → 2 ⊤ : B1 → 2

CMCS, Paphos, 26/03/10

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Example

9

a

a

− → nil x

a

− → y x

a

− → y x ⊗ x

a

− → y ⊗ y φ ::= ⊤ | aφ

• liftings:

B

Σ-liftings:

(none)

a : B2 → 2 ⊤ : B1 → 2

CMCS, Paphos, 26/03/10

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Example

9

a

a

− → nil x

a

− → y x

a

− → y x ⊗ x

a

− → y ⊗ y φ ::= ⊤ | aφ

• liftings:

B

Σ-liftings:

(none)

a : B2 → 2 ⊤ : B1 → 2 ⊤ = ?

CMCS, Paphos, 26/03/10

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Example

9

a

a

− → nil x

a

− → y x

a

− → y x ⊗ x

a

− → y ⊗ y φ ::= ⊤ | aφ

• liftings:

B

Σ-liftings:

a : B2 → 2 ⊤ : B1 → 2 ⊤ = ? T : Σ1 → 2

CMCS, Paphos, 26/03/10

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Example

9

a

a

− → nil x

a

− → y x

a

− → y x ⊗ x

a

− → y ⊗ y φ ::= ⊤ | aφ

• liftings:

B

Σ-liftings:

a : B2 → 2 ⊤ : B1 → 2 T : Σ1 → 2 ⊤ = T

CMCS, Paphos, 26/03/10

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Example

9

a

a

− → nil x

a

− → y x

a

− → y x ⊗ x

a

− → y ⊗ y φ ::= ⊤ | aφ

• liftings:

B

Σ-liftings:

a : B2 → 2 ⊤ : B1 → 2 T : Σ1 → 2 ⊤ = T aT = ?

CMCS, Paphos, 26/03/10

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Example

9

a

a

− → nil x

a

− → y x

a

− → y x ⊗ x

a

− → y ⊗ y φ ::= ⊤ | aφ

• liftings:

B

Σ-liftings:

a : B2 → 2 ⊤ : B1 → 2 T : Σ1 → 2 ⊤ = T aT = ? a ∨ [⊗] : Σ2 → 2

CMCS, Paphos, 26/03/10

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Example

9

a

a

− → nil x

a

− → y x

a

− → y x ⊗ x

a

− → y ⊗ y φ ::= ⊤ | aφ

• liftings:

B

Σ-liftings:

a : B2 → 2 ⊤ : B1 → 2 T : Σ1 → 2 ⊤ = T a ∨ [⊗] : Σ2 → 2 aT = a ∨ [⊗]a⊤

CMCS, Paphos, 26/03/10

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Example

9

a

a

− → nil x

a

− → y x

a

− → y x ⊗ x

a

− → y ⊗ y φ ::= ⊤ | aφ

• liftings:

B

Σ-liftings:

a : B2 → 2 ⊤ : B1 → 2 T : Σ1 → 2 ⊤ = T a ∨ [⊗] : Σ2 → 2 aT = a ∨ [⊗]a⊤ a⊤ : B1 → 2

CMCS, Paphos, 26/03/10

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Example

9

a

a

− → nil x

a

− → y x

a

− → y x ⊗ x

a

− → y ⊗ y φ ::= ⊤ | aφ

• liftings:

B

Σ-liftings:

a : B2 → 2 ⊤ : B1 → 2 T : Σ1 → 2 ⊤ = T a ∨ [⊗] : Σ2 → 2 aT = a ∨ [⊗]a⊤ a⊤ : B1 → 2 a⊤ = a ∨ [⊗]a⊤

CMCS, Paphos, 26/03/10

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Example

9

a

a

− → nil x

a

− → y x

a

− → y x ⊗ x

a

− → y ⊗ y φ ::= ⊤ | aφ

• liftings:

B

Σ-liftings:

a : B2 → 2 ⊤ : B1 → 2 T : Σ1 → 2 ⊤ = T a ∨ [⊗] : Σ2 → 2 aT = a ∨ [⊗]a⊤ a⊤ : B1 → 2 a⊤ = a ∨ [⊗]a⊤ a(b ∨ [⊗]x) = ?

