Generic Trace Theory Ichiro Hasuo, Bart Jacobs and Ana Sokolova SOS - - PowerPoint PPT Presentation

Generic Trace Theory

Ichiro Hasuo, Bart Jacobs and Ana Sokolova

SOS group - Radboud University Nijmegen

CMCS’06, Generic traces – p.1/22

Talk about...

• systems as coalgebras

states

• CMCS’06, Generic traces – p.2/22

Talk about...

• systems as coalgebras

states + transitions

• a
• b
• a
• c
• c
• CMCS’06, Generic traces – p.2/22

Talk about...

• systems as coalgebras

states + transitions S, α : S → FS, for F a functor

CMCS’06, Generic traces – p.2/22

Talk about...

• systems as coalgebras

states + transitions S, α : S → FS, for F a functor

• semantic relations represent behaviour

CMCS’06, Generic traces – p.2/22

Talk about...

• systems as coalgebras

states + transitions S, α : S → FS, for F a functor

• semantic relations represent behaviour

LT/BT spectrum

CMCS’06, Generic traces – p.2/22

Talk about...

• systems as coalgebras

states + transitions S, α : S → FS, for F a functor

• semantic relations represent behaviour

LT/BT spectrum ... linear-time behaviour via trace semantics

CMCS’06, Generic traces – p.2/22

LT/BT spectrum

Are these non-deterministic systems equal ?

• x

a

• y

a

• a
• b
• c
• b
• c
• CMCS’06, Generic traces – p.3/22

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

CMCS’06, Generic traces – p.3/22

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}

CMCS’06, Generic traces – p.3/22

Traces - LTS

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

CMCS’06, Generic traces – p.4/22

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∗

CMCS’06, Generic traces – p.4/22

Traces - generative

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

CMCS’06, Generic traces – p.5/22

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

· · ·

CMCS’06, Generic traces – p.5/22

Trace of a coalgebra ?

CMCS’06, Generic traces – p.6/22

Trace of a coalgebra ?

• Power&Turi ’99 - P(1 + Σ ×

)

• Jacobs ’04 - PF
• Hasuo&Jacobs CALCO ’05 - PF, shapely F
• Hasuo&Jacobs CALCO Jnr ’05 - DF, shapely F
• Generic Trace Theory - T F, order-enriched setting

CMCS’06, Generic traces – p.6/22

Trace of a coalgebra ?

• Power&Turi ’99 - P(1 + Σ ×

)

• Jacobs ’04 - PF
• Hasuo&Jacobs CALCO ’05 - PF, shapely F
• Hasuo&Jacobs CALCO Jnr ’05 - DF, shapely F
• Generic Trace Theory - T F, order-enriched setting

main idea: coinduction in a Kleisli category

CMCS’06, Generic traces – p.6/22

Coinduction

FX

F(beh)

• FZ

X

α

• beh
• Z

∼ =

• system

final coalgebra

CMCS’06, Generic traces – p.7/22

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

CMCS’06, Generic traces – p.7/22

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

CMCS’06, Generic traces – p.7/22

Types of systems

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

c

→ T F X

CMCS’06, Generic traces – p.8/22

Types of systems

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

c

→ T F X monad - branching type

CMCS’06, Generic traces – p.8/22

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

CMCS’06, Generic traces – p.8/22

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

CMCS’06, Generic traces – p.8/22

Distributive law

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

CMCS’06, Generic traces – p.9/22

Distributive law

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

• x

a

• a
• a
• b
• b
• CMCS’06, Generic traces – p.9/22

Distributive law

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

• x

a

• a
• a
• b
• b
• x

ab

• aa
• CMCS’06, Generic traces – p.9/22

Distributive law

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

• x

a

• a
• a
• b
• b
• x

ab

• aa
• X

c

→ PFX

CMCS’06, Generic traces – p.9/22

Distributive law

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

• x

a

• a
• a
• b
• b
• x

ab

• aa
• X

c

→ PFX

CMCS’06, Generic traces – p.9/22

Distributive law

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

• x

a

• a
• a
• b
• b
• x

ab

• aa
• X

c

→ PFX

PFc

→ PFPFX

CMCS’06, Generic traces – p.9/22

Distributive law

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

• x

a

• a
• a
• b
• b
• x

ab

• aa
• X

c

→ PFX

PFc

→ PFPFX

CMCS’06, Generic traces – p.9/22

Distributive law

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

• x

a

• a
• a
• b
• b
• x

ab

• aa
• X

c

→ PFX

PFc

→ PFPFX

d.l.

