Graded ML
1/53
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
A coalgebraic approach to graded modal logic
and graded bisimilarity
Joshua Sack
This talk is based on joint work with Luca Aceto and Anna
Ingolfsdottir
2014, November 11
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Graded modal language
The graded modal language Lgml is defined by
ϕ ::= true | ¬ϕ | ϕ ∧ ϕ | 3n ϕ (n ∈ N)
We derive 2n ϕ ≡ ¬3n ¬ϕ, and have the following basic readings:
3n ϕ reads “ϕ has weight at least n.”
2n ϕ reads “¬ϕ has weight less than n.”
What is meant by “weight” depends on different semantics, which
we now define.
2/53
Graded ML
3/53
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Semantics of graded modal logic
There are many choices of semantics. We focus on three
1
standard frame-semantics
(on Kripke frames)
2
multi-frame semantics
(on multi-frames, a type of weighted frames)
3
predicate-lifting coalgebraic semantics
(on coalgebras that are equivalent to multi-frames)
Behavioral equiv
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Graded modal logic
Definition (Kripke Frame)
A Kripke frame is a tuple F = (S, R), such that
S is a set (of “states”)
R ⊆ S 2 is a binary relation
A pair (F , s), s ∈ S is a pointed Kripke frame.
(F , s) |= 3n ϕ if and only if
there are at least n s-successors t, such that (F , t) |= ϕ.
(ϕ holds in at least n-successor states.)
(F , s) |= 2n ϕ if and only if
for all sets A of n s-successors, there is a t ∈ A such that
(F , t) |= ϕ.
(ϕ fails to hold in fewer than n-successor states.)
Note that 21 and 31 coincide with the standard modal 2 and 3.
4/53
Graded ML
5/53
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Frame semantics example
a
b
c
d
a |= 32 true ∧ 23 ¬ true ∧ 21 21 ¬ true is read
“There are at least 2 successors, fewer than 3 successors, and
every successor from here has no successor.”
c |= 32 32 true ∧ 31 21 ¬ true
“There are at least 2 successors with at least 2 successors and
there is at least one successor with no successors.”
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Multi-frames and weighted modal logic
Definition (Multiframe (aka multigraph))
A multi-frame is a pair M = (S, Σ), such that
S is a set (lets assume it is countable)
Σ consists of, for each s ∈ S, a multiset function σ s : S → N
(here N = {0, 1, 2, . . . }).
(M, s) |= 3n ϕ if and only if
P s
{σ (t) | (M, t) |= ϕ} ≥ n.
P s
(M, s) |= 2n ϕ if and only if {σ (t) | (M, t) |= ¬ϕ} < n.
6/53
Graded ML
7/53
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Multi-frame semantics example
a
b
2
3
2
1
c
1
d
a |= 33 true ∧ 24 ¬ true ∧ 21 21 ¬ true is read
“The weighted out-degree is at least 3, and fewer than 4, and
every successor from here has no successor.”
c |= 32 32 true ∧ 31 21 ¬ true
“There is a weight of at least 2 of states whose weighted
out-degree is at least 2 and there is at least one successor
with no successors.”
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Translations between frame and multi-frame
Definition (Translation M)
Given a Kripke frame F = (S, R), we can translate it into the
multi-frame M(F ) = (S, Σ), where for each s ∈ S,
1 sRt
s
σ (t) =
0 otherwise
Definition (Translation K)
Given a multi frame M = (S, Σ), we can translate it into a Kripke
frame K(M) = (T , R), where
S
T = s∈S {(s, n) | 1 ≤ n ≤ supa∈S {1, σ a (s)}}
(s, n)R(t, m) if and only if 1 ≤ m ≤ σ s (t).
8/53
Graded ML
9/53
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Example of translation M
a
a
b
M
c
d
1
b
1
1
−→
1
c
1
d
Graded ML
10/53
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Example of translation K
b2
2
a
1
b
2
2
c
1
a1
b1
c2
d2
c1
d1
K
−→
d
1
Behavioral equiv
Graded ML
11/53
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Relationship between frame and multi-frame
For any pointed frame or multiframe P, let
L(P) = {ϕ ∈ Lgml | P |= ϕ}.
