Operational Semantics Coalgebraically Bartek Klin University of - - PowerPoint PPT Presentation

Operational Semantics Coalgebraically

Bartek Klin University of Cambridge, Warsaw University

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Summary

2

Structural Operational Semantics is about Defining transition relations by induction

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Summary

2

Structural Operational Semantics is about Defining transition relations by induction

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Summary

2

Structural Operational Semantics Distributing syntax over behaviour is about

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I. FROM COINDUCTIVE DEFINITIONS TO BIALGEBRAS

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A coinductive definition

4

Example: streams

BX = A × X

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A coinductive definition

4

Example: streams

BX = A × X Z = Aω z = hd, tl : Z → BZ

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A coinductive definition

4

Example: streams

BX = A × X Z = Aω z = hd, tl : Z → BZ alt : Z2 → Z

Define:

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A coinductive definition

4

Example: streams

BX = A × X Z = Aω z = hd, tl : Z → BZ alt : Z2 → Z

Define:

a1, a2, a3, a4, a5, . . . b1, b2, b3, b4, b5, . . .

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A coinductive definition

4

Example: streams

BX = A × X Z = Aω z = hd, tl : Z → BZ alt : Z2 → Z

Define:

a1, a2, a3, a4, a5, . . . b1, b2, b3, b4, b5, . . .

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A coinductive definition

4

Example: streams

BX = A × X Z = Aω z = hd, tl : Z → BZ alt : Z2 → Z

Define:

a1, a2, a3, a4, a5, . . . b1, b2, b3, b4, b5, . . . alt a1, b2, a3, b4, a5, . . .

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A coinductive definition

4

Example: streams

BX = A × X Z = Aω z = hd, tl : Z → BZ alt : Z2 → Z

Define:

a1, a2, a3, a4, a5, . . . b1, b2, b3, b4, b5, . . . alt a1, b2, a3, b4, a5, . . . Z2

h

• alt
• B(Z2)

B(alt)

• Z

z

BZ

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A coinductive definition

4

Example: streams

BX = A × X Z = Aω z = hd, tl : Z → BZ alt : Z2 → Z

Define:

a1, a2, a3, a4, a5, . . . b1, b2, b3, b4, b5, . . . alt a1, b2, a3, b4, a5, . . . Z2

h

• alt
• B(Z2)

B(alt)

• Z

z

BZ = h(σ, τ) hd(σ), (tl(τ), tl(σ))

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A coinductive definition

4

Example: streams

BX = A × X Z = Aω z = hd, tl : Z → BZ alt : Z2 → Z

Define:

a1, a2, a3, a4, a5, . . . b1, b2, b3, b4, b5, . . . alt a1, b2, a3, b4, a5, . . . = Z2

alt

• z2 (BZ)2

λ

B(Z2)

B(alt)

• Z

z

BZ h(σ, τ) hd(σ), (tl(τ), tl(σ))

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A coinductive definition

4

Example: streams

BX = A × X Z = Aω z = hd, tl : Z → BZ alt : Z2 → Z

Define:

a1, a2, a3, a4, a5, . . . b1, b2, b3, b4, b5, . . . alt a1, b2, a3, b4, a5, . . . = Z2

alt

• z2 (BZ)2

λ

B(Z2)

B(alt)

• Z

z

BZ λ(a, σ, b, τ) a, (τ, σ)

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A coinductive definition

4

Example: streams

BX = A × X Z = Aω z = hd, tl : Z → BZ alt : Z2 → Z

Define:

a1, a2, a3, a4, a5, . . . b1, b2, b3, b4, b5, . . . alt a1, b2, a3, b4, a5, . . . = Z2

alt

• z2 (BZ)2

λ

B(Z2)

B(alt)

• Z

z

BZ λ(a, σ, b, τ) a, (τ, σ)

is natural.

