[PPT] - On Semantic Relations: From probabilistic systems to coalgebras and PowerPoint Presentation

SLIDE 1

On Semantic Relations:

From probabilistic systems to coalgebras and back

Ana Sokolova

SOS group, Radboud University Nijmegen

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.1/30

SLIDE 2

Outline

Introduction - probabilistic systems and coalgebras

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.2/30

SLIDE 3

Outline

Introduction - probabilistic systems and coalgebras
Bisimilarity - the strong end of the spectrum

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.2/30

SLIDE 4

Outline

Introduction - probabilistic systems and coalgebras
Bisimilarity - the strong end of the spectrum
Application - expressiveness hierarchy

(older result)

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.2/30

SLIDE 5

Outline

Introduction - probabilistic systems and coalgebras
Bisimilarity - the strong end of the spectrum
Application - expressiveness hierarchy

(older result)

Trace semantics - the weak end of the spectrum

(new !)

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.2/30

SLIDE 6

Systems

are formal objects, transition systems (e.g. LTS), that serve as models of real (software, hardware,...) systems

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.3/30

SLIDE 7

Systems

are formal objects, transition systems (e.g. LTS), that serve as models of real (software, hardware,...) systems

GEOCAL

’06, Probabilistic Systems Workshop, AS – p.3/30

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Systems

are formal objects, transition systems (e.g. LTS), that serve as models of real (software, hardware,...) systems

a
b
a
c
c
GEOCAL

’06, Probabilistic Systems Workshop, AS – p.3/30

SLIDE 9

Probabilistic systems

arise by enriching transition systems with (discrete) probabilities as labels on the transitions. Examples:

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.4/30

SLIDE 10

Probabilistic systems

arise by enriching transition systems with (discrete) probabilities as labels on the transitions. Examples:

1

3 2 3

⇒
1
GEOCAL

’06, Probabilistic Systems Workshop, AS – p.4/30

SLIDE 11

Probabilistic systems

arise by enriching transition systems with (discrete) probabilities as labels on the transitions. Examples:

a[ 1

3]

b[ 2

3]

⇒
a[1]
GEOCAL

’06, Probabilistic Systems Workshop, AS – p.4/30

SLIDE 12

Probabilistic systems

arise by enriching transition systems with (discrete) probabilities as labels on the transitions. Examples:

♦

1 3 2 3

⇒
a
GEOCAL

’06, Probabilistic Systems Workshop, AS – p.4/30

SLIDE 13

Probabilistic systems

arise by enriching transition systems with (discrete) probabilities as labels on the transitions. Examples:

a
b
⇒

♦

1

GEOCAL

’06, Probabilistic Systems Workshop, AS – p.4/30

SLIDE 14

Probabilistic systems

arise by enriching transition systems with (discrete) probabilities as labels on the transitions. Examples:

⇒
1
a
b
1
a
GEOCAL

’06, Probabilistic Systems Workshop, AS – p.4/30

SLIDE 15

Probabilistic systems

arise by enriching transition systems with (discrete) probabilities as labels on the transitions. Examples:

⇒
a
1

3 2 3

b
1
GEOCAL

’06, Probabilistic Systems Workshop, AS – p.4/30

SLIDE 16

Coalgebras

are an elegant generalization of transition systems with states + transitions

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.5/30

SLIDE 17

Coalgebras

are an elegant generalization of transition systems with states + transitions as pairs S, α : S → FS, for F a functor

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.5/30

SLIDE 18

Coalgebras

are an elegant generalization of transition systems with states + transitions as pairs S, α : S → FS, for F a functor

based on category theory
provide a uniform way of treating transition systems
provide general notions and results e.g. a generic notion of

bisimulation

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.5/30

SLIDE 19

Examples

A TS is a pair S, α : S → PS !! coalgebra of the powerset functor P

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.6/30

SLIDE 20

Examples

A TS is a pair S, α : S → PS !! coalgebra of the powerset functor P An LTS is a pair S, α : S → PSA !!! coalgebra of the functor PA Note: PA ∼ = P(A × )

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.6/30

SLIDE 21

More examples

Thanks to the probability distribution functor D

DS = {µ : S → [0, 1], µ[S] = 1}, µ[X] =

s∈X µ(x)

