Categorical approaches to bisimilaritygroup-mmm.org/~dubut/presentations/pps19.pdf ·...

Categorical approaches to bisimilarityPPS’ seminar, IRIF, Paris 7

Jérémy Dubut

National Institute of InformaticsJapanese-French Laboratory for Informatics

April 2nd

Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 1 / 40

Bisimilarity of Transition Systems


Transition Systems

Transition system :A TS T = (Q, i ,∆) on the alphabet Σ is the following data:

a set Q (of states);an initial state i ∈ Q;a set of transitions ∆ ⊆ Q × Σ× Q.

Σ = {a, b, c},Q = {0, 1, 2, 3},i = 0,∆ = {(0, a, 0), (0, b, 1), (0, a, 2),(1, c , 2), (2, b, 0), (2, a, 3)}.

0

1

2

3

ab

a

b c

a


Bisimulations of Transition Systems

Bisimulations [Park81] :A bisimulation between T1 = (Q1, i1,∆1) and T2 = (Q2, i2,∆2) is a relationR ⊆ Q1 × Q2 such that:(i) (i1, i2) ∈ R;(ii) if (q1, q2) ∈ R and (q1, a, q

′1) ∈ ∆1 then there is q′2 ∈ Q2 such that

(q2, a, q′2) ∈ ∆2 and (q′1, q

′2) ∈ R;

(iii) if (q1, q2) ∈ R and (q2, a, q′2) ∈ ∆2 then there is q′1 ∈ Q1 such that

(q1, a, q′1) ∈ ∆1 and (q′1, q

′2) ∈ R.

• •

•

•

•

•

•

•

•

a

c

b a

a

c

b


Several Characterisations of Bisimilarity

Bisimilarity:Given two TS T and T ′, the following are equivalent:

[Park81] There is a bisimulation between T and T ′.

[Stirling96] Defender has a strategy to never loose in a 2-player game on Tand T ′.

[Hennessy80] T and T ′ satisfy the same formulae of the Hennessy-Milnerlogic.

In this case, we say that T and T ′ are bisimilar.


Morphisms of Transition Systems

Morphism of TS:A morphism of TS f : T1 = (Q1, i1,∆1) −→ T2 = (Q2, i2,∆2) is a function

f : Q1 −→ Q2

such that:preserving the initial state: f (i1) = i2,preserving the transitions: for every (p, a, q) ∈ ∆1, (f (p), a, f (q)) ∈ ∆2.

TS(Σ) = category of transition systems and morphisms

•

• •

• •

•

•

• •

a a

b c

a

b c

Morphisms are functionalsimulations:Morphisms are precisely functions fbetween states whose graph{(q, f (q)) | q ∈ Q1} is a simulation.


Categorical Characterisations

Bisimilarity, using morphisms:Two TS T and T ′ are bisimilar iff there is a span of functional bisimulationsbetween them.

T ′′

T T ′

f g


Bisimilarity from Coalgebra

J. Rutten. Universal coalgebra: a theory of systems.Theoretical Computer Science 249(1), 3–80 (2000)


Transition systems, as pointed coalgebras

Set of transitions, as functions:There is a bijection between sets of transitions ∆ ⊂ Q × Σ× Q and functions oftype:

δ : Q −→ P(Σ× Q)

where P(X ) is the powerset {U | U ⊂ X}.

Initial states, as functions:There is a bijection between initial states i ∈ Q and functions of type:

ι : ∗ −→ Q

where ∗ is a singleton.


Example

Σ = {a, b, c},Q = {0, 1, 2, 3},i = 0,∆ = {(0, a, 0), (0, b, 1), (0, a, 2),(1, c , 2), (2, b, 0), (2, a, 3)}.