→ PPFFX

CMCS’06, Generic traces – p.9/22

Distributive law

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

• x

a

• a
• a
• b
• b
• x

ab

• aa
• X

c

→ PFX

PFc

→ PFPFX

d.l.

→ PPFFX

CMCS’06, Generic traces – p.9/22

Distributive law

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

• x

a

• a
• a
• b
• b
• x

ab

• aa
• X

c

→ PFX

PFc

→ PFPFX

d.l.

→ PPFFX

m.m.

→ PFFX

CMCS’06, Generic traces – p.9/22

Distributive law

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

• x

a

• a
• a
• b
• b
• x

ab

• aa
• X

c

→ PFX

PFc

→ PFPFX

d.l.

→ PPFFX

m.m.

→ PFFX

CMCS’06, Generic traces – p.9/22

Distributive law

is needed for X

c

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

CMCS’06, Generic traces – p.10/22

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

CMCS’06, Generic traces – p.10/22

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 ).

CMCS’06, Generic traces – p.10/22

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 ) !!!

CMCS’06, Generic traces – p.10/22

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

CMCS’06, Generic traces – p.10/22

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

CMCS’06, Generic traces – p.10/22

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

CMCS’06, Generic traces – p.10/22

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 → · · ·

CMCS’06, Generic traces – p.10/22

Main Theorem

If ♣, then FK

ℓ(T )A ηA◦α ∼ =

• FK

ℓ(T )A

A A

ηFA◦α−1 ∼ =

• is initial

is final in Kℓ(T )

CMCS’06, Generic traces – p.11/22

Main Theorem

If ♣, then FK

ℓ(T )A ηA◦α ∼ =

• FK

ℓ(T )A

A A

ηFA◦α−1 ∼ =

• is initial

is final in Kℓ(T ) [α : FA

∼ =

→ A denotes the initial F-algebra in Sets]

CMCS’06, Generic traces – p.11/22

Main Theorem

If ♣, then FK

ℓ(T )A ηA◦α ∼ =

• FK

ℓ(T )A

A A

ηFA◦α−1 ∼ =

• is initial

is final in Kℓ(T ) [α : FA

∼ =

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

CMCS’06, Generic traces – p.11/22

The assumptions ♣:

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

left-strict composition

CMCS’06, Generic traces – p.12/22

The assumptions ♣:

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

left-strict composition

• A functor F that preserves ω-colimits

CMCS’06, Generic traces – p.12/22

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 )