Df : DS → DT, Df(µ)(t) = µ[f −1({t})]

the probabilistic systems are also coalgebras

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.7/30

SLIDE 22

More examples

Thanks to the probability distribution functor D

DS = {µ : S → [0, 1], µ[S] = 1}, µ[X] =

s∈X µ(x)

Df : DS → DT, Df(µ)(t) = µ[f −1({t})]

the probabilistic systems are also coalgebras ... of functors built by the following syntax F ::= | A | P | D | G + H | G × H | GA | G ◦ H

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.7/30

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reactive, generative

evolve from LTS - functor P (A × ) ∼ = P

A

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.8/30

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reactive, generative

evolve from LTS - functor P (A × ) ∼ = P

A

reactive systems: functor (D + 1)A

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.8/30

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reactive, generative

evolve from LTS - functor P (A × ) ∼ = P

A

reactive systems: functor (D + 1)A generative systems: functor (D + 1)(A × ) = D(A × ) + 1

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.8/30

SLIDE 26

reactive, generative

evolve from LTS - functor P (A × ) ∼ = P

A

reactive systems: functor (D + 1)A generative systems: functor (D + 1)(A × ) = D(A × ) + 1 note: in the probabilistic case (D + 1)A ∼ = D(A × ) + 1

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.8/30

SLIDE 27

Probabilistic system types

MC D DLTS ( + 1)A LTS P(A × ) ∼ = PA React (D + 1)A Gen D(A × ) + 1 Str D + (A × ) + 1 Alt D + P(A × ) Var D(A × ) + P(A × ) SSeg P(A × D) Seg PD(A × ) . . . . . .

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.9/30

SLIDE 28

Probabilistic system types

MC D DLTS ( + 1)A LTS P(A × ) ∼ = PA React (D + 1)A Gen D(A × ) + 1 Str D + (A × ) + 1 Alt D + P(A × ) Var D(A × ) + P(A × ) SSeg P(A × D) Seg PD(A × ) . . . . . .

a[ 2

3 ]

a[ 1

3 ]

b[1]

b[1]
a[1]
GEOCAL

’06, Probabilistic Systems Workshop, AS – p.9/30

SLIDE 29

Probabilistic system types

MC D DLTS ( + 1)A LTS P(A × ) ∼ = PA React (D + 1)A Gen D(A × ) + 1 Str D + (A × ) + 1 Alt D + P(A × ) Var D(A × ) + P(A × ) SSeg P(A × D) Seg PD(A × ) . . . . . .

a[ 1

4 ]

a[ 1

2 ]

b[ 1

4 ]

c[1]
c[1]
GEOCAL

’06, Probabilistic Systems Workshop, AS – p.9/30

SLIDE 30

Probabilistic system types

MC D DLTS ( + 1)A LTS P(A × ) ∼ = PA React (D + 1)A Gen D(A × ) + 1 Str D + (A × ) + 1 Alt D + P(A × ) Var D(A × ) + P(A × ) SSeg P(A × D) Seg PD(A × ) . . . . . . ♦

1 4 3 4

b
a
♦

1 2 1 2

GEOCAL

’06, Probabilistic Systems Workshop, AS – p.9/30

SLIDE 31

Probabilistic system types

MC D DLTS ( + 1)A LTS P(A × ) ∼ = PA React (D + 1)A Gen D(A × ) + 1 Str D + (A × ) + 1 Alt D + P(A × ) Var D(A × ) + P(A × ) SSeg P(A × D) Seg PD(A × ) . . . . . .

a
a
b
1

4 3 4

1

1 3

2

3

GEOCAL

’06, Probabilistic Systems Workshop, AS – p.9/30

SLIDE 32

Probabilistic system types

MC D DLTS ( + 1)A LTS P(A × ) ∼ = PA React (D + 1)A Gen D(A × ) + 1 Str D + (A × ) + 1 Alt D + P(A × ) Var D(A × ) + P(A × ) SSeg P(A × D) Seg PD(A × ) . . . . . .

a[ 1

4 ]

b[ 3

4 ]

a[ 1

3 ]

a[ 2

3 ]

GEOCAL

’06, Probabilistic Systems Workshop, AS – p.9/30

SLIDE 33

Bisimulation - LTS

Consider the LTS

s0
a
b
t0

b

a
t2
a
b
s1
t1
t3

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.10/30

SLIDE 34

Bisimulation - LTS

Consider the LTS

s0
a
b
t0

b

a
t2
a
b
s1
t1
t3

The states s0 and t0 are bisimilar since there is a bisimulation R relating them...