λ : (B−)2 = ⇒ B(−)2

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Another example

5

BX = A × X Z = Aω

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Another example

5

BX = A × X Z = Aω

Define:

+ : A2 → A

Assume:

⊕ : Z2 → Z

as pointwise

+

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Another example

5

BX = A × X Z = Aω

Define:

+ : A2 → A

Assume:

⊕ : Z2 → Z

as pointwise

+ Z2

h

• ⊕
• B(Z2)

B(⊕)

• Z

z

BZ

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Another example

5

BX = A × X = Z = Aω

Define:

+ : A2 → A

Assume:

⊕ : Z2 → Z

as pointwise

+ Z2

h

• ⊕
• B(Z2)

B(⊕)

• Z

z

BZ h(σ, τ) hd(σ) + hd(τ), (tl(σ), tl(τ))

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Another example

5

BX = A × X = Z = Aω

Define:

+ : A2 → A

Assume:

⊕ : Z2 → Z

as pointwise

+ Z2

⊕

• z2 (BZ)2

λ

B(Z2)

B(⊕)

• Z

z

BZ h(σ, τ) hd(σ) + hd(τ), (tl(σ), tl(τ))

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Another example

5

BX = A × X = Z = Aω

Define:

+ : A2 → A

Assume:

⊕ : Z2 → Z

as pointwise

+ Z2

⊕

• z2 (BZ)2

λ

B(Z2)

B(⊕)

• Z

z

BZ λ(a, σ, b, τ) a + b, (σ, τ)

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Another example

5

BX = A × X = Z = Aω

Define:

+ : A2 → A

Assume:

⊕ : Z2 → Z

as pointwise

+ Z2

⊕

• z2 (BZ)2

λ

B(Z2)

B(⊕)

• Z

z

BZ λ(a, σ, b, τ) a + b, (σ, τ)

Again, natural.

λ : (B−)2 = ⇒ B(−)2

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Distributive laws

6

To define g : ΣZ → Z provide λ : ΣB =

⇒ BΣ

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Distributive laws

6

To define g : ΣZ → Z provide λ : ΣB =

⇒ BΣ

a distributive law

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Distributive laws

6

To define g : ΣZ → Z provide λ : ΣB =

⇒ BΣ

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Distributive laws

6

To define g : ΣZ → Z provide λ : ΣB =

⇒ BΣ ΣZ

g

• Σz ΣBZ

λZ BΣZ Bg

• Z

z

BZ

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Distributive laws

6

To define g : ΣZ → Z provide λ : ΣB =

⇒ BΣ ΣZ

g

• Σz ΣBZ

λZ BΣZ Bg

• Z

z

BZ X

h BX

ΣX

Σh ΣBX λX BΣX

Σλ

Benefits of naturality:

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More benefits

7

ΣZ

g

• Σz ΣBZ

λZ BΣZ Bg

• Z

z

BZ λ : ΣB = ⇒ BΣ

• perations
• n final -coalgebras

B

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More benefits

7

ΣZ

g

• Σz ΣBZ

λZ BΣZ Bg

• Z

z

BZ λ : ΣB = ⇒ BΣ

• perations
• n final -coalgebras

B

but also:

ΣA

a

• Σh
• A

h

• ΣBA

λA

BΣA

Ba

BA

behaviour

• f initial -algebras

Σ

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Behaviour of terms

8

BX = A × X ΣX = A + X2 alt a

binary constant, for a ∈ A

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Behaviour of terms

8

BX = A × X ΣX = A + X2 alt a

binary constant, for a ∈ A

(a, x′, b, y′) → a, (y′, x′) () → a, () λa

X : 1 → BΣX

λalt

X : (BX)2 → BΣX

λX : ΣBX → BΣX

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Behaviour of terms

8

BX = A × X ΣX = A + X2 alt a

binary constant, for a ∈ A

x

a

→ x′ y

b

→ y′ alt(x, y)

a

→ alt(y′, x′) a

a

→ a (a, x′, b, y′) → a, (y′, x′) () → a, () λa

X : 1 → BΣX

λalt

X : (BX)2 → BΣX

λX : ΣBX → BΣX

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Behaviour of terms

8

BX = A × X ΣX = A + X2 alt a

binary constant, for a ∈ A

x

a

→ x′ y

b

→ y′ alt(x, y)

a

→ alt(y′, x′) a

a

→ a

Then, e.g.,

alt(a, b)

a

→ alt(b, a)

etc.

