SOS group, Radboud University Nijmegencs.uni-salzburg.at/~anas/papers/Dortmund-Presentation.pdf · 2020-06-04 · Probabilistic systems a place where categories meet probability Ana

Probabilistic systemsa place where categories meet probability

Ana Sokolova

SOS group, Radboud University Nijmegen

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

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)

















Systemsare formal objects, transition systems (e.g. LTS), that serveas models of real (software, hardware,...) systems



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



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

Coalgebrasare an elegant generalization of transition systems with

states + transitions

• based on category theory

• provide a uniform way of treating transition systems

• provide general notions and results e.g. a generic notion ofbisimulation



states + transitionsas pairs

〈S, α : S → FS〉, for F a functor






states + transitionsas pairs

〈S, α : S → FS〉, for F a functor





Examples

A TS is a pair 〈S, α : S → PS〉

!! coalgebra of the powerset functor P


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× )


More examplesThanks 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


More examplesThanks 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 functorsbuilt by the following syntax

F ::= | A | P | D | G +H | G ×H | GA | G ◦H


reactive, generative

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

reactive systems:functor (D + 1)A

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

note: in the probabilistic case

(D + 1)A 6∼= D(A× ) + 1







(D + 1)A 6∼= D(A× ) + 1







(D + 1)A 6∼= D(A× ) + 1







(D + 1)A 6∼= D(A× ) + 1


Probabilistic system typesMC 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


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× )

. . . . . .

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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× )

. . . . . .

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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× )

. . . . . .

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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× )

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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× )

. . . . . .

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Bisimulation - LTSConsider the LTS

•s0EDGFa

@A//

b²²

•t0

b²²

a // •t2GFED

a

BCoo

b²²

•s1 •t1 •t3



•s0EDGFa

@A//

b²²

•t0

b²²

a // •t2GFED

a

BCoo

b²²

•s1 •t1 •t3

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



•s0EDGFa

@A//

b²²

•t0

b²²

a // •t2GFED

a

BCoo

b²²

•s1 •t1 •t3

Transfer condition: 〈s, t〉 ∈ R =⇒

sa→ s′ ⇒ (∃t′) t

a→ t′, 〈s′, t′〉 ∈ R,

ta→ t′ ⇒ (∃s′) s

a→ s′, 〈s′, t′〉 ∈ R


Bisimulation - generativeConsider the generative systems

•s0EDGF

a[ 12]

@A//

b[ 12]

²²

•t0

b[ 12]

²²

a[ 12]

''•t2GFEDa[ 1

6]

BCoo

a[ 13]

gg

b[ 12]

²²•s1 •t1 •t3



•s0EDGF

a[ 12]

@A//

b[ 12]

²²

•t0

b[ 12]

²²

a[ 12]

''•t2GFEDa[ 1

6]

BCoo

a[ 13]

gg

b[ 12]

²²•s1 •t1 •t3

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



•s0EDGF

a[ 12]

@A//

b[ 12]

²²

•t0

b[ 12]

²²

a[ 12]

''•t2GFEDa[ 1

6]

BCoo

a[ 13]

gg

b[ 12]

²²•s1 •t1 •t3


s;µ⇒ (∃µ′) t;µ′, µ ≡R,A µ′


Bisimulation - simple SegalaConsider the simple Segala systems

•s0EDGFa

@A//

b²²

•t0

b²²

a //

13

|| _Âe%j*o/t4y9 23 ///o/o/o •t2

GFEDa

BCoo

b²²

•s1 •t1 •t3



•s0EDGFa

@A//

b²²

•t0

b²²

a //

13

|| _Âe%j*o/t4y9 23 ///o/o/o •t2

GFEDa

BCoo

b²²

•s1 •t1 •t3

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



•s0EDGFa

@A//

b²²

•t0

b²²

a //

13

|| _Âe%j*o/t4y9 23 ///o/o/o •t2

GFEDa

BCoo

b²²

•s1 •t1 •t3


sa→ µ⇒ (∃µ′) t

a→ µ′, µ ≡R µ′


Coalgebraic bisimulationA bisimulation between

〈S, α : S → FS〉 and 〈T , β : S → FS〉

is R ⊆ S × T such that ∃ γ:



〈S, α : S → FS〉 and 〈T , β : S → FS〉


S

α

²²

R

γ

²²

π1oo π2 // T

β

²²FS FR

Fπ1

ooFπ2

// FT



〈S, α : S → FS〉 and 〈T , β : S → FS〉


S

α

²²

R

γ

²²

π1oo π2 // T

β

²²FS FR

Fπ1

ooFπ2

// FT


〈α(s), β(t)〉 ∈ Rel(F)(R)



〈S, α : S → FS〉 and 〈T , β : S → FS〉


S

α

²²

R

γ

²²

π1oo π2 // T

β

²²FS FR

Fπ1

ooFπ2

// FT

Theorem: Coalgebraic and concrete bisimilarity coincide !


Expressiveness

simple Segala system → Segala system

•a

²²p1

ww w7 w7 w7 w7 w7 w7 w7

p2ÄÄ Ä?Ä?

Ä? pn

(((h(h(h(h(h(h(h(h

• • ... •


Expressiveness


•a

²²p1

ww w7 w7 w7 w7 w7 w7 w7

p2ÄÄ Ä?Ä?

Ä? pn

(((h(h(h(h(h(h(h(h

• • ... •

•

²²a[p1]

ww w7 w7 w7 w7 w7 w7 w7

a[p2]ÄÄ Ä?Ä?

Ä? a[pn]

(((h(h(h(h(h(h(h(h

• • ... •


Expressiveness


•a

²²p1

ww w7 w7 w7 w7 w7 w7 w7

p2ÄÄ Ä?Ä?

Ä? pn

(((h(h(h(h(h(h(h(h

• • ... •

•

²²a[p1]

ww w7 w7 w7 w7 w7 w7 w7

a[p2]ÄÄ Ä?Ä?

Ä? a[pn]

(((h(h(h(h(h(h(h(h

• • ... •

When do we consider one type of systems moreexpressive than another?


Comparison criterion

CoalgF → CoalgG

if there is a mapping 〈S, α : S → FS〉T7→ 〈S, α̃ : S → GS〉

that preserves and reflects bisimilarity



CoalgF → CoalgG



s〈S,α〉 ∼ t〈T,β〉 ⇐⇒ sT 〈S,α〉 ∼ tT 〈T,β〉



CoalgF → CoalgG



Theorem: An injective natural transformation F ⇒ G issufficient for CoalgF → CoalgG



CoalgF → CoalgG



Theorem: An injective natural transformation F ⇒ G issufficient for CoalgF → CoalgG

proof via cocongruences - behavioral equivalence


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


Example


injective for

(A×D)τ⇒D(A× )

given by

τX(〈a, µ〉) = δa × µ, where

µ× µ′(〈x, x′〉) = µ(x) · µ′(x′)



Example


injective for

(A×D)τ⇒D(A× )


µ× µ′(〈x, x′〉) = µ(x) · µ′(x′)



Example


injective for

(A×D)τ⇒D(A× )


µ× µ′(〈x, x′〉) = µ(x) · µ′(x′)



Example


injective for

(A×D)τ⇒D(A× )


µ× µ′(〈x, x′〉) = µ(x) · µ′(x′)



The hierarchy...

MG

PZ

::ttttt

Seg

::uuuuuBun

ddIIIII

SSeg

99sssssVar

ccHHHHH::vvvvv

Alt

[[6666666666666666

React

88qqqqqqLTS

eeLLLLL::uuuuu

Gen

ddIIIIIStr

OO

DLTS

ggNNNNNN88rrrrr

MC

OO


The hierarchy...

MG

PZ

::ttttt

Seg

::uuuuuBun

ddIIIII

SSeg

99sssssVar

ccHHHHH::vvvvv

Alt

[[6666666666666666

React

88qqqqqqLTS

eeLLLLL::uuuuu

Gen

ddIIIIIStr

OO

DLTS

ggNNNNNN88rrrrr

MC

OO

* Falk Bartels, AS, Erik de VinkUniversity Dortmund, CS Kolloquium, 12.6.6 – p.17/32

LT/BT spectrumBisimilarity is not the only semantics...


LT/BT spectrumAre these non-deterministic systems equal ?