Then
Given a pointed Kripke frame (F , s)
L(F , s) = L(M(F ), s).
Given a pointed multi-frame (M, s)
L(M, s) = L(K(M), (s, 1)).
These are proved by a straightforward induction on formulas.
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Coalgebraic modal logic background
Definition (Finite powerset functor)
The finite powerset functor P : Set → Set is given by
For each object (set) X , P(X ) consists of all subsets of X
For each morphism (function) f : X → Y , Pf : Z 7→ f [Z ] for
each finite subset Z ⊆ X .
Definition (P-coalgebra)
A P coalgebra is a pair (A, α), where A is a set and α : X → PX
is a morphism (function).
Each P-coalgebra defines a Kripke frame (directed graph).
12/53
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Predicate lifting background
Definition (Contravariant powerset functor)
The contravariant power set functor 2 : Set → Set is given by
For each X , 2(X ) = P(X ) the set of all subsets of X
For each morphism f : X → Y , 2f : Z 7→ f −1 [Z ] for each
subset Z of Y .
13/53
Graded ML
14/53
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Natural transformation for contravariant functors
A natural transformation from contravariant functor F to G on
Set, is a collection of morphisms λX for each set X , such that
whenever f : X → Y , the following commutes.
FY
λX
F (f )
GY
/ FX
G (f )
λY
/ GX
Graded ML
15/53
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Predicate lifting semantics for powerset functor
Definition
A predicate lifting for P is a natural transformation λ : 2 → 2 ◦ P,
where 2 is the contravariant powerset functor.
For each set S, we define
λS : A 7→ {B ∈ P(S) | A ∩ B 6= ∅} .
Given a coalgebra X = (S, α) and s ∈ S,
(X , s) |= 3ϕ if and only if α(s) ∈ λS [[ϕ]],
where [[ϕ]] = {t | (X , t) |= ϕ}.
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Multi-set functor
Definition (Finite mulit-set functor)
The finite multiset functor B : Set → Set is given by
For each set X , B(X ) consists of all finite multisets on X
(functions σ : X → N with finite support).
P
For each morphism f , Bf σ : y 7→ {σ(x) | f (x) = y }.
((Bf σ)(y ) gives σ’s “weighted sum” σ(f −1 [y ]) of f −1 [y ])
Definition (B-coalgebra)
A B coalgebra is a pair (A, α), where A is a set and α : X → BX is
a morphism (function).
Each B-coalgebra defines a multiframe (aka multigraph).
16/53
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Predicate lifting (semantics)
Definition
A predicate lifting for B is a natural transformation λ : 2 → 2 ◦ B,
where 2 is the contravariant powerset functor.
For each set S and n ∈ N, we define
(
)
X
λnS : A 7→ σ ∈ B(S) σ(x) ≥ n .
x∈A
Given a coalgebra X = (S, α) and s ∈ S,
(X , s) |= 3n ϕ if and only if α(s) ∈ λnS [[ϕ]],
where [[ϕ]] = {t | (X , t) |= ϕ}.
17/53
Graded ML
18/53
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Relationship between multi-frame and coalgebra
Let
M map each coalgebra X to its multiframe M(X ), and
C map each multi-frame M to its coalgebra C(M).
Then
Given a pointed coalgebra (X , s)
L(X , s) = L(M(X ), s).
Given a pointed multiframe (M, s)
L(M, s) = L(C(M), s).
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Bisimulation: basic definition
Definition (Bisimulation)
A bisimulation between F1 = (S1 , R1 ) and F2 = (S2 , R2 ) is a
relation N ⊆ S1 × S2 , such that whenever xN y ,
1
whenever xR1 x 0 , there is a y 0 ∈ R2 (y ), such that x 0 N y 0 .
2
whenever yR2 y 00 , there is an x 00 ∈ R1 (x), such that x 00 N y 00 .
x0 o
R1
N
y0 o
R1
x
N
R2
y
/ x 00
N
R2
/ y 00
The largest bisimulation, denoted -, is called bisimilarity
19/53
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Examples concerning bisimulations
a
c
b
Every point is bisimilar to each other. In particular, a - b.
x
y
z
No point is bisimilar to a distinct other. In particular, x 6- y .