0

1

2

3

ab

a

b c

a

ι : ∗ −→ Q∗ 7→ 0

δ : Q −→ P(Σ× Q)0 7→ {(a, 0), (b, 1), (a, 2)}1 7→ {(c , 2)}2 7→ {(b, 0)}3 7→ ∅


Pointed coalgebras

Pointed coalgebras:Given an endofunctor G : C −→ C and an object I ∈ C, a pointed coalgebra isthe following data:

an object Q ∈ C,a morphism ι : I −→ Q of C,a morphism σ : Q −→ G (Q) of C.

G is often decomposed as T ◦ F , where:T : “branching type”, e.g, non-deterministic, probabilistic, weighted.

For TS: T = P.

F : “transition type”.For TS: F = Σ×_.

I is often the final object, but we will see other examples.For TS: I = ∗, the final object.


Morphisms of TS, using Pointed Coalgebras

Morphisms of TS are lax morphisms of pointed coalgebrasA morphism of TS, seen as pointed coalgebras T = (Q1, ι1, δ1) andT ′ = (Q2, ι2, δ2) is the same as a function

f : Q1 −→ Q2

satisfying

I Q1

Q2

P(Σ× Q1)

P(Σ× Q2)

� ⊆f P(Σ× f )

ι1 σ1

ι2

σ2


Lax Morphisms of Pointed Coalgebras

Lax Morphisms:Assume there is an order � on every Hom-set of the form C(X ,G (Y )). A laxmorphism from (Q1, ι1, δ1) to (Q2, ι2, δ2) is a morphism

f : Q1 −→ Q2

of C satisfying

I Q1

Q2

G (Q1)

G (Q2)

� �f G(f )

ι1 σ1

ι2

σ2

Coallax(G , I ) = category of pointed coalgebras and lax morphisms.


What about functional bisimulations?Functional bisimulations are homomorphisms of pointed coalgebrasFor two TS, seen as pointed coalgebras T = (Q1, ι1, δ1) and T ′ = (Q2, ι2, δ2),and for a function of the form f : Q1 −→ Q2, the following are equivalent:

The graph {(q, f (q)) | q ∈ Q1} of f is a bisimulation.f is a homomorphism of pointed coalgebras, that is, the following diagramcommutes:

I Q1

Q2

P(Σ× Q1)

P(Σ× Q2)

��f P(Σ× f )

ι1 σ1

ι2

σ2

Bisimilarity, using homomorphisms of pointed coalgebrasFor two TS T and T ′, the following are equivalent:

T and T ′ are bisimilar.There is a span of homomorphisms of pointed coalgebras between T and T ′.


Homomorphisms of Pointed Coalgebras

Morphisms:A homomorphism from (Q1, ι1, δ1) to (Q2, ι2, δ2) is a morphism

f : Q1 −→ Q2

of C satisfying

I Q1

Q2

G (Q1)

G (Q2)

��f G(f )

ι1 σ1

ι2

σ2

Coal(G , I ) = category of pointed coalgebras and homomorphisms.


Summary

[Wißman, D., Katsumata, Hasuo – FoSSaCS’19]−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→Non-deterministic branching

coalgebra open maps

data type G : C → C, I ∈ CJ : P →M� on C(X ,G (Y ))

systems pointed coalgebras objects ofM

functional simulations lax morphisms morphisms ofM

functional bisimulations homomorphisms open maps

bisimilarity existence of a span offunctional bisimulations

[Lasota’02]←−−−−−−−−−−−−−−−Small category of paths


Bisimilarity from Open Maps

A. Joyal, M. Nielsen, G. Winskel. Bisimulation from Open Maps.Information and Computation 127, 164–185 (1996)


Runs in a Transition System

RunA run in a transition system (Q, i ,∆) is sequence written as:

q0a1−−→ q1

a2−−→ . . .an−−→ qn

with:qj ∈ Q and aj ∈ Σ

q0 = i

for every j , (qj , aj+1, qj+1) ∈ ∆

q0a−−→ q0

b−−→ q1c−−→ q2

a−−→ q3q0

q1

q2

q3

ab

a

b c

a


Runs, CategoricallyFinite Linear Systems:A finite linear system is a TS of the form 〈a1, . . . , an〉 = ([n], 0,∆) where:

[n] is the set {0, . . . , n};∆ is of the form {(i , ai+1, i + 1) | i ∈ [n − 1]} for some a1, ..., an in Σ.

