[PPT] - Probabilistic systems a place where categories meet probability Ana PowerPoint Presentation

SLIDE 1

Probabilistic systems

a place where categories meet probability

Ana Sokolova

SOS group, Radboud University Nijmegen

University Dortmund, CS Kolloquium, 12.6.6 – p.1/32

SLIDE 2

Outline

Introduction - probabilistic systems and coalgebras

University Dortmund, CS Kolloquium, 12.6.6 – p.2/32

SLIDE 3

Outline

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

University Dortmund, CS Kolloquium, 12.6.6 – p.2/32

SLIDE 4

Outline

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

(older result)

University Dortmund, CS Kolloquium, 12.6.6 – p.2/32

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

(newer result)

University Dortmund, CS Kolloquium, 12.6.6 – p.2/32

SLIDE 6

Systems

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

University Dortmund, CS Kolloquium, 12.6.6 – p.3/32

SLIDE 7

Systems

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

University Dortmund, CS Kolloquium, 12.6.6 – p.3/32

SLIDE 8

Systems

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

a
b
a
c
c
University Dortmund, CS Kolloquium, 12.6.6 – p.3/32

SLIDE 9

Probabilistic systems

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

University Dortmund, CS Kolloquium, 12.6.6 – p.4/32

SLIDE 10

Probabilistic systems

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

1

3 2 3

⇒
1
University Dortmund, CS Kolloquium, 12.6.6 – p.4/32

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]
University Dortmund, CS Kolloquium, 12.6.6 – p.4/32

SLIDE 12

Probabilistic systems

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

♦

1 3 2 3

⇒
a
University Dortmund, CS Kolloquium, 12.6.6 – p.4/32

SLIDE 13

Probabilistic systems

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

a
b
⇒

♦

1

University Dortmund, CS Kolloquium, 12.6.6 – p.4/32

SLIDE 14

Probabilistic systems

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

⇒
1
a
b
1
a
University Dortmund, CS Kolloquium, 12.6.6 – p.4/32

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
University Dortmund, CS Kolloquium, 12.6.6 – p.4/32

SLIDE 16

Coalgebras

are an elegant generalization of transition systems with states + transitions

University Dortmund, CS Kolloquium, 12.6.6 – p.5/32

SLIDE 17

Coalgebras

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

University Dortmund, CS Kolloquium, 12.6.6 – p.5/32

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

University Dortmund, CS Kolloquium, 12.6.6 – p.5/32

SLIDE 19

Examples

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

University Dortmund, CS Kolloquium, 12.6.6 – p.6/32

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 × )

University Dortmund, CS Kolloquium, 12.6.6 – p.6/32

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

University Dortmund, CS Kolloquium, 12.6.6 – p.7/32

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

University Dortmund, CS Kolloquium, 12.6.6 – p.7/32

SLIDE 23

reactive, generative

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

A

University Dortmund, CS Kolloquium, 12.6.6 – p.8/32

SLIDE 24

reactive, generative

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

A

reactive systems: functor (D + 1)A

University Dortmund, CS Kolloquium, 12.6.6 – p.8/32

SLIDE 25

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

University Dortmund, CS Kolloquium, 12.6.6 – p.8/32

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

University Dortmund, CS Kolloquium, 12.6.6 – p.8/32

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 × ) . . . . . .

University Dortmund, CS Kolloquium, 12.6.6 – p.9/32

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]
University Dortmund, CS Kolloquium, 12.6.6 – p.9/32

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]
University Dortmund, CS Kolloquium, 12.6.6 – p.9/32

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

University Dortmund, CS Kolloquium, 12.6.6 – p.9/32

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

University Dortmund, CS Kolloquium, 12.6.6 – p.9/32

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 ]

University Dortmund, CS Kolloquium, 12.6.6 – p.9/32

SLIDE 33

Bisimulation - LTS

Consider the LTS

s0
a
b
t0

b

a
t2
a
b
s1
t1
t3

University Dortmund, CS Kolloquium, 12.6.6 – p.10/32

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...

University Dortmund, CS Kolloquium, 12.6.6 – p.10/32

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

University Dortmund, CS Kolloquium, 12.6.6 – p.10/32

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

University Dortmund, CS Kolloquium, 12.6.6 – p.11/32

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...

University Dortmund, CS Kolloquium, 12.6.6 – p.11/32

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′

University Dortmund, CS Kolloquium, 12.6.6 – p.11/32

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

University Dortmund, CS Kolloquium, 12.6.6 – p.12/32

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...