CMCS’06, Generic traces – p.12/22

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

CMCS’06, Generic traces – p.12/22

Proof sketch

In Sets

¡

F0

F ¡

· · ·Fn0

Fn ¡

Fn+10

Fn+1 ¡

· · ·

CMCS’06, Generic traces – p.13/22

Proof sketch

In Sets · · ·

Fn−1 ¡ Fn0 Fn ¡ Fn+10 Fn+1 ¡

· · ·

CMCS’06, Generic traces – p.13/22

Proof sketch

In Sets A · · ·

Fn−1 ¡ Fn0 αn

• Fn+10

αn+1

• · · ·

CMCS’06, Generic traces – p.13/22

Proof sketch

In Sets A · · ·

Fn−1 ¡ Fn0 αn

• Fαn−1
• Fn+10

αn+1

• Fαn
• · · ·

FA

CMCS’06, Generic traces – p.13/22

Proof sketch

In Sets A

α−1 ∼ =

• · · ·

Fn−1 ¡ Fn0 αn

• Fαn−1
• Fn+10

αn+1

• Fαn
• · · ·

FA

α

• CMCS’06, Generic traces – p.13/22

Proof sketch

In Kℓ(T) A

Jα−1 ∼ =

• · · ·

JFn−1 ¡

Fn0

Jαn

• JFαn−1
• Fn+10

Jαn+1

• JFαn
• · · ·

FA

Jα

• CMCS’06, Generic traces – p.13/22

Proof sketch

In Kℓ(T) A

Jα−1 ∼ =

• · · ·

¯ Fn−1 ¡

Fn0

Jαn

• ¯

FJαn−1

• Fn+10

Jαn+1

• ¯

FJαn

• · · ·

FA

Jα

• CMCS’06, Generic traces – p.13/22

Proof sketch

In Kℓ(T) A

(Jα−1)P ∼ =

• · · ·

( ¯ Fn−1 ¡ )P

• Fn0

(Jαn)P

• ( ¯

FJαn−1)P

• Fn+10

(Jαn+1)P

• ( ¯

FJαn)P

• · · ·

FA

(Jα)P

• CMCS’06, Generic traces – p.13/22

Proof sketch

In Kℓ(T) A

Jα ∼ =

• · · ·

¯ Fn−1 !

• Fn0

(Jαn)P

• ¯

F(Jαn−1)P

• Fn+10

(Jαn+1)P

• ¯

F(Jαn)P

• · · ·

FA

Jα−1

• CMCS’06, Generic traces – p.13/22

Corollary (♣)

For X

c

→ F K

ℓ(T )X in Kℓ(T )

CMCS’06, Generic traces – p.14/22

Corollary (♣)

For X

c

→ F K

ℓ(T )X in Kℓ(T )

... X

c

→ T FX in Sets

CMCS’06, Generic traces – p.14/22

Corollary (♣)

For X

c

→ F K

ℓ(T )X in Kℓ(T )

... X

c

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

CMCS’06, Generic traces – p.14/22

Corollary (♣)

For X

c

→ F K

ℓ(T )X in Kℓ(T )

... X

c

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

ℓ(T )X

F K

ℓ(T )(trc)

• FK

ℓ(T )A

X

c

• trc
• A

∼ =

• CMCS’06, Generic traces – p.14/22

It works for...

• branching types:

* lift monad 1 + systems with non-termination, exception * powerset monad P non-deterministic systems * subdistribution monad D probabilistic systems

CMCS’06, Generic traces – p.15/22

It works for...

• branching types:

* lift monad 1 + systems with non-termination, exception * powerset monad P non-deterministic systems * subdistribution monad D probabilistic systems DX = {µ : X → [0, 1] |

x∈X µ(x) ≤ 1}

CMCS’06, Generic traces – p.15/22

It works for...

• branching types:

* lift monad 1 + systems with non-termination, exception * powerset monad P non-deterministic systems * subdistribution monad D probabilistic systems all with pointwise order !

CMCS’06, Generic traces – p.15/22

together with...

• linear I/O types:

CMCS’06, Generic traces – p.16/22

together with...

• linear I/O types:

shapely functors

CMCS’06, Generic traces – p.16/22

together with...

• linear I/O types:

shapely functors F = id | Σ | F × F |

i Fi

CMCS’06, Generic traces – p.16/22

together with...

• linear I/O types:

shapely functors F = id | Σ | F × F |

i Fi

* modular distributive law between commutative monads and shapely functors * our monads are commutative

CMCS’06, Generic traces – p.16/22

Hence, it works...

• for LTS with explicit termination

P(1 + Σ × )

CMCS’06, Generic traces – p.17/22

Hence, it works...

• for LTS with explicit termination

P(1 + Σ × )

• for generative systems with explicit termination

D(1 + Σ × )

CMCS’06, Generic traces – p.17/22

Hence, it works...