CMCS, Paphos, 26/03/10

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Example

9

a

a

− → nil x

a

− → y x

a

− → y x ⊗ x

a

− → y ⊗ y φ ::= ⊤ | aφ

• liftings:

B

Σ-liftings:

a : B2 → 2 ⊤ : B1 → 2 T : Σ1 → 2 ⊤ = T a ∨ [⊗] : Σ2 → 2 aT = a ∨ [⊗]a⊤ a⊤ : B1 → 2 a⊤ = a ∨ [⊗]a⊤ a(b ∨ [⊗]x) = ? [⊗] : Σ2 → 2

CMCS, Paphos, 26/03/10

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Example

9

a

a

− → nil x

a

− → y x

a

− → y x ⊗ x

a

− → y ⊗ y φ ::= ⊤ | aφ

• liftings:

B

Σ-liftings:

a : B2 → 2 ⊤ : B1 → 2 T : Σ1 → 2 ⊤ = T a ∨ [⊗] : Σ2 → 2 aT = a ∨ [⊗]a⊤ a⊤ : B1 → 2 a⊤ = a ∨ [⊗]a⊤ [⊗] : Σ2 → 2 a(b ∨ [⊗]x) = [⊗]ax

CMCS, Paphos, 26/03/10

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Example

9

a

a

− → nil x

a

− → y x

a

− → y x ⊗ x

a

− → y ⊗ y φ ::= ⊤ | aφ

• liftings:

B

Σ-liftings:

a : B2 → 2 ⊤ : B1 → 2 T : Σ1 → 2 ⊤ = T a ∨ [⊗] : Σ2 → 2 aT = a ∨ [⊗]a⊤ a⊤ : B1 → 2 a⊤ = a ∨ [⊗]a⊤ [⊗] : Σ2 → 2 a(b ∨ [⊗]x) = [⊗]ax a[⊗]x = [⊗]ax

CMCS, Paphos, 26/03/10

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Gobbledygook

10

Set

• 2−
• ⊥

Σ

• B
• Set
• 2−
• L
• Γ
• ρ

λ ζ χ

CMCS, Paphos, 26/03/10

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Gobbledygook

10

Set

• 2−
• ⊥

Σ

• B
• Set
• 2−
• L
• Γ
• ρ

λ ζ χ

(PωX)A

CMCS, Paphos, 26/03/10

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Gobbledygook

10

Set

• 2−
• ⊥

Σ

• B
• Set
• 2−
• L
• Γ
• ρ

λ ζ χ

(PωX)A 1 + A + X2

CMCS, Paphos, 26/03/10

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Gobbledygook

10

Set

• 2−
• ⊥

Σ

• B
• Set
• 2−
• L
• Γ
• ρ

λ ζ χ

(PωX)A

x

a

− → y x

a

− → y x ⊗ x

a

− → y ⊗ y

1 + A + X2

CMCS, Paphos, 26/03/10

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Gobbledygook

10

Set

• 2−
• ⊥

Σ

• B
• Set
• 2−
• L
• Γ
• ρ

λ ζ χ

(PωX)A

x

a

− → y x

a

− → y x ⊗ x

a

− → y ⊗ y

1 + A + X2 ⊤, a

CMCS, Paphos, 26/03/10

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Gobbledygook

10

Set

• 2−
• ⊥

Σ

• B
• Set
• 2−
• L
• Γ
• ρ

λ ζ χ

(PωX)A

x

a

− → y x

a

− → y x ⊗ x

a

− → y ⊗ y

1 + A + X2 T, a, [⊗] ⊤, a

CMCS, Paphos, 26/03/10

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Gobbledygook

10

Set

• 2−
• ⊥

Σ

• B
• Set
• 2−
• L
• Γ
• ρ

λ ζ χ

(PωX)A

x

a

− → y x

a

− → y x ⊗ x

a

− → y ⊗ y

1 + A + X2 T, a, [⊗] ⊤, a a[⊗]x [⊗]ax =

CMCS, Paphos, 26/03/10

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Future work

11

• Guidelines for choosing syntactic liftings/equations
• Study complex, GSOS-like equations