University Dortmund, CS Kolloquium, 12.6.6 – p.12/32

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′

University Dortmund, CS Kolloquium, 12.6.6 – p.12/32

SLIDE 42

Coalgebraic bisimulation

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

University Dortmund, CS Kolloquium, 12.6.6 – p.13/32

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

University Dortmund, CS Kolloquium, 12.6.6 – p.13/32

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)

University Dortmund, CS Kolloquium, 12.6.6 – p.13/32

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 !

University Dortmund, CS Kolloquium, 12.6.6 – p.13/32

SLIDE 46

Expressiveness

simple Segala system → Segala system

a
p1

p2 pn

...
University Dortmund, CS Kolloquium, 12.6.6 – p.14/32

SLIDE 47

Expressiveness

simple Segala system → Segala system

a
p1

p2 pn

...
a[p1]

a[p2] a[pn]

...
University Dortmund, CS Kolloquium, 12.6.6 – p.14/32

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?

University Dortmund, CS Kolloquium, 12.6.6 – p.14/32

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

University Dortmund, CS Kolloquium, 12.6.6 – p.15/32

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

University Dortmund, CS Kolloquium, 12.6.6 – p.15/32

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

University Dortmund, CS Kolloquium, 12.6.6 – p.15/32

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

University Dortmund, CS Kolloquium, 12.6.6 – p.15/32

SLIDE 53

Example

Indeed SSeg → Seg since P(A × D)

Pτ

⇒ PD(A × ) is injective for

University Dortmund, CS Kolloquium, 12.6.6 – p.16/32

SLIDE 54

Example

Indeed SSeg → Seg since P(A × D)

Pτ

⇒ PD(A × ) is injective for (A × D)

τ

⇒ D(A × ) given by

University Dortmund, CS Kolloquium, 12.6.6 – p.16/32

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

University Dortmund, CS Kolloquium, 12.6.6 – p.16/32

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′)

University Dortmund, CS Kolloquium, 12.6.6 – p.16/32

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

University Dortmund, CS Kolloquium, 12.6.6 – p.16/32

SLIDE 58

The hierarchy...

MG PZ

Seg
Bun
SSeg
Var
Alt
React
LTS
Gen
Str
DLTS
MC
University Dortmund, CS Kolloquium, 12.6.6 – p.17/32

SLIDE 59

The hierarchy...

MG PZ

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

University Dortmund, CS Kolloquium, 12.6.6 – p.17/32

SLIDE 60

LT/BT spectrum

Bisimilarity is not the only semantics...

University Dortmund, CS Kolloquium, 12.6.6 – p.18/32

SLIDE 61

LT/BT spectrum

Are these non-deterministic systems equal ?

x

a

y

a

a
b
c
b
c
University Dortmund, CS Kolloquium, 12.6.6 – p.18/32

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

University Dortmund, CS Kolloquium, 12.6.6 – p.18/32

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}

University Dortmund, CS Kolloquium, 12.6.6 – p.18/32

SLIDE 64

Traces - LTS

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

University Dortmund, CS Kolloquium, 12.6.6 – p.19/32

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∗

University Dortmund, CS Kolloquium, 12.6.6 – p.19/32

SLIDE 66

Traces - generative

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

University Dortmund, CS Kolloquium, 12.6.6 – p.20/32

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

· · ·

University Dortmund, CS Kolloquium, 12.6.6 – p.20/32

SLIDE 68

Trace of a coalgebra ?

University Dortmund, CS Kolloquium, 12.6.6 – p.21/32

SLIDE 69

Trace of a coalgebra ?

Power&Turi ’99
Jacobs ’04
Hasuo& Jacobs ’05
Ichiro Hasuo, Bart Jacobs, AS:

Generic Trace Theory, CMCS’06

University Dortmund, CS Kolloquium, 12.6.6 – p.21/32

SLIDE 70

Trace of a coalgebra ?