0 1 2 . . . n − 1 na1 a2 an

Runs are morphismsThere is a bijection between runs of T and morphisms of TS between a finitelinear system to T .

43210a b c a q0

q1

q2

q3

ab

a

b c

a


Functional Bisimulations, from Lifting Properties of PathsFunctional bisimulations are open maps:For a morphism f of TS from T to T ′, the following are equivalent:

The reachable graph of f , that is, {(q, f (q)) | q reachable} is a bisimulation.f has the right lifting property w.r.t. path extensions, that for everycommutative square (in plain):

〈a1, . . . , an〉 T

〈a1, . . . , an, an+1, . . . , an+p〉 T ′

ρ

f

ρ′

injθ

there is a lifting (in dot), making the two triangles commute.

Bisimilarity, using open mapsFor two TS T and T ′, the following are equivalent:

T and T ′ are bisimilar.There is a span of open maps between T and T ′.


Open mapsOpen map situation:An open map situation is a categoryM (of systems) together with asubcategory J : P ↪→M (of paths).

M = category of systems (Ex : TS(Σ)),P = sub-category of finite linear systems.

Open maps:A morphism f : T −→ T ′ ofM is said to be open if for every commutativesquare (in plain):

J(P) T

J(Q) T ′

ρ

f

ρ′

J(p)θ

where p : P −→ Q is a morphism of P, there is a lifting (in dot) making the twotriangles commute.


Summary


coalgebra open maps

data type G : C → C, I ∈ CJ : P ↪→M� on C(X ,G (Y ))







From Open Maps to Coalgebra

S. Lasota. Coalgebra morphisms subsume open maps.Theor. Comput. Sci. 280(1–2): 123–135 (2002)


Problem Setting

Input: An open map situation P ↪→M such that:P is small,P andM has a common initial object 0

Problem: Constructa coalgebra situation:

I G : C −→ C,I I ∈ C,I � on C(X ,G(Y )).

a functor Beh :M−→ Coallax(G , I )

such that

f is an open map iff Beh(f ) is a homomorphism.


SolutionC = Setob(P),

G ((XP)P∈ob(P)) = (∏

Q∈P(P(XQ))P(P,Q)

)P∈P

I0 = ∗, IP = ∅ otherwise,

� point-wise inclusion,

Beh(X ):I XP = set of runs labelled by P, i.e., the set M(P,X ),I ι : (IP) −→ (XP) maps ∗ to unique morphism from 0 to X ,I σP = (σP,Q)Q : XP −→

∏Q∈P

(P(XQ))

P(P,Q), where σP,Q maps a run ρlabelled by P to the set of runs labelled by Q that extend ρ.

Theorem [Lasota02]:f is an open map iff Beh(f ) is a homomorphism.


From Coalgebra to Open Maps

T. Wißman, J. Dubut, S. Katsumata, I. Hasuo. Path Category For Free – OpenMorphisms From Coalgebras With Non-Deterministic Branching. FoSSaCS’19


Problem Setting

Input: A coalgebra situation:G = T ◦ F : C −→ C,I ∈ C,� on C(X ,G (Y )).

satisfying some axioms.

Problem: Construct an open map situation J : P ↪→ Coallax(G , I ) such that laxhomomorphism f : c1 −→ c2:

if f is a homomorphism then f is openif f is open and c2 is reachable, then f is a homomorphism


THE key notion: F -precise morphisms

F -precise morphisms:A morphism s : S → FR of C is F -precise if for all f , g , h:

S FC

FR FD

f

s Fh

Fg

∃d=⇒

S FC

FR

f

sFd

&C

R D

h

g

d

Intuition: a morphism s : S → FR is precise iff every element of R is used exactlyonce in the definition of s.