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.10/30

SLIDE 35

Bisimulation - LTS

Consider the LTS

s0
a
b
t0

b

a
t2
a
b
s1
t1
t3

Transfer condition: s, t ∈ R = ⇒ s

a

→ s′ ⇒ (∃t′) t

a

→ t′, s′, t′ ∈ R, t

a

→ t′ ⇒ (∃s′) s

a

→ s′, s′, t′ ∈ R

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.10/30

SLIDE 36

Bisimulation - generative

Consider the generative systems

s0
a[ 1

2]

b[ 1

2]

t0

b[ 1

2 ]

a[ 1

2 ]

t2
a[ 1

6]

a[ 1

3 ]

b[ 1

2 ]

s1
t1
t3

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.11/30

SLIDE 37

Bisimulation - generative

Consider the generative systems

s0
a[ 1

2]

b[ 1

2]

t0

b[ 1

2 ]

a[ 1

2 ]

t2
a[ 1

6]

a[ 1

3 ]

b[ 1

2 ]

s1
t1
t3

The states s0 and t0 are bisimilar, and so are s0 and t2, since there is a bisimulation R relating them...

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.11/30

SLIDE 38

Bisimulation - generative

Consider the generative systems

s0
a[ 1

2]

b[ 1

2]

t0

b[ 1

2 ]

a[ 1

2 ]

t2
a[ 1

6]

a[ 1

3 ]

b[ 1

2 ]

s1
t1
t3

Transfer condition: s, t ∈ R = ⇒ s ❀ µ ⇒ (∃µ′) t ❀ µ′, µ ≡R,A µ′

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.11/30

SLIDE 39

Bisimulation - simple Segala

Consider the simple Segala systems

s0
a
b
t0

b

a
1

3

2

3

t2
a
b
s1
t1
t3

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.12/30

SLIDE 40

Bisimulation - simple Segala

Consider the simple Segala systems

s0
a
b
t0

b

a
1

3

2

3

t2
a
b
s1
t1
t3

The states s0 and t0 are bisimilar, since there is a bisimulation R relating them...

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.12/30

SLIDE 41

Bisimulation - simple Segala

Consider the simple Segala systems

s0
a
b
t0

b

a
1

3

2

3

t2
a
b
s1
t1
t3

Transfer condition: s, t ∈ R = ⇒ s

a

→ µ ⇒ (∃µ′) t

a

→ µ′, µ ≡R µ′

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.12/30

SLIDE 42

Coalgebraic bisimulation

A bisimulation between S, α : S → FS and T, β : S → FS is R ⊆ S × T such that ∃ γ:

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.13/30

SLIDE 43

Coalgebraic bisimulation

A bisimulation between S, α : S → FS and T, β : S → FS is R ⊆ S × T such that ∃ γ: S

α

R

γ

π1
π2

T

β

FS

FR

Fπ1

Fπ2

FT

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.13/30

SLIDE 44

Coalgebraic bisimulation

A bisimulation between S, α : S → FS and T, β : S → FS is R ⊆ S × T such that ∃ γ: S

α

R

γ

π1
π2

T

β

FS

FR

Fπ1

Fπ2

FT Transfer condition: s, t ∈ R = ⇒ α(s), β(t) ∈ Rel(F)(R)

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.13/30

SLIDE 45

Coalgebraic bisimulation

A bisimulation between S, α : S → FS and T, β : S → FS is R ⊆ S × T such that ∃ γ: S

α

R

γ

π1
π2

T

β

FS

FR

Fπ1

Fπ2

FT

Theorem: Coalgebraic and concrete bisimilarity coincide !