Power&Turi ’99
Jacobs ’04
Hasuo& Jacobs ’05
Ichiro Hasuo, Bart Jacobs, AS:

Generic Trace Theory, CMCS’06 main idea: coinduction in a Kleisli category

University Dortmund, CS Kolloquium, 12.6.6 – p.21/32

SLIDE 71

Coinduction

FX

F(beh)

FZ

X

α

beh
Z

∼ =

system

final coalgebra

University Dortmund, CS Kolloquium, 12.6.6 – p.22/32

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

University Dortmund, CS Kolloquium, 12.6.6 – p.22/32

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

University Dortmund, CS Kolloquium, 12.6.6 – p.22/32

SLIDE 74

Types of systems

For trace semantics systems are suitably modelled as coalgebras in Sets X

c

→ T F X

University Dortmund, CS Kolloquium, 12.6.6 – p.23/32

SLIDE 75

Types of systems

For trace semantics systems are suitably modelled as coalgebras in Sets X

c

→ T F X monad - branching type

University Dortmund, CS Kolloquium, 12.6.6 – p.23/32

SLIDE 76

Types of systems

For trace semantics systems are suitably modelled as coalgebras in Sets X

c

→ T F X monad - branching type functor - linear i/o type

University Dortmund, CS Kolloquium, 12.6.6 – p.23/32

SLIDE 77

Types of systems

For trace semantics systems are suitably modelled as coalgebras in Sets X

c

→ T F X monad - branching type functor - linear i/o type needed: distributive law FT ⇒ T F

University Dortmund, CS Kolloquium, 12.6.6 – p.23/32

SLIDE 78

Distributive law

is needed since branching is irrelevant: LTS with - PF = P(1 + A × )

University Dortmund, CS Kolloquium, 12.6.6 – p.24/32

SLIDE 79

Distributive law

is needed since branching is irrelevant: LTS with - PF = P(1 + A × )

x

a

a
a
b
b
University Dortmund, CS Kolloquium, 12.6.6 – p.24/32

SLIDE 80

Distributive law

is needed since branching is irrelevant: LTS with - PF = P(1 + A × )

x

a

a
a
b
b
x

ab

aa
University Dortmund, CS Kolloquium, 12.6.6 – p.24/32

SLIDE 81

Distributive law

is needed since branching is irrelevant: LTS with - PF = P(1 + A × )

x

a

a
a
b
b
x

ab

aa
X

c

→ PFX

University Dortmund, CS Kolloquium, 12.6.6 – p.24/32

SLIDE 82

Distributive law

is needed since branching is irrelevant: LTS with - PF = P(1 + A × )

x

a

a
a
b
b
x

ab

aa
X

c

→ PFX

University Dortmund, CS Kolloquium, 12.6.6 – p.24/32

SLIDE 83

Distributive law

is needed since branching is irrelevant: LTS with - PF = P(1 + A × )

x

a

a
a
b
b
x

ab

aa
X

c

→ PFX

PFc

→ PFPFX

University Dortmund, CS Kolloquium, 12.6.6 – p.24/32

SLIDE 84

Distributive law

is needed since branching is irrelevant: LTS with - PF = P(1 + A × )

x

a

a
a
b
b
x

ab

aa
X

c

→ PFX

PFc

→ PFPFX

University Dortmund, CS Kolloquium, 12.6.6 – p.24/32

SLIDE 85

Distributive law

is needed since branching is irrelevant: LTS with - PF = P(1 + A × )

x

a

a
a
b
b
x

ab

aa
X

c

→ PFX

PFc

→ PFPFX

d.l.

→ PPFFX

University Dortmund, CS Kolloquium, 12.6.6 – p.24/32

SLIDE 86

Distributive law

is needed since branching is irrelevant: LTS with - PF = P(1 + A × )

x

a

a
a
b
b
x

ab

aa
X

c

→ PFX

PFc

→ PFPFX

d.l.

→ PPFFX

University Dortmund, CS Kolloquium, 12.6.6 – p.24/32

SLIDE 87

Distributive law

is needed since branching is irrelevant: LTS with - PF = P(1 + A × )

x

a

a
a
b
b
x

ab

aa
X

c

→ PFX

PFc

→ PFPFX

d.l.

→ PPFFX

m.m.

→ PFFX

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SLIDE 88

Distributive law

is needed since branching is irrelevant: LTS with - PF = P(1 + A × )

x

a

a
a
b
b
x

ab

aa
X

c

→ PFX

PFc

→ PFPFX

d.l.

→ PPFFX

m.m.

→ PFFX

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SLIDE 89

Distributive law

is needed for X

c

→ T FX to be a coalgebra in the Kleisli category Kℓ(T )..

University Dortmund, CS Kolloquium, 12.6.6 – p.25/32

SLIDE 90

Distributive law

is needed for X

c

→ T FX to be a coalgebra in the Kleisli category Kℓ(T )..

objects - sets
arrows - X

f

→ Y are functions f : X → T Y

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SLIDE 91

Distributive law

is needed for X

c

→ T FX to be a coalgebra in the Kleisli category Kℓ(T ).. FT ⇒ T F : F lifts to F K

ℓ(T ) on Kℓ(T ).