•xa

²²

•ya

¢¢¥¥¥¥ a

ÀÀ:::

:

•b

¡¡££££ c

ÁÁ<<<

< •b ²²

•c

²²• • • •



•xa

²²

•ya

¢¢¥¥¥¥ a

ÀÀ:::

:

•b

¡¡££££ c

ÁÁ<<<

< •b ²²

•c

²²• • • •

x and y are:

• different wrt. bisimilarity



•xa

²²

•ya

¢¢¥¥¥¥ a

ÀÀ:::

:

•b

¡¡££££ c

ÁÁ<<<

< •b ²²

•c

²²• • • •

x and y are:

• different wrt. bisimilarity, but

• equivalent wrt. trace semanticstr(x) = tr(y) = {ab, ac}


Traces - LTSFor LTS with explicit termination (NA)

trace = the set of all possiblelinear behaviors


Traces - LTSFor LTS with explicit termination (NA)

trace = the set of all possiblelinear behaviors

Example:

•xEDGFa

@A// a // •y

GFEDb

BCoo

²²X

tr(y) = b∗, tr(x) = a+ · tr(y) = a+ · b∗


Traces - generativeFor generative probabilistic systems with ex. termination

trace =sub-probability distribution overpossible linear behaviors


Traces - generativeFor generative probabilistic systems with ex. termination

trace =sub-probability distribution overpossible linear behaviors

Example:

•x

b[ 13]

²²

a[ 13]

//13

&&LLLLL

LLLLLL

•yGFEDa[ 1

2]

BCoo

12

²²•zBC@A

a[1]

GF // X

tr(x) : 〈〉 7→ 13

a 7→ 13· 1

2

a2 7→ 13· 1

2· 1

2

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

Trace of a coalgebra ?

• Power&Turi ’99

• Jacobs ’04

• Hasuo& Jacobs ’05

• Ichiro Hasuo, Bart Jacobs, AS:Generic Trace Theory, CMCS’06




• Jacobs ’04






• Jacobs ’04



main idea: coinduction in a Kleisli category


Coinduction

FXF(beh) //_______ FZ

X

α

OO

beh//_______ Z

∼=

OO

system final coalgebra


Coinduction

FXF(beh) //_______ FZ

X

α

OO

beh//_______ Z

∼=

OO


• finality = ∃!(morphism for any F - coalgebra)

• beh gives the behavior of the system

• this yields final coalgebra semantics


Coinduction

FXF(beh) //_______ FZ

X

α

OO

beh//_______ Z

∼=

OO


• f.c.s. in Sets = bisimilarity

• f.c.s. in a Kleisli category = trace semantics


Types of systemsFor trace semantics systems are suitably modelled ascoalgebras in Sets

Xc→ T F X

monad - branching type

functor - linear i/o type

needed: distributive law FT ⇒ T F



Xc→ T F X






Xc→ T F X






Xc→ T F X





Distributive lawis needed since branching is irrelevant:

LTS with X- PF = P(1 + A× )




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Xc→ PFX

PFc→ PFPFX




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ab

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Xc→ PFX

PFc→ PFPFX

d.l.→ PPFFX




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X X X

•x

ab

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»»X

Xc→ PFX

PFc→ PFPFX

d.l.→ PPFFX




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b²²

X X X

•x

ab

§§

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Xc→ PFX

PFc→ PFPFX

d.l.→ PPFFX

m.m.→ PFFX




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b²²

X X X

•x

ab

§§

aa

»»X

Xc→ PFX

PFc→ PFPFX

d.l.→ PPFFX

m.m.→ PFFX


Distributive lawis needed for X

c→ T FX to be a coalgebra in the Kleisli

category K`(T )..




category K`(T )..

• objects - sets

• arrows - Xf→ Y are functions f : X → T Y




category K`(T )..

FT ⇒ T F : F lifts to F K̀ (T ) on K`(T ).




category K`(T )..