(a, x′, b, y′) → a, (y′, x′) () → a, () λa

X : 1 → BΣX

λalt

X : (BX)2 → BΣX

λX : ΣBX → BΣX

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More benefits

9

ΣZ

g

• Σz ΣBZ

λZ BΣZ Bg

• Z

z

BZ λ : ΣB = ⇒ BΣ

• perations
• n final -coalgebras

B

but also:

ΣA

a

• Σh
• A

h

• ΣBA

λA

BΣA

Ba

BA

behaviour

• f initial -algebras

Σ

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More benefits

9

ΣZ

g

• Σz ΣBZ

λZ BΣZ Bg

• Z

z

BZ λ : ΣB = ⇒ BΣ

• perations
• n final -coalgebras

B

but also:

ΣA

a

• Σh
• A

h

• ΣBA

λA

BΣA

Ba

BA

behaviour

• f initial -algebras

Σ

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Bialgebras

10

• bialgebra:

λ ΣX

Σh

• g

X

h

BX ΣBX

λX

BΣX

Bg

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Bialgebras

10

• bialgebra:

λ

morphisms:

ΣX

Σh

• g

X

h

BX ΣBX

λX

BΣX

Bg

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Bialgebras

10

• bialgebra:

λ

morphisms:

ΣX

g

X

h BX

ΣY

k

Y

l

BY ΣX

Σh

• g

X

h

BX ΣBX

λX

BΣX

Bg

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Bialgebras

10

• bialgebra:

λ

morphisms:

ΣX

g

X

h f

• BX

ΣY

k

Y

l

BY ΣX

Σh

• g

X

h

BX ΣBX

λX

BΣX

Bg

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Bialgebras

10

• bialgebra:

λ

morphisms:

ΣX

g

• Σf
• X

h f

• BX

Bf

• ΣY

k

Y

l

BY ΣX

Σh

• g

X

h

BX ΣBX

λX

BΣX

Bg

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Bialgebras

10

• bialgebra:

λ

morphisms:

ΣX

g

• Σf
• X

h f

• BX

Bf

• ΣY

k

Y

l

BY

Bialgebras are (co)algebras:

λ-bialg ∼ = Σλ-alg ΣX

Σh

• g

X

h

BX ΣBX

λX

BΣX

Bg

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Bialgebras

10

• bialgebra:

λ

morphisms:

ΣX

g

• Σf
• X

h f

• BX

Bf

• ΣY

k

Y

l

BY

Bialgebras are (co)algebras:

λ-bialg ∼ = Σλ-alg ∼ = Bλ-coalg ΣX

Σh

• g

X

h

BX ΣBX

λX

BΣX

Bg

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More benefits

11

ΣZ

g

• Σz ΣBZ

λZ BΣZ Bg

• Z

z

BZ λ : ΣB = ⇒ BΣ ΣA

a

• Σh
• A

h

• ΣBA

λA

BΣA

Ba

BA

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More benefits

11

ΣZ

g

• Σz ΣBZ

λZ BΣZ Bg

• Z

z

BZ λ : ΣB = ⇒ BΣ ΣA

a

• Σh
• A

h

• ΣBA

λA

BΣA

Ba

BA

the final bialgebra the initial bialgebra

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Bisimilarity as a congruence

12

ΣA

a ∼ =

• Σf
• A

h f

• BA

Bf

• ΣZ

g

Z

z ∼ = BZ

The kernel relation of is:

• bisimilarity on ,
• a congruence on .

f h a

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Bisimilarity as a congruence

12

ΣA

a ∼ =

• Σf
• A

h f

• BA

Bf

• ΣZ

g

Z

z ∼ = BZ

The kernel relation of is:

• bisimilarity on ,
• a congruence on .

f h a

Bisimilarity is a congruence

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Bisimilarity as a congruence