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SLIDE 92

Distributive law

is needed for X

c

→ T FX to be a coalgebra in the Kleisli category Kℓ(T ).. FT ⇒ T F : F lifts to F K

ℓ(T ) on Kℓ(T ).

Hence: coalgebra X

c

→ FK

ℓ(T )X in Kℓ(T ) !!!

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SLIDE 93

Distributive law

is needed for X

c

→ T FX to be a coalgebra in the Kleisli category Kℓ(T ).. FT ⇒ T F : F lifts to F K

ℓ(T ) on Kℓ(T ).

Hence: coalgebra X

c

→ FK

ℓ(T )X in Kℓ(T ) !!!

in Kℓ(T ) : X

c

→ F K

ℓ(T )X

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SLIDE 94

Distributive law

is needed for X

c

→ T FX to be a coalgebra in the Kleisli category Kℓ(T ).. FT ⇒ T F : F lifts to F K

ℓ(T ) on Kℓ(T ).

Hence: coalgebra X

c

→ FK

ℓ(T )X in Kℓ(T ) !!!

in Kℓ(T ) : X

c

→ F K

ℓ(T )X

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SLIDE 95

Distributive law

is needed for X

c

→ T FX to be a coalgebra in the Kleisli category Kℓ(T ).. FT ⇒ T F : F lifts to F K

ℓ(T ) on Kℓ(T ).

Hence: coalgebra X

c

→ FK

ℓ(T )X in Kℓ(T ) !!!

in Kℓ(T ) : X

c

→ F K

ℓ(T )X FK

ℓ(T )c

→ F K

ℓ(T )FK ℓ(T )X

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SLIDE 96

Distributive law

is needed for X

c

→ T FX to be a coalgebra in the Kleisli category Kℓ(T ).. FT ⇒ T F : F lifts to F K

ℓ(T ) on Kℓ(T ).

Hence: coalgebra X

c

→ FK

ℓ(T )X in Kℓ(T ) !!!

in Kℓ(T ) : X

c

→ F K

ℓ(T )X FK

ℓ(T )c

→ FK

ℓ(T )FK ℓ(T )X → · · ·

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SLIDE 97

Main theorem - traces

If ♣, then FK

ℓ(T )I ηI◦α ∼ =

FK

ℓ(T )I

I I

ηFI◦α−1 ∼ =

is initial

is final in Kℓ(T )

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SLIDE 98

Main theorem - traces

If ♣, then FK

ℓ(T )I ηI◦α ∼ =

FK

ℓ(T )I

I I

ηFI◦α−1 ∼ =

is initial

is final in Kℓ(T ) [α : FI

∼ =

→ I denotes the initial F-algebra in Sets]

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SLIDE 99

Main theorem - traces

If ♣, then FK

ℓ(T )I ηI◦α ∼ =

FK

ℓ(T )I

I I

ηFI◦α−1 ∼ =

is initial

is final in Kℓ(T ) [α : FI

∼ =

→ I denotes the initial F-algebra in Sets] proof: via limit-colimit coincidence Smyth&Plotkin ’82

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SLIDE 100

The assumptions ♣:

A monad T s.t. Kℓ(T ) is DCpo⊥-enriched

left-strict composition

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SLIDE 101

The assumptions ♣:

A monad T s.t. Kℓ(T ) is DCpo⊥-enriched

left-strict composition

A functor F that preserves ω-colimits

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SLIDE 102

The assumptions ♣:

A monad T s.t. Kℓ(T ) is DCpo⊥-enriched

left-strict composition

A functor F that preserves ω-colimits
A distributive law FT ⇒ T F:

lifting FK

ℓ(T )

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SLIDE 103

The assumptions ♣:

A monad T s.t. Kℓ(T ) is DCpo⊥-enriched

left-strict composition

A functor F that preserves ω-colimits
A distributive law FT ⇒ T F:

lifting FK

ℓ(T )

FK

ℓ(T ) should be locally monotone

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SLIDE 104

Corollary (♣)

For X

c

→ F K

ℓ(T )X in Kℓ(T )

University Dortmund, CS Kolloquium, 12.6.6 – p.28/32

SLIDE 105

Corollary (♣)

For X

c

→ F K

ℓ(T )X in Kℓ(T )

... X

c

→ T FX in Sets

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SLIDE 106

Corollary (♣)

For X

c

→ F K

ℓ(T )X in Kℓ(T )

... X

c

→ T FX in Sets ∃! finite trace map trc : X → T I in Sets:

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SLIDE 107

Corollary (♣)

For X

c

→ F K

ℓ(T )X in Kℓ(T )

... X

c

→ T FX in Sets ∃! finite trace map trc : X → T I in Sets: in Kℓ(T ) FK

ℓ(T )X F K

ℓ(T )(trc)

FK

ℓ(T )I

X

c

trc
I

∼ =

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SLIDE 108

It works for...

lift, powerset, sub-distribution monad

University Dortmund, CS Kolloquium, 12.6.6 – p.29/32

SLIDE 109

It works for...

