Bisimulations
Codensity
Composing
Mayuko Kori" ?

Kazuki Watanabe, Jurriaan Rot Shin-ya Katsumata

1. National Institute of Informatics
The Graduate University for Advanced Studies

S Kyoto SangyoUniversity
University
Radboud,(SOKENTAI)Coalgebraic bisimulation (Hermida Jacobs&FGE~Sets

fibrationPmepf Spatial

FGB

Yets

F-

bisimulation

structure.

O/FX
transition

2g.

Osystem,Inf

Standard bisimulation↳ GEReld
PG Set-·R.

On-aS

*

equiv

RESXS

(stiPSis↓ (1 2) +R.

a.

catFr

↓ Coalg(p)Coalg(F)

Kripke frame SPR-.

ofisimulations
Coalq(F).

-Comput 1983.

Cat. ofF-coalgs.

-bisimulation on S
clES = EytSy KiyER& y ESy.

=Ex'(fx (x) y)ER.

(language equivalence of det automata.

bisimilarity metric for MDP.

1

viaof distributive law
bisimulations
lifting
Composing[Bonchit Act info 17]

pe,.

.

bisimulations.
by liftings F T X,

I Coalg (i)[Coalg()along p↓↓Coalg(p)B T B
B[Coalg(F) Coalg(F)*,of F T X.

byF B +:.

.

=:,whereNTX,-1

distributive law X : TFV = FT

Tx
Tx==X0+( )XoTHsystems
Composeproduct composition of standard bisimulations.
Tx
product of Kripke frames S1 Se

2.
g.

.

...

Tx
Prol

7-...

productof bisimulations R1 R& Rat Coalg
Ri +-.

.

Coalg(F)s,(F) S2bisim on Si==>R x Rz,R2) + Coalg (F) +x

bisim on Tx/S 82)(81 82).

..

2

of our work
Overview.
FT X
liftings
IgivenbyCoalg()"
Coalg(F)]u ensured,,byu↓*Coalg(F(WeandadoptE↓
Coalg(F)Y

suf cond.
.

codensity liftings.

] given by F,T x,

codensity lifting for F [Sprunger JLC21]
lized codensity lifting for T
new !
genera+.

&sufficient condition-provideadefined byof liftability of X

that gives various composition of bisimulations.

Results are based on 2-categorical extension of Beohart's decomposition .

[Beohar Gurke Konig Messing Forster Schroder Wild STACS'24],,,,,,,

3

Comparison

related works.

to1. Abstract GSOS rules
not2.

.

(Turia Plotkin LICs'97] etal.
.

directly apply to similarity

behavioural metricor&Our results cover.
ofX
in [Bonchit Act. info 17)
Liftability.

,

fibration#+

proof ofliftability

I[S6 Bonchit 17]
,

Rel

-

Set

Rel(F) RfI
,

Sym,

,

good cond
all distributive laws

~.

lift

I I
[S8 Bonchi'+,17]~ generalvariousvariouscase-by-caseours

Sweet.

spot !

variouscodensity liftings.
by sufficient cond4

Outline
1. Codensity-lifting

Ordinaryone.

*SFt.

GEFGB-Generalizing...

for system's behavior

R.
codensity liftingIENE E
pN ↓

14

B B

...

for composing coalg carriers

liftability of distributive laws
C
[F +J ST +Jo[T ] ([F ]MN

2. Sufficient condition of**,.

=,i

TFN

FT.

.

Codensity lifting [F t]&,DefI

truth value

Given[F TJR.

:-FGK

RE Er

Frenmodality

FGC
canI

↑

lifting of FS[FiTIREE- E is the largest

CFTJTCIETP
It[Sprunger +, JLC'21]

bewritten

For Eneas.

.

[F T]R(1),=t.

Iisliftable.

Style
Kantorovich
1 (ToF E)*
REPE(I M),E.
g.

Lifting for Standard bisim.

Kantorovich metric ,

...

5

[]
[FGeneralizing codensity lifting,original Beohart's decomposition (Beohart STAcs'24]
R
IED[F T]
.

↓S

ne4Il

Il[oFC)

IBPF

Def

Rit

Given REEr

E----#B
1P

[F T]

S-.

:=

GT

R.

-m·

17 T.

.

is thein

Sp(Fit)oLIEIl*R,

T : FR-eIIl

Im [F TJ,

TEH
modality,

RT

RepEt,RepEl- M)-R)truth values

[F t]

.

[ 1(ToFASTOR)

RPRoSp(Fit)oLPR

=.

[oFC)

C

S=Il

RepEt -R),,,!

CAflatSp
2-functor:[1//CAT +-1(ToFaL#RepEl- M),

largest lifting of FS t I is liftable.
.

.

-)
6

Outline
1. Codensity-lifting

Ordinaryone.

*SFt.

GE

FGB-Generalizing...

for system's behavior

R.
codensity lifting

IENE EpN ↓14B B...

for composing coalg carriers

liftability of distributive laws
C
[F +J ST +Jo[T ] ([F ]MN

2. Sufficient condition of*

*,.