12

ΣA

a ∼ =

• Σf
• A

h f

• BA

Bf

• ΣZ

g

Z

z ∼ = BZ

The kernel relation of is:

• bisimilarity on ,
• a congruence on .

f h a

Bisimilarity is a congruence For example,

alt(a, a) ≈ a

hence

C[alt(a, a)] ≈ C[a]

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II. MORE DISTRIBUTIVE LAWS

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Another coinductive definition

14

BX = A × X Z = Aω

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Another coinductive definition

14

BX = A × X Z = Aω

Define: zip : Z2 → Z

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Another coinductive definition

14

BX = A × X Z = Aω a1, a2, a3, a4, a5, . . . b1, b2, b3, b4, b5, . . .

Define: zip : Z2 → Z

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Another coinductive definition

14

BX = A × X Z = Aω a1, a2, a3, a4, a5, . . . b1, b2, b3, b4, b5, . . .

Define: zip : Z2 → Z

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Another coinductive definition

14

BX = A × X Z = Aω a1, a2, a3, a4, a5, . . . b1, b2, b3, b4, b5, . . .

Define: zip : Z2 → Z

zip a1, b1, a2, b2, a3, . . .

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Another coinductive definition

14

BX = A × X Z = Aω a1, a2, a3, a4, a5, . . . b1, b2, b3, b4, b5, . . .

Define: zip : Z2 → Z

Z2

zip

• z2 (BZ)2

λ

B(Z2)

B(zip)

• Z

z

BZ zip a1, b1, a2, b2, a3, . . .

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Another coinductive definition

14

BX = A × X Z = Aω a1, a2, a3, a4, a5, . . . b1, b2, b3, b4, b5, . . .

Define: zip : Z2 → Z

Z2

zip

• z2 (BZ)2

λ

B(Z2)

B(zip)

• Z

z

BZ λ(a, σ, b, τ) a, (b :: τ, σ) = zip a1, b1, a2, b2, a3, . . .

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Another coinductive definition

14

BX = A × X Z = Aω a1, a2, a3, a4, a5, . . . b1, b2, b3, b4, b5, . . .

Define: zip : Z2 → Z

Z2

zip

• z2 (BZ)2

λ

B(Z2)

B(zip)

• Z

z

BZ λ(a, σ, b, τ) a, (b :: τ, σ) = x

a

→ x′ zip(x, y)

a

→ zip(y, x′) zip a1, b1, a2, b2, a3, . . .

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Another coinductive definition

14

BX = A × X Z = Aω a1, a2, a3, a4, a5, . . . b1, b2, b3, b4, b5, . . .

Define: zip : Z2 → Z

Z2

zip

• z2 (BZ)2

λ

B(Z2)

B(zip)

• Z

z

BZ λ(a, σ, b, τ) a, (b :: τ, σ) = x

a

→ x′ zip(x, y)

a

→ zip(y, x′)

But this is not natural!

zip a1, b1, a2, b2, a3, . . .

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Copointed coalgebras

15

A distributive law definition of :

zip Z2

zip

• id,z2 (Z × BZ)2

λ

B(Z2)

B(zip)

• Z

z

BZ (x, a, x′, y, b, y′) → a, (y, x′) x

a

→ x′ zip(x, y)

a

→ zip(y, x′)

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Copointed coalgebras

15

A distributive law definition of :

zip Z2

zip

• id,z2 (Z × BZ)2

λ

B(Z2)

B(zip)

• Z

z

BZ (x, a, x′, y, b, y′) → a, (y, x′) x

a

→ x′ zip(x, y)

a

→ zip(y, x′) λ : Σ(Id × B) = ⇒ BΣ

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Copointed coalgebras

15

A distributive law definition of :

zip Z2

zip

• id,z2 (Z × BZ)2

λ

B(Z2)

B(zip)

• Z

z

BZ (x, a, x′, y, b, y′) → a, (y, x′) x

a

→ x′ zip(x, y)

a

→ zip(y, x′) λ : Σ(Id × B) = ⇒ BΣ

To gain the benefits, work with copointed coalgebras.