• for LTS with explicit termination

P(1 + Σ × )

• for generative systems with explicit termination

D(1 + Σ × ) Note: Initial 1 + Σ ×

• algebra is

Σ∗

[nil,cons] ∼ =

1 + Σ × Σ∗

CMCS’06, Generic traces – p.17/22

Finite traces - LTS with

the finality diagram in Kℓ(P) FK

ℓ(P)X

F K

ℓ(P)(trc)

• FK

ℓ(P)Σ∗

X

c

• trc
• Σ∗

∼ =

• CMCS’06, Generic traces – p.18/22

Finite traces - LTS with

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

(1+Σ× )K

ℓ(P)(trc)

• 1 + Σ × Σ∗

X

c

• trc
• Σ∗

∼ =

• CMCS’06, Generic traces – p.18/22

Finite traces - LTS with

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

(1+Σ× )K

ℓ(P)(trc)

• 1 + Σ × Σ∗

X

c

• trc
• Σ∗

∼ =

• amounts to
• ∈ trc(x)

⇐ ⇒ ∈ c(x)

• a · w ∈ trc(x)

⇐ ⇒ (∃x′)a, x′ ∈ c(x), w ∈ trc(x′)

CMCS’06, Generic traces – p.18/22

Finite traces - generative

the finality diagram in Kℓ(D) FK

ℓ(D)X

F K

ℓ(D)(trc)

• FK

ℓ(D)Σ∗

X

c

• trc
• Σ∗

∼ =

• CMCS’06, Generic traces – p.19/22

Finite traces - generative

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

(1+Σ× )K

ℓ(D)(trc)

• 1 + Σ × Σ∗

X

c

• trc
• Σ∗

∼ =

• CMCS’06, Generic traces – p.19/22

Finite traces - generative

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

(1+Σ× )K

ℓ(D)(trc)

• 1 + Σ × Σ∗

X

c

• trc
• Σ∗

∼ =

• amounts to trc(x) :
• → c(x)()
• a · w →

y∈X c(x)(a, y) · c(y)(w)

CMCS’06, Generic traces – p.19/22

Parallel composition

For u, v ∈ P(Σ∗) 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

CMCS’06, Generic traces – p.20/22

Parallel composition

For u, v ∈ P(Σ∗) 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 Also: Equations u v = v u, (u v) w = u (v w), . . . can be proved by coinduction

CMCS’06, Generic traces – p.20/22

Conclusions

• Systems as coalgebras
• Behaviour via coinduction

CMCS’06, Generic traces – p.21/22

Conclusions

• Systems as coalgebras
• Behaviour via coinduction

* generic trace semantics: coinduction in Kℓ(T ) FK

ℓ(T )X

F K

ℓ(T )(trc)

• FK

ℓ(T )A

X

c

• trc
• A

∼ =

• CMCS’06, Generic traces – p.21/22

Conclusions

• Systems as coalgebras
• Behaviour via coinduction

* generic trace semantics: coinduction in Kℓ(T ) FK

ℓ(T )X

F K

ℓ(T )(trc)

• FK

ℓ(T )A

X

c

• trc
• A

∼ =

• Main technical result: initial algebra = final coalgebra

in an order enriched setting

CMCS’06, Generic traces – p.21/22

Future work

• Combined monads:

CMCS’06, Generic traces – p.22/22

Future work

• Combined monads:

* non-determinism + probability [Vardi ’85, Segala & Lynch ’95] monad/order structure yet to be found [Varacca & Winskel ’05]

CMCS’06, Generic traces – p.22/22

Future work

• Combined monads:

* non-determinism + probability [Vardi ’85, Segala & Lynch ’95] monad/order structure yet to be found [Varacca & Winskel ’05] * PP [Kupke & Venema ’05]

CMCS’06, Generic traces – p.22/22

Future work

• Combined monads:

* non-determinism + probability [Vardi ’85, Segala & Lynch ’95] monad/order structure yet to be found [Varacca & Winskel ’05] * PP [Kupke & Venema ’05]

• Between bisimilarity and trace in the spectrum

CMCS’06, Generic traces – p.22/22

Future work

• Combined monads:

* non-determinism + probability [Vardi ’85, Segala & Lynch ’95] monad/order structure yet to be found [Varacca & Winskel ’05] * PP [Kupke & Venema ’05]

• Between bisimilarity and trace in the spectrum
• of probabilistic languages

CMCS’06, Generic traces – p.22/22