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

University Dortmund, CS Kolloquium, 12.6.6 – p.29/32

SLIDE 110

It works for...

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

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SLIDE 111

It works for...

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

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

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SLIDE 112

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 × )

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SLIDE 113

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 × ) Note: Initial 1 + A ×

algebra is

A∗

[nil,cons] ∼ =

1 + A × A∗

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SLIDE 114

Finite traces - LTS with

the finality diagram in Kℓ(P) FK

ℓ(P)X

F K

ℓ(P)(trc)

FK

ℓ(P)A∗

X

c

trc
A∗

∼ =

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SLIDE 115

Finite traces - LTS with

the finality diagram in Kℓ(P) 1 + A × X

(1+A× )K

ℓ(P)(trc)

1 + A × A∗

X

c

trc
A∗

∼ =

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SLIDE 116

Finite traces - LTS with

the finality diagram in Kℓ(P) 1 + A × X

(1+A× )K

ℓ(P)(trc)

1 + A × A∗

X

c

trc
A∗

∼ =

amounts to
∈ trc(x)

⇐ ⇒ ∈ c(x)

a · w ∈ trc(x)

⇐ ⇒ (∃x′)a, x′ ∈ c(x), w ∈ trc(x′)

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SLIDE 117

Finite traces - generative

the finality diagram in Kℓ(D) FK

ℓ(D)X

F K

ℓ(D)(trc)

FK

ℓ(D)A∗

X

c

trc
A∗

∼ =

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SLIDE 118

Finite traces - generative

the finality diagram in Kℓ(D) 1 + A × X

(1+A× )K

ℓ(D)(trc)

1 + A × A∗

X

c

trc
A∗

∼ =

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SLIDE 119

Finite traces - generative

the finality diagram in Kℓ(D) 1 + A × X

(1+A× )K

ℓ(D)(trc)

1 + A × A∗

X

c

trc
A∗

∼ =

amounts to trc(x) :
→ c(x)()
a · w →

y∈X c(x)(a, y) · c(y)(w)

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SLIDE 120

Conclusions & future work

Coalgebras allow for a unified treatment and expressiveness

study of (P)TS

University Dortmund, CS Kolloquium, 12.6.6 – p.32/32

SLIDE 121

Conclusions & future work

Coalgebras allow for a unified treatment and expressiveness

study of (P)TS

Coinduction gives us semantic relations:

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SLIDE 122

Conclusions & future work

Coalgebras allow for a unified treatment and expressiveness

study of (P)TS

Coinduction gives us semantic relations:

* bisimilarity for F-systems in Sets

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SLIDE 123

Conclusions & future work

Coalgebras allow for a unified treatment and expressiveness

study of (P)TS

Coinduction gives us semantic relations:

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

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SLIDE 124

Conclusions & future work

Coalgebras allow for a unified treatment and expressiveness

study of (P)TS

Coinduction gives us semantic relations:

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

Other monads e.g. PD ? ... suitable monad/order structure yet to

be found (Varacca&Winskel)

University Dortmund, CS Kolloquium, 12.6.6 – p.32/32

SLIDE 125

Conclusions & future work

Coalgebras allow for a unified treatment and expressiveness

study of (P)TS

Coinduction gives us semantic relations:

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

Other monads e.g. PD ? ... suitable monad/order structure yet to

be found (Varacca&Winskel)

Other semantics ? .. between bisimilarity and trace in the

spectrum

University Dortmund, CS Kolloquium, 12.6.6 – p.32/32

SLIDE 126

Conclusions & future work

Coalgebras allow for a unified treatment and expressiveness

study of (P)TS

Coinduction gives us semantic relations:

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

Other monads e.g. PD ? ... suitable monad/order structure yet to

be found (Varacca&Winskel)

Other semantics ? .. between bisimilarity and trace in the

spectrum

University Dortmund, CS Kolloquium, 12.6.6 – p.32/32