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Pointed algebras

16

a/− : Z → Z b1, b2, b3, b4, b5, . . . a, b2, b3, b4, b5, . . . a/−

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Pointed algebras

16

a/− : Z → Z b1, b2, b3, b4, b5, . . . a, b2, b3, b4, b5, . . . a/−

?

Z

a/−

• BZ

B(a/−)

• Z

z

BZ

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Pointed algebras

16

a/− : Z → Z b1, b2, b3, b4, b5, . . . a, b2, b3, b4, b5, . . . a/−

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Pointed algebras

16

a/− : Z → Z b1, b2, b3, b4, b5, . . . a, b2, b3, b4, b5, . . . a/− ΣZ

g

• Σz ΣBZ

λ

B(Z + ΣZ)

B[id,g]

• Z

z

BZ

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Pointed algebras

16

a/− : Z → Z b1, b2, b3, b4, b5, . . . a, b2, b3, b4, b5, . . . a/− ΣZ

g

• Σz ΣBZ

λ

B(Z + ΣZ)

B[id,g]

• Z

z

BZ λ : BZ → B(Z + Z) x

b

− → x′ a/x

a

− → x′ b, x′ → a, ι1(x′)

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Pointed algebras

16

a/− : Z → Z b1, b2, b3, b4, b5, . . . a, b2, b3, b4, b5, . . . a/− ΣZ

g

• Σz ΣBZ

λ

B(Z + ΣZ)

B[id,g]

• Z

z

BZ λ : BZ → B(Z + Z) x

b

− → x′ a/x

a

− → x′ b, x′ → a, ι1(x′)

To gain the benefits, work with pointed algebras.

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Complex successors

17

x

a

− → x′ f(x)

a

− → zip(a, f(x′))

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Complex successors

17

x

a

− → x′ f(x)

a

− → zip(a, f(x′)) λ : ΣB = ⇒ BTΣ

free monad over Σ

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Complex successors

17

x

a

− → x′ f(x)

a

− → zip(a, f(x′)) λ : ΣB = ⇒ BTΣ

free monad over Σ

ΣZ

g

• Σz ΣBZ

λ

BTΣ

Bg♯

• Z

z

BZ

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Complex successors

17

x

a

− → x′ f(x)

a

− → zip(a, f(x′)) λ : ΣB = ⇒ BTΣ

free monad over Σ

ΣZ

g

• Σz ΣBZ

λ

BTΣ

Bg♯

• Z

z

BZ

Here, work with E-M algebras for the monad TΣ

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Complex successors

17

x

a

− → x′ f(x)

a

− → zip(a, f(x′)) λ : ΣB = ⇒ BTΣ

free monad over Σ Together with :

zip λ : Σ(Id × B) = ⇒ BTΣ ΣZ

g

• Σz ΣBZ

λ

BTΣ

Bg♯

• Z

z

BZ

Here, work with E-M algebras for the monad TΣ

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Look-ahead

18

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Look-ahead

18

• dd : Z → Z

a1, a2, a3, a4, a5, . . . a1, a3, a5, a7, ...

• dd

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Look-ahead

18

• dd : Z → Z

a1, a2, a3, a4, a5, . . . a1, a3, a5, a7, ...

• dd

x

a

− → y

b

− → z

• dd(x)

a

− → odd(z)

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Look-ahead

18

• dd : Z → Z

a1, a2, a3, a4, a5, . . . a1, a3, a5, a7, ...

• dd

x

a

− → y

b

− → z

• dd(x)

a

− → odd(z) λ : ΣDB = ⇒ BΣ

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Look-ahead

18

• dd : Z → Z

a1, a2, a3, a4, a5, . . . a1, a3, a5, a7, ...

• dd

x

a

− → y

b

− → z

• dd(x)

a

− → odd(z) λ : ΣDB = ⇒ BΣ

cofree comonad over B

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Look-ahead

18

• dd : Z → Z

a1, a2, a3, a4, a5, . . . a1, a3, a5, a7, ...