=,i

TFN

FT.

.

law
distributiveofLiftability

eneralizedone

N- the,Consider &L EERENB

T:Fi+ 1inT:

Tli ... d)+,

We tackle theU↓,bein B,,

[Tir]

>
-EPinatR[F T],↓Pp

BN-BDF

problemwhen X TFFT is liftable to::

X [TirT /[F TJM)N&:.

*=[FiTJ ST J↑?
.

m
It gives

Rbisimulations
compose~*T])*<Coalg([FCoalg([F.

T]+].

↓

Coalg(F)*[<↓Coalg(p) ~ compose systems

Coalg(F)7

Our sufficient condition
How do we lift X to [T,t]*(+, +],↓(FY YoRoSp(T .)=.

/[F TJ on) N[T . + ]=(F iT** => [FiT] (T +J *?
**·LoRoSp(FYtYoLROS(2)=Ihm[F T]*,

X TFE FT

is:<TJ.+by L + R&liftable if(1) and (2) hold.

.

(1)

X isa

2-cell (T.)·(FY*) => (F t)0 (T 5).

.

(2) The last equality holds.

liff Sp(T r).

·in

1/1CAT.

4)*for each
LP is approximating to [F TJ
in the sense of [Komoridat LICS'21].

.

S

of composition ofExamples

bisimulations

Compositionof bisim ([T ,])of systems (TX)

Composition.

.

7Standard bisim.

product-Larg equivalence·ofdet automata.

-Bisimilarity pseudometricfor det.

automata.

-Similarity

parallelcompositionmax2)-

Bisimulation
for MDP

metricgiven byTlaib)
I (l a) (l b)
=--

-

Y

Satis

fy

Our

sufficientcondunsolved
if itsatisfies9Conclusions

Coalg)(F.

T]*[Tr]Coalg((F T])
2.

↓Coalg(p) ~ compose systems↓Coalg(F)-Tx*>-Coalg(F)Generalizing codensity liftings for [T +J*: EN.

·Sufficent condition of liftability
4-.
blu [and TS approximation property
commutation

bisimulations
compose~.

-E

FTJMN [FiTJ CT It
*[T .

,:

↑F

* FT

Committed in this talk) A composition of codensity games via modalities.
It preserves

invariants

under our sufficient condition.
10Appendix

Example Composing:

bisimulations for Kripke frames

Bef A bisimulation on a Kripke frame 8 S-4S
is R & SXS S .
:.

t.

E( ii) and y (x -ax + x)

RPop R.

=>and R2,=>

areR XR2 is,bisima.

onbisim.

S , and S2

on

Kripkeframes 8.,

82

Bisimulations R R2..

aProductsaby.

,#coalgebras C C2=>>

liftings of C C2 along

compose by a product-respectively

the product of S and S2.

Coalgebraic reformulation-,,

Compose by a distributive law,

afibration.

lifting of .
compose by aX/

our focus

1Composing coalgebras
Givenfunctors F B

+:

via

distributive law*

1B T B -B:.

,

distributive law X TFEFT,a:

we have

TX

Coalg (F(N:& Xi En FXiJicEli

Coalg(F)

>-Tx

To TI

My ↳-:

T:=X:

Set- Set

Then Tx (C (2).

·:Xxi (A B),+:=>FI*).
composite systemcomponent systemXi= product of Kripke frames [CiX
-DXihi=1.2[(a b) /atA be BY,,>
P(X XX2)
X , XX2 ,a.

b↳[laid') /a Ca becb]=,

How do we construct a bisimulation on Tx (c) ?

composite system3decomposition Fibrationally

Beohar et al 's,.

decomposition of [F T]↑as.

Im [F +]R.

.

=E(-IlIl&Spi↓,B)1(To FM),)ExpEriM)

M)

v)oP

Eit,RepEG

ToFH

S+=S

where.

being sandwiched blu adjoints.

RPRoSp(Fit)oLPMF(Beohart StAcs'24]--fibred lifting of Fop(S [ (B(X allr)@Sp(Fit)
Sp(B[ToFEIGES]
CLateRed↓-

B(- 2)0P,

.

PSet,

RRR is the right adjoint of Li .

↓:=-fib.

BBDF[B(FX M2),

E essential

part-

given by F t.

11

Codensity liftings of T

lesp product functors).

.

.
Example Pseudo-metric lifting by a modality &diPMeE PMet↓↓

Euclidean distance,Set

T

Set

di:=:.

ife g.
.

,.

SMetColECo 1) x 10 1) > Co 1).

.

diProp [X +] [dx di) ((x y) ky)
.

.

,=+-,

functionis a.

(dx (i) di (1 %'),,,t is monotone+ (0 , 0↓ (a b).

+(a b),:=-0 .

5 (c d) = +=F/la-c 1b-dl)

for eachb)(1 a) (l b)max (a&.

.

,

1--a.

b c d,.

C0 13,-They give various non-trivial liftings.

13