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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
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 2 / 40
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
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 3 / 40
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
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 4 / 40
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.
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 5 / 40
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.
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 6 / 40
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
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 7 / 40
Bisimilarity from Coalgebra
J. Rutten. Universal coalgebra: a theory of systems.Theoretical Computer Science 249(1), 3–80 (2000)
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 8 / 40
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.
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 9 / 40
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→ ∅
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 10 / 40
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.
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 11 / 40
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
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 12 / 40
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.
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 13 / 40
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 ′.
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 14 / 40
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.
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 15 / 40
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
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 16 / 40
Bisimilarity from Open Maps
A. Joyal, M. Nielsen, G. Winskel. Bisimulation from Open Maps.Information and Computation 127, 164–185 (1996)
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 17 / 40
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
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 18 / 40
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
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 19 / 40
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 ′.
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 20 / 40
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.
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 21 / 40
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
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 22 / 40
From Open Maps to Coalgebra
S. Lasota. Coalgebra morphisms subsume open maps.Theor. Comput. Sci. 280(1–2): 123–135 (2002)
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 23 / 40
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.
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 24 / 40
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.
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 25 / 40
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
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 26 / 40
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
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 27 / 40
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.
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 28 / 40
Examples
FX = X × X +⊥
X Y X Y ′
not precise precise
⊥
⊥
⊥
⊥
f f ′
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 29 / 40
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
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 30 / 40
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
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 31 / 40
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
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 32 / 40
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.
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 33 / 40
Instances
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 34 / 40
Labelled Transition Systems
C = Set,
F (X ) = Σ× X ,
T = P,
I = ∗.
Paths are given by words.
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 35 / 40
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
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 36 / 40
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.
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 37 / 40
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.
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 38 / 40
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 = ∗.
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 39 / 40
Conclusion
[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
Jérémy Dubut (NII & JFLI) Categorical approaches to bisimilarity April 2nd 40 / 40