• dd

x

a

− → y

b

− → z

• dd(x)

a

− → odd(z) λ : ΣDB = ⇒ BΣ

cofree comonad over B

ΣZ

g

• Σz♭ ΣDBZ

λZ

BΣZ

Bg

• Z

z

BZ

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Monad over comonad

19

The general case:

λ : TΣDB = ⇒ DBTΣ

subject to laws

TΣZ

g

• TΣz TΣDBZ

λz

DBTΣZ

DBg

• Z

z

DBZ

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Monad over comonad

19

The general case:

λ : TΣDB = ⇒ DBTΣ

subject to laws This subsumes all previous cases.

TΣZ

g

• TΣz TΣDBZ

λz

DBTΣZ

DBg

• Z

z

DBZ

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III. STRUCTURAL OPERATIONAL SEMANTICS

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III. STRUCTURAL OPERATIONAL SEMANTICS

BX = (PωX)A

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Toy SOS example

21

a

a

− → nil ΣX = 1 + A + X2 BX = (PωX)A x

a

− → x′ y

a

− → y′ x ⊗ y

a

− → x′ ⊗ y′

This defines: λ : ΣB =

⇒ BΣ

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Toy SOS example

21

a

a

− → nil ΣX = 1 + A + X2 BX = (PωX)A x

a

− → x′ y

a

− → y′ x ⊗ y

a

− → x′ ⊗ y′

This defines: λ : ΣB =

⇒ BΣ

Fact: the LTS induced by the rules is the initial -bialgebra.

λ

Hence, bisimilarity on it is a congruence.

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GSOS

22

Laws correspond to rules:

λ : Σ(Id × B) = ⇒ BTΣ

• , are all distinct

{xi

aij

− → yij}1≤i≤n

1≤j≤mi

{xi

bik

− →}1≤i≤n

1≤k≤li

f(x1, . . . , xn)

c

− → t xi yij

• no other variables occur in t

where

• there are finitely many rules for every ,

f c

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GSOS

22

Laws correspond to rules:

λ : Σ(Id × B) = ⇒ BTΣ

• , are all distinct

{xi

aij

− → yij}1≤i≤n

1≤j≤mi

{xi

bik

− →}1≤i≤n

1≤k≤li

f(x1, . . . , xn)

c

− → t xi yij

• no other variables occur in t

where

• there are finitely many rules for every ,

f c

no lookahead

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GSOS

22

Laws correspond to rules:

λ : Σ(Id × B) = ⇒ BTΣ

• , are all distinct

{xi

aij

− → yij}1≤i≤n

1≤j≤mi

{xi

bik

− →}1≤i≤n

1≤k≤li

f(x1, . . . , xn)

c

− → t xi yij

• no other variables occur in t

where

• there are finitely many rules for every ,

f c

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GSOS

22

Laws correspond to rules:

λ : Σ(Id × B) = ⇒ BTΣ

• , are all distinct

{xi

aij

− → yij}1≤i≤n

1≤j≤mi

{xi

bik

− →}1≤i≤n

1≤k≤li

f(x1, . . . , xn)

c

− → t xi yij

• no other variables occur in t

where

• there are finitely many rules for every ,

f c

This is GSOS, where bisimilarity is a congruence.

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Safe ntree

23

Laws correspond to rules:

λ : ΣDB = ⇒ B(Id + Σ)

• , are all distinct

xi

• no other variables occur in the rule

where

• there are finitely many rules for every ,

f c

• has depth at most

{zi

ai

− → yi}i∈I {wj

bj

− →}j∈J f(x1, . . . , xn)

c

− → t t 1 yi

• the graph of premises is well-founded

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Safe ntree

23

Laws correspond to rules:

λ : ΣDB = ⇒ B(Id + Σ)

• , are all distinct

xi

• no other variables occur in the rule

where

• there are finitely many rules for every ,

f c

• has depth at most

{zi

ai

− → yi}i∈I {wj

bj

− →}j∈J f(x1, . . . , xn)

c

− → t t 1 yi

• the graph of premises is well-founded

This is the “safe” ntree format.