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.13/30

SLIDE 46

Expressiveness

simple Segala system → Segala system

a
p1

p2 pn

...
GEOCAL

’06, Probabilistic Systems Workshop, AS – p.14/30

SLIDE 47

Expressiveness

simple Segala system → Segala system

a
p1

p2 pn

...
a[p1]

a[p2] a[pn]

...
GEOCAL

’06, Probabilistic Systems Workshop, AS – p.14/30

SLIDE 48

Expressiveness

simple Segala system → Segala system

a
p1

p2 pn

...
a[p1]

a[p2] a[pn]

...
When do we consider one type of systems more

expressive than another?

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.14/30

SLIDE 49

Comparison criterion

CoalgF → CoalgG if there is a mapping S, α : S → FS

T

→ S, ˜ α : S → GS that preserves and reflects bisimilarity

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.15/30

SLIDE 50

Comparison criterion

CoalgF → CoalgG if there is a mapping S, α : S → FS

T

→ S, ˜ α : S → GS that preserves and reflects bisimilarity sS,α ∼ tT,β ⇐ ⇒ sT S,α ∼ tT T,β

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.15/30

SLIDE 51

Comparison criterion

CoalgF → CoalgG if there is a mapping S, α : S → FS

T

→ S, ˜ α : S → GS that preserves and reflects bisimilarity Theorem: An injective natural transformation F ⇒ G is sufficient for CoalgF → CoalgG

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.15/30

SLIDE 52

Comparison criterion

CoalgF → CoalgG if there is a mapping S, α : S → FS

T

→ S, ˜ α : S → GS that preserves and reflects bisimilarity Theorem: An injective natural transformation F ⇒ G is sufficient for CoalgF → CoalgG

proof via cocongruences - behavioral equivalence

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.15/30

SLIDE 53

Example

Indeed SSeg → Seg since P(A × D)

Pτ

⇒ PD(A × ) is injective for

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.16/30

SLIDE 54

Example

Indeed SSeg → Seg since P(A × D)

Pτ

⇒ PD(A × ) is injective for (A × D)

τ

⇒ D(A × ) given by

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.16/30

SLIDE 55

Example

Indeed SSeg → Seg since P(A × D)

Pτ

⇒ PD(A × ) is injective for (A × D)

τ

⇒ D(A × ) given by τX(a, µ) = δa × µ, where

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.16/30

SLIDE 56

Example

Indeed SSeg → Seg since P(A × D)

Pτ

⇒ PD(A × ) is injective for (A × D)

τ

⇒ D(A × ) given by τX(a, µ) = δa × µ, where µ × µ′(x, x′) = µ(x) · µ′(x′)

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.16/30

SLIDE 57

Example

Indeed SSeg → Seg since P(A × D)

Pτ

⇒ PD(A × ) is injective for (A × D)

τ

⇒ D(A × ) given by τX(a, µ) = δa × µ, where µ × µ′(x, x′) = µ(x) · µ′(x′) and δa is Dirac distribution for a

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.16/30

SLIDE 58

The hierarchy...

MG PZ

Seg
Bun
SSeg
Var
Alt
React
LTS
Gen
Str
DLTS
MC
GEOCAL

’06, Probabilistic Systems Workshop, AS – p.17/30

SLIDE 59

The hierarchy...

MG PZ

Seg
Bun
SSeg
Var
Alt
React
LTS
Gen
Str
DLTS
MC
* Falk Bartels, AS, Erik de Vink

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SLIDE 60

LT/BT spectrum

Bisimilarity is not the only semantics...

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.18/30

SLIDE 61

LT/BT spectrum

Are these non-deterministic systems equal ?

x

a

y

a

a
b
c
b
c
GEOCAL

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SLIDE 62

LT/BT spectrum

Are these non-deterministic systems equal ?

x

a

y

a

a
b
c
b
c
x and y are:
different wrt. bisimilarity

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.18/30

SLIDE 63

LT/BT spectrum

Are these non-deterministic systems equal ?

x

a

y

a

a
b
c
b
c
x and y are:
different wrt. bisimilarity, but
equivalent wrt. trace semantics

tr(x) = tr(y) = {ab, ac}

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.18/30

SLIDE 64

Traces - LTS

For LTS with explicit termination (NA) trace = the set of all possible linear behaviors