Examples

FX = X × X +⊥

X Y X Y ′

not precise precise

⊥

⊥

⊥

⊥

f f ′


The Path Category P = Path(I ,F )

A path consists in:a finite sequence P0, . . . , Pn of objects of C with P0 = I ,a finite sequence of F + 1-precise maps:

fk : Pk −→ FPk+1 + 1

A morphism between paths, from (Pk , pk)k≤n to (Qj , qj)j≤m consists in asequence of isomorphisms φk : Pk −→ Qk such that:

Pk FPk+1 + 1

Qk FQk+1 + 1

φk

pk

Fφk+1+1

qk


ExamplesFX = {a} × X + X × X , I = ∗, pk : Pk → FPk+1 + {⊥}

P0

P1

P2 P3

P4

a

a

⊥ ⊥

⊥

p0

p1p2

p3

P0

P1 P2 P3 P4

a

a ⊥

⊥

aa

p0p1 p2 p3


The Functor JAssumptions on C and T :

1 C has finite coproducts,2 η : IdC −→ T ,3 ⊥ : 1 −→ T such that ⊥X ∈ C(1,T (X )) is the least element for �,4 some others.

Typical example: the powerset functor PSet has disjoint unions and empty set,η is the unit ηX (x) = {x},⊥ is given by the empty subset ⊥X (∗) = ∅,. . .

Theorem:There is a functor J : Path(I ,F ) −→ Coallax(TF , I ) given by J(Pk , pk) :=

Iin0−−→

∐k≤n

Pk

[inl·[F ink+1·pk ]k<n,inr·!

]−−−−−−−−−−−−−−−→ F

∐k≤n

Pk + 1[η,⊥]� TF

∐k≤n

Pk


Wrapping up

Theorem:A homomorphism of pointed coalgebras in Coallax(TF , I ) is open.

Proposition-Definition:For a pointed coalgebra c = (Q, ι, σ), the following are equivalent:

c has no proper subcoalgebra,the set of all morphisms of the form J(Pk , pk) −→ c is jointly epic.

In this case, we say that c is reachable.

Theorem:An open map h : c −→ c ′ in Coallax(TF , I ) where c is reachable is ahomomorphism.


Instances


Labelled Transition Systems

C = Set,

F (X ) = Σ× X ,

T = P,

I = ∗.

Paths are given by words.


Various Tree-like AutomataC = Set,

F analytic, i.e., F (X ) =∐

σ/n∈Σ

X n/Gσ,

T = P,

I = ∗.

Paths are given by “partial” trees.

P0

P1

P2 P3

P4

a

a

⊥ ⊥

⊥

p0

p1p2

p3


Multi-Sorted Transition Systems [Lasota’02]C = Setob(P),

F ((XP)P∈P) =( ∐Q∈P

P(P,Q)× XQ

)P∈P,

T ((XP)P∈P) = (P(XP))P∈P,

I0 = ∗, IP = ∅.

Paths are given by sequences of path extensions from the initial path category:

0 m1−−→ P1m2−−→ P2 · · ·

mn−−→ Pn

Consequence:We cannot expect a more general translation from coalgebra to open maps.


Regular Nondeterministic Nominal Automata [Schröder etal.’17]

C = Nom,

F (X ) = 1 + A× X + [A]X , where [A]_ is a binding operator,

T = Pufs , the set of uniformly finitely supported,

I = A#n, the set of n-tuples of distincts atoms.


General Kripke Frames [Kupke et al.’04]

C = Stone, the category of Stone spaces

F = Id,

T = V, the Vietoris topology on the set of compact subsets,

I = ∗.


Conclusion


coalgebra open maps

data type G : C → C, I ∈ CJ : P ↪→M� on C(X ,G (Y ))







Categorical approaches to bisimilaritygroup-mmm.org/~dubut/presentations/pps19.pdf ·...

Documents

Transcript of Categorical approaches to bisimilaritygroup-mmm.org/~dubut/presentations/pps19.pdf ·...