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Monad over comonad?

24

λ : Σ(Id × B) = ⇒ BTΣ

complex successors

λ : ΣDB = ⇒ B(Id + Σ)

look-ahead

λ : TΣDB = ⇒ DBTΣ

both?

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Monad over comonad?

24

λ : Σ(Id × B) = ⇒ BTΣ

complex successors

λ : ΣDB = ⇒ B(Id + Σ)

look-ahead

λ : TΣDB = ⇒ DBTΣ

both?

x

a

− → y y

b

− → f(x)

b

− → k k

a

− → f(k)

But: does not induce an LTS!

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IV . FURTHER DEVELOPMENTS

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Probabilistic (reactive) GSOS

26

λ : Σ(Id × B) = ⇒ BTΣ

Probabilistic bisimilarity is a congruence.

⇐ ⇒ BX = (1 + DωX)A DωX = {φ : X → R+

0 | ♯supp(φ) < ω, x φ(x) = 1}

• xi

a

− →

• a∈Di
• xi

a

− →

• a∈Bi
• xij

bj

− → yj

• 1≤j≤k

f(x1, . . . , xn)

c,w

− → t

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Stochastic GSOS

27

BX = (FX)A FX = {φ : X → R+

0 | ♯supp(φ) < ω}

λ : Σ(Id × B) = ⇒ BTΣ

• xi

a@wai

− →

• a∈Di,1≤i≤n
• xij

bj

− → yj

• 1≤j≤k

f(x1, . . . , xn)

c,w

− → t

Stochastic bisimilarity is a congruence.

⇐ ⇒

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Weighted GSOS

28

BX = (FX)A λ : Σ(Id × B) = ⇒ BTΣ

Weighted bisimilarity is a congruence.

FX = {φ : X → W | ♯supp(φ) < ω}

• xi

a

− → wa,i

• a∈Di,1≤i≤n
• xij

bj

− → yj

• 1≤j≤k

f(x1, . . . , xn)

c,β

− → t = ⇒

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Weighted GSOS

28

BX = (FX)A λ : Σ(Id × B) = ⇒ BTΣ

Weighted bisimilarity is a congruence.

FX = {φ : X → W | ♯supp(φ) < ω}

any comm. monoid

• xi

a

− → wa,i

• a∈Di,1≤i≤n
• xij

bj

− → yj

• 1≤j≤k

f(x1, . . . , xn)

c,β

− → t = ⇒

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Timed SOS

29

BX = {T ⇀ X | . . .} T a time domain

(a comm. monoid with cancellation and linear specialization order)

λ : TΣDB = ⇒ DBTΣ ⇐ ⇒ {xi(0)

ti

− → xi(ti) | ti ∈ T ∧ xi(ti)↓}1≤i≤n f(x1(0), . . . , xn(0))

t

− → θt

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Name-passing GSOS

30

Another underlying category: Nom (actually a bit more complicated...)

λ : Σ(| −|× B| −| ) = ⇒ BTΣ| −| = ⇒

“Wide open” bisimilarity a congruence.

x

c!d

− → x′ y

c?(a)

− → y′ x||y

τ

− → y||([d/a]y′)

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Other work

31

• The “microcosm” interpretation

GSOS rules

• perations on coalgebras

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Other work

31

• The “microcosm” interpretation

GSOS rules

• perations on coalgebras

Parallel composition of states vs. Parallel composition of coalgebras

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Other work

31

• The “microcosm” interpretation

GSOS rules

• perations on coalgebras

Parallel composition of states vs. Parallel composition of coalgebras

• tyft/tyxt categorically

t1

a1

− → u1 t2

a2

− → u2 f(x1, . . . , xn)

a

− → t

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V . SELECTED OPEN PROBLEMS

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Rules/premises/conclusions

33

collection of rules distributive law

≈

rule

≈

?