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.19/30

SLIDE 65

Traces - LTS

For LTS with explicit termination (NA) trace = the set of all possible linear behaviors Example:

x
a
a
y
b
tr(y) = b∗,

tr(x) = a+ · tr(y) = a+ · b∗

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.19/30

SLIDE 66

Traces - generative

For generative probabilistic systems with ex. termination trace = sub-probability distribution over possible linear behaviors

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.20/30

SLIDE 67

Traces - generative

For generative probabilistic systems with ex. termination trace = sub-probability distribution over possible linear behaviors Example:

x

b[ 1

3 ]

a[ 1

3 ]

1

3

y
a[ 1

2]

1

2

z
a[1]
tr(x) :

→ 1

3

a → 1

3 · 1 2

a2 → 1

3 · 1 2 · 1 2

· · ·

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.20/30

SLIDE 68

Trace of a coalgebra ?

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.21/30

SLIDE 69

Trace of a coalgebra ?

Power&Turi ’99
Jacobs ’04
Hasuo& Jacobs ’05
Hasuo, Jacobs, AS: Generic Trace Theory, CMCS’06

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.21/30

SLIDE 70

Trace of a coalgebra ?

Power&Turi ’99
Jacobs ’04
Hasuo& Jacobs ’05
Hasuo, Jacobs, AS: Generic Trace Theory, CMCS’06

main idea: coinduction in a Kleisli category

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.21/30

SLIDE 71

Coinduction

FX

F(beh)

FZ

X

α

beh
Z

∼ =

system

final coalgebra

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.22/30

SLIDE 72

Coinduction

FX

F(beh)

FZ

X

α

beh
Z

∼ =

system

final coalgebra

finality = ∃!(morphism for any F- coalgebra)
beh gives the behavior of the system
this yields final coalgebra semantics

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.22/30

SLIDE 73

Coinduction

FX

F(beh)

FZ

X

α

beh
Z

∼ =

system

final coalgebra

f.c.s. in Sets = bisimilarity
f.c.s. in a Kleisli category = trace semantics

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.22/30

SLIDE 74

When does it work?

Monad T s.t. Kℓ(T ) is DCpo⊥-enriched

left-strict composition

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SLIDE 75

When does it work?

Monad T s.t. Kℓ(T ) is DCpo⊥-enriched

left-strict composition

Functor F and a distributive law π: FT ⇒ T F:

lifting Kℓ(F) of F

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.23/30

SLIDE 76

When does it work?

Monad T s.t. Kℓ(T ) is DCpo⊥-enriched

left-strict composition

Functor F and a distributive law π: FT ⇒ T F:

lifting Kℓ(F) of F

Kℓ(F) is locally monotone

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.23/30

SLIDE 77

When does it work?

Monad T s.t. Kℓ(T ) is DCpo⊥-enriched

left-strict composition

Functor F and a distributive law π: FT ⇒ T F:

lifting Kℓ(F) of F

Kℓ(F) is locally monotone
F preserves ω-colimits

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.23/30

SLIDE 78

Main Theorem

If • • •• and a : FI

∼ =

→ I denotes the initial Sets-algebra, then Kℓ(F)I

ηI◦a ∼ =

Kℓ(F)I

I I

ηFI◦a−1 ∼ =

is initial

is final in Kℓ(T )

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.24/30

SLIDE 79

Main Theorem

If • • •• and a : FI

∼ =

→ I denotes the initial Sets-algebra, then Kℓ(F)I

ηI◦a ∼ =

Kℓ(F)I

I I

ηFI◦a−1 ∼ =

is initial

is final in Kℓ(T ) proof: via limit-colimit coincidence Smyth&Plotkin ’82

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.24/30

SLIDE 80

Corollary

Let • • •• and a : FI

∼ =

→ I denote the initial Sets-algebra. For α : X → Kℓ(F)X in Kℓ(T ) i.e.