Aside comment
However, x and y are mutually similar, in that each simulates the
other.
20/53
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Examples concerning bisimulations
a
c
b
Every point is bisimilar to each other. In particular, a - b.
x
y
z
No point is bisimilar to a distinct other. In particular, x 6- y .
Aside comment
However, x and y are mutually similar, in that each simulates the
other.
20/53
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Examples concerning bisimulations
a
c
b
Every point is bisimilar to each other. In particular, a - b.
x
y
z
No point is bisimilar to a distinct other. In particular, x 6- y .
Aside comment
However, x and y are mutually similar, in that each simulates the
other.
20/53
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Graded bisimulation
Definition (Graded bisimulation)
A g -bisimulation relation between F1 = (S1 , R1 ) and F2 = (S2 , R2 )
is a relation Z⊆ P <ω (S1 ) × P <ω (S2 ) satisfying
1 whenever X Z Y ,
1
2
3
2
|X | = |Y |,
for each x ∈ X , there is a y ∈ Y such that {x} Z {y }, and
for each y ∈ Y , there is an x ∈ X such that {x} Z {y };
whenever {x} Z {y }
1
2
if X ⊆ R1 (x) is finite, then there exists some finite Y ⊆ R2 (y )
such that X Z Y , and
if Y ⊆ R2 (y ) is finite, then there exists some finite X ⊆ R1 (x)
such that X Z Y .
The largest g -bisimulation, written -g , is called g -bisimilarity.
We will in general focus on image-finite Kripke frames.
21/53
Graded ML
22/53
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Graded bisimulation in pictures
{x 0 }
⊆
Z
Z
{y 0 }
⊆
X0 o
R1
Z
Y0 o
⊇
X
Y
{x}
Z
⊇
{y 00 }
R1
/ X 00
Z
R2
{y }
{x 00 }
Z
R2
/ Y 00
Behavioral equiv
Graded ML
23/53
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Relationship between bisimulation and g -bisimulation
Given any graded bisimulation Z, the set
N = {(x, y ) | ({x}, {y }) ∈ Z}
is a bisimulation.
The converse need not be true:
c
b
a
d
N = {b, c}2 ∪ {a, d}2 is a bisimulation
a and d are not g -bisimilar
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Resource bisimulation
Definition (Resource bisimulation)
An r -bisimulation between F1 = (S1 , R1 ) and F2 = (S2 , R2 ) is a
relation R ⊆ S1 × S2 satisfying
whenever (s1 , s2 ) ∈ R, there is a bijective function
fs1 ,s2 : R1 (s1 ) → R2 (s2 )
such that
(s, fs1 ,s2 (s)) ∈ R
for each s ∈ R1 (s1 ).
The largest r -bisimulation, written -r , is called r -bisimilarity.
24/53
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Relationship between g and r bisimulation
Given an r -bisimulation R, there exists a g -bisimulation Z, such
that
{({x}, {y }) ∈ Z | (x, y ) ∈ R}.
In particular,
Z=
[
xRy
{({x}, {y })} ∪
[
{(A, fx,y (A))}.
A∈P <ω (R1 (x))
Conversely, is it the case that if Z is a g -bisimulation, then
{(x, y ) | ({x}, {y }) ∈ Z} is an r -bisimulation?
But what about g -bisimilarity and r -bisimilarity?
Goal
We aim to answer the second question.
25/53
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Relationship between g and r bisimulation
Given an r -bisimulation R, there exists a g -bisimulation Z, such
that
{({x}, {y }) ∈ Z | (x, y ) ∈ R}.
In particular,
Z=
[
xRy
{({x}, {y })} ∪
[
{(A, fx,y (A))}.
A∈P <ω (R1 (x))
Conversely, is it the case that if Z is a g -bisimulation, then
{(x, y ) | ({x}, {y }) ∈ Z} is an r -bisimulation?
But what about g -bisimilarity and r -bisimilarity?