Hence: coalgebra Xc→ F K̀ (T )X in K`(T ) !!!




category K`(T )..



in K`(T ) : Xc→ F K̀ (T )X




category K`(T )..



in K`(T ) : Xc→ F K̀ (T )X




category K`(T )..



in K`(T ) : Xc→ F K̀ (T )X

FK̀ (T )c→ F K̀ (T )F K̀ (T )X




category K`(T )..



in K`(T ) : Xc→ F K̀ (T )X

FK̀ (T )c→ F K̀ (T )F K̀ (T )X → · · ·


Main theorem - tracesIf ♣, then

F K̀ (T )I

ηI◦α ∼=

²²

F K̀ (T )I

I I

ηFI◦α−1 ∼=

OO

is initial is final in K`(T )



F K̀ (T )I

ηI◦α ∼=

²²

F K̀ (T )I

I I

ηFI◦α−1 ∼=

OO


[α : FI∼=→ I denotes the initial F-algebra in Sets]



F K̀ (T )I

ηI◦α ∼=

²²

F K̀ (T )I

I I

ηFI◦α−1 ∼=

OO


[α : FI∼=→ I denotes the initial F-algebra in Sets]

proof: via limit-colimit coincidence Smyth&Plotkin ’82


The assumptions ♣:• A monad T s.t. K`(T ) is DCpo⊥-enriched

left-strict composition

• A functor F that preserves ω-colimits

• A distributive law FT ⇒ T F : lifting F K̀ (T )

• F K̀ (T ) should be locally monotone




















Corollary (♣)

For Xc→ F K̀ (T )X in K`(T )


Corollary (♣)

For Xc→ F K̀ (T )X in K`(T ) ... X

c→ T FX in Sets


Corollary (♣)


c→ T FX in Sets

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


Corollary (♣)


c→ T FX in Sets

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

in K`(T ) F K̀ (T )XF K̀ (T )(trc) //______ F K̀ (T )I

X

c

OO

trc

//__________ I

∼=

OO


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 terminationD(1 + A× )

Note: Initial 1 + A× - algebra is

A∗[nil,cons]

∼=// 1 + A× A∗




• Hence:




A∗[nil,cons]

∼=// 1 + A× A∗




• Hence:




A∗[nil,cons]

∼=// 1 + A× A∗




• Hence:




A∗[nil,cons]

∼=// 1 + A× A∗




• Hence:




A∗[nil,cons]

∼=// 1 + A× A∗




• Hence:




A∗[nil,cons]

∼=// 1 + A× A∗


Finite traces - LTS with X

the finality diagram in K`(P)

F K̀ (P)XF K̀ (P)(trc) //______ F K̀ (P)A

∗

X

c

OO

trc

//__________ A∗

∼=

OO




1 + A×X(1+A× )K̀ (P)(trc) //_________ 1 + A× A∗

X

c

OO

trc

//_____________ A∗

∼=

OO




1 + A×X(1+A× )K̀ (P)(trc) //_________ 1 + A× A∗

X

c

OO

trc

//_____________ A∗

∼=

OO

amounts to

• 〈〉 ∈ trc(x) ⇐⇒ X ∈ c(x)

• a · w ∈ trc(x) ⇐⇒ (∃x′)〈a, x′〉 ∈ c(x), w ∈ trc(x′)


Finite traces - generative X

the finality diagram in K`(D)

F K̀ (D)XF K̀ (D)(trc) //_______ F K̀ (D)A

∗

X

c

OO

trc

//__________ A∗

∼=

OO




1 + A×X(1+A× )K̀ (D)(trc) //_________ 1 + A× A∗

X

c

OO

trc

//_____________ A∗

∼=

OO




1 + A×X(1+A× )K̀ (D)(trc) //_________ 1 + A× A∗

X

c

OO

trc

//_____________ A∗

∼=

OO

amounts to trc(x) :

• 〈〉 7→ c(x)(X)

• a · w 7→∑

y∈X c(x)(a, y) · c(y)(w)


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 K`(T )

• Other monads e.g. PD ? ... suitable monad/order structure yet tobe found (Varacca&Winskel)

• Other semantics ? .. between bisimilarity and trace in thespectrum



study of (P)TS








study of (P)TS








study of (P)TS








study of (P)TS








study of (P)TS








study of (P)TS







Documents

SOS group, Radboud University Nijmegencs.uni-salzburg.at/~anas/papers/Dortmund-Presentation.pdf · 2020-06-04 · Probabilistic systems a place where categories meet probability Ana