α : X → T FX in Sets

∃! trace map tr(α) : X → T I such that in Kℓ(T ): Kℓ(F)X

K ℓ(F)(tr(α))

Kℓ(F)I

X

α

tr(α)
I

∼ =

GEOCAL

’06, Probabilistic Systems Workshop, AS – p.25/30

SLIDE 81

It works for...

lift, powerset, sub-distribution monad

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.26/30

SLIDE 82

It works for...

lift, powerset, sub-distribution monad
shapely functors - almost all polynomial

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SLIDE 83

It works for...

lift, powerset, sub-distribution monad
shapely functors - almost all polynomial
Hence:

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.26/30

SLIDE 84

It works for...

lift, powerset, sub-distribution monad
shapely functors - almost all polynomial
Hence:

* for LTS with explicit termination P(1 + A × )

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.26/30

SLIDE 85

It works for...

lift, powerset, sub-distribution monad
shapely functors - almost all polynomial
Hence:

* for LTS with explicit termination P(1 + A × ) * for generative systems with explicit termination D(1 + A × )

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.26/30

SLIDE 86

Finite traces - LTS with

the finality diagram in Kℓ(P) Kℓ(F)X

K ℓ(F)(tr(α))

Kℓ(F)A∗

X

α

tr(α)
A∗

∼ =

amounts to
∈ tr(α)(x)

⇐ ⇒ ∈ α(x)

a · w ∈ tr(α)(x)

⇐ ⇒ (∃x′)a, x′ ∈ α(x), w ∈ tr(α)(x′)

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.27/30

SLIDE 87

Finite traces - generative

the finality diagram in Kℓ(D) Kℓ(F)X

K ℓ(F)(tr(α))

Kℓ(F)A∗

X

α

tr(α)
A∗

∼ =

amounts to tr(α)(x) :
→ α(x)()
a · w →

y∈X α(x)(a, y) · tr(α)(y)(w)

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.28/30

SLIDE 88

Parallel composition

For u, v ∈ P(A∗) the (shuffle) parallel composition u v: ∈ u v

def

⇐ ⇒ ∈ u and ∈ v a · w ∈ u v

def

⇐ ⇒ w ∈ ∂au v

r

w ∈ u ∂av for ∂au = {w ∈ Σ∗ | a · w ∈ u} can be defined by coinduction

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.29/30

SLIDE 89

Conclusions & future work

Coalgebras allow for a unified treatment of (probabilistic)

transition systems

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.30/30

SLIDE 90

Conclusions & future work

Coalgebras allow for a unified treatment of (probabilistic)

transition systems

Coinduction gives us semantic relations:

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.30/30

SLIDE 91

Conclusions & future work

Coalgebras allow for a unified treatment of (probabilistic)

transition systems

Coinduction gives us semantic relations:

* bisimilarity for F-systems in Sets

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.30/30

SLIDE 92

Conclusions & future work

Coalgebras allow for a unified treatment of (probabilistic)

transition systems

Coinduction gives us semantic relations:

* bisimilarity for F-systems in Sets * trace semantics for T F-systems in Kℓ(T)

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.30/30

SLIDE 93

Conclusions & future work

Coalgebras allow for a unified treatment of (probabilistic)

transition systems

Coinduction gives us semantic relations:

* bisimilarity for F-systems in Sets * trace semantics for T F-systems in Kℓ(T)

Different monads e.g. PD - suitable monad/order structure yet to

be found (Varacca&Winskel)

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.30/30

SLIDE 94

Conclusions & future work

Coalgebras allow for a unified treatment of (probabilistic)

transition systems

Coinduction gives us semantic relations:

* bisimilarity for F-systems in Sets * trace semantics for T F-systems in Kℓ(T)

Different monads e.g. PD - suitable monad/order structure yet to

be found (Varacca&Winskel)

Other semantics that are between bisimilarity and trace in the

spectrum

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.30/30

SLIDE 95

Conclusions & future work

Coalgebras allow for a unified treatment of (probabilistic)

transition systems

Coinduction gives us semantic relations:

* bisimilarity for F-systems in Sets * trace semantics for T F-systems in Kℓ(T)

Different monads e.g. PD - suitable monad/order structure yet to

be found (Varacca&Winskel)

Other semantics that are between bisimilarity and trace in the

spectrum

Parallel composition of ”probabilistic languages"

GEOCAL ’06, Probabilistic Systems Workshop, AS – p.30/30