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Rules/premises/conclusions

33

collection of rules distributive law

≈

rule

≈

?

x

a

− → x′ y

b

− → f(x, y)

c

− → g(x′, y)

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x | = ax′ y | = [b]ff

Rules/premises/conclusions

33

collection of rules distributive law

≈

rule

≈

?

x

a

− → x′ y

b

− → f(x, y)

c

− → g(x′, y)

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x | = ax′ y | = [b]ff

Rules/premises/conclusions

33

collection of rules distributive law

≈

rule

≈

?

x

a

− → x′ y

b

− → f(x, y)

c

− → g(x′, y)

premise

≈

modal formula conclusion

≈

?

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Connections to modal logic

34

Distributive law bisimilarity a congruence.

= ⇒

What about other equivalences?

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Connections to modal logic

34

Distributive law bisimilarity a congruence.

= ⇒

What about other equivalences? Idea: see them as logical equivalences, use the logic.

⊤ = T aT = a ∨ [⊗]a⊤ a⊤ = a ∨ [⊗]a⊤ a(b ∨ [⊗]x) = [⊗]ax a[⊗]x = [⊗]ax

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Translations

35

Morphisms of ditributive laws:

• well understood abstractly
• very few examples worked out

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Translations

35

Morphisms of ditributive laws:

• well understood abstractly
• very few examples worked out

What is a translation of GSOS specifications?

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Translations

35

Morphisms of ditributive laws:

• well understood abstractly
• very few examples worked out

What is a translation of GSOS specifications? How to translate CCS into timed CCS? Or into -calculus?

π

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Translations

35

Morphisms of ditributive laws:

• well understood abstractly
• very few examples worked out

What is a translation of GSOS specifications? How to translate CCS into timed CCS? Or into -calculus?

π

Logics can be translated too!

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Parametricity

36

Various form of parametricity:

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Parametricity

36

Various form of parametricity:

• parametrized by :

x

a

− → x′ loop(x)

a

− → x′; loop(x) loop ;

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Parametricity

36

Various form of parametricity:

• parametrized by :

x

a

− → x′ loop(x)

a

− → x′; loop(x) loop ;

• CCS parametrized by a set of actions

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Parametricity

36

Various form of parametricity:

• parametrized by :

x

a

− → x′ loop(x)

a

− → x′; loop(x) loop ;

• CCS parametrized by a set of actions

What is a parametric specification?

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Parametricity

36

Various form of parametricity:

• parametrized by :

x

a

− → x′ loop(x)

a

− → x′; loop(x) loop ;

• CCS parametrized by a set of actions

What is a parametric specification? E.g. pushout parametricity:

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Parametricity

36

Various form of parametricity:

• parametrized by :

x

a

− → x′ loop(x)

a

− → x′; loop(x) loop ;

• CCS parametrized by a set of actions

What is a parametric specification? E.g. pushout parametricity:

λp

• λb

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Parametricity

36

Various form of parametricity:

• parametrized by :

x

a

− → x′ loop(x)

a

− → x′; loop(x) loop ;

• CCS parametrized by a set of actions

What is a parametric specification? E.g. pushout parametricity:

λp

• λb

parameter body

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Parametricity

36

Various form of parametricity:

• parametrized by :

x

a

− → x′ loop(x)

a

− → x′; loop(x) loop ;

• CCS parametrized by a set of actions

What is a parametric specification? E.g. pushout parametricity:

λp

• λa

λb

parameter body

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Parametricity

36

Various form of parametricity:

• parametrized by :

x

a

− → x′ loop(x)

a

− → x′; loop(x) loop ;

• CCS parametrized by a set of actions

What is a parametric specification? E.g. pushout parametricity:

λp

• λa

λb

parameter body argument

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Parametricity

36

Various form of parametricity:

• parametrized by :

x

a

− → x′ loop(x)

a

− → x′; loop(x) loop ;

• CCS parametrized by a set of actions

What is a parametric specification? E.g. pushout parametricity:

λp

• λa
• λb

λ

parameter body argument

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Modular SOS formalism

37

Language for defining SOS specifications

• building blocks: distributive laws
• ways of combining them:

= translation = parametricity / instantiation = ways of writing them down (rules?)