Goal
We aim to answer the second question.
25/53
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Relationship between g and r bisimulation
Given an r -bisimulation R, there exists a g -bisimulation Z, such
that
{({x}, {y }) ∈ Z | (x, y ) ∈ R}.
In particular,
Z=
[
xRy
{({x}, {y })} ∪
[
{(A, fx,y (A))}.
A∈P <ω (R1 (x))
Conversely, is it the case that if Z is a g -bisimulation, then
{(x, y ) | ({x}, {y }) ∈ Z} is an r -bisimulation?
But what about g -bisimilarity and r -bisimilarity?
Goal
We aim to answer the second question.
25/53
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Relationship between g and r bisimulation
Given an r -bisimulation R, there exists a g -bisimulation Z, such
that
{({x}, {y }) ∈ Z | (x, y ) ∈ R}.
In particular,
Z=
[
xRy
{({x}, {y })} ∪
[
{(A, fx,y (A))}.
A∈P <ω (R1 (x))
Conversely, is it the case that if Z is a g -bisimulation, then
{(x, y ) | ({x}, {y }) ∈ Z} is an r -bisimulation?
But what about g -bisimilarity and r -bisimilarity?
Goal
We aim to answer the second question.
25/53
Graded ML
26/53
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Goal
We want, for any pointed Kripke frames (F , x) and (K , y ), that
(F , x) -r (K , y )
iff
(F , x) -g (K , y ).
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Hennessy-Milner Property for g -bisimilarity
Definition (Hennessy-Milner property)
A bisimilarity - has the Hennessy-Milner property if
(F1 , s1 ) - (F2 , s2 ) if and only if L(F1 , s1 ) = L(F2 , s2 ).
Theorem
g -bisimilarity satisfies the Hennessy-Milner Property on
image-finite Kripke frames.
27/53
Behavioral equiv
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Proof sketch (right-to-left)
Define
X1 Z X2 if and only if each of the following holds:
|X1 | = |X2 |,
for each x1 ∈ X1 , there is x2 ∈ X2 such that L(x1 ) = L(x2 ),
and
for each x2 ∈ X2 , there is x1 ∈ X1 such that L(x1 ) = L(x2 ).
Given L(w1 ) = L(w2 ), we have that {w1 } Z {w2 }.
28/53
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Recall relationships among semantics
Connection between frame and multi-frame
Given a pointed Kripke frame (F , s)
L(F , s) = L(M(F ), s).
The pointed multi-frame (M(F ), s) satisfies the same formulas as
the pointed frame (M, s).
Connection between multi-frame and coalgebra
Given a pointed multiframe (M, s)
L(M, s) = L(C(M), s).
The pointed coalgebra (C(M), s) satisfies the same formulas as the
pointed multi-frame (M, s).
29/53
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Progress so far
The following are equivalent
L(CM(F ), x) = L(CM(K ), y )
L(M(F ), x) = L(M(K ), y )
Basic Induction
L(F , x) = L(K , y ).
Basic Induction
(F , x) -g (K , y ).
30/53
HM property
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
r -bisimulation on multiframes
An r -bisimulation between multiframes M1 and M2 is the first
coordinate projection of an ordinary frame r -bisimulation R
between K(M1 ) and K(M2 ) that has the following constraint:
If (x, n)R(y , m), then (x, n0 )R(y , m0 ).
We have an equivalent formulation on the next slide.
31/53
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
r -bisimulation on multiframes
An r -bisimulation between multiframes M1 and M2 is the first
coordinate projection of an ordinary frame r -bisimulation R
between K(M1 ) and K(M2 ) that has the following constraint:
If (x, n)R(y , m), then (x, n0 )R(y , m0 ).
We have an equivalent formulation on the next slide.
31/53
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
r -bisimulation on multiframes definition
Definition (r -bisimulation on multiframes)
A (multiframe) r -bisimulation between M1 = (S1 , Σ1 ) and
M2 = (S2 , Σ2 ) is a relation R ⊆ S1 × S2 such that
whenever (s1 , s2 ) ∈ R, there is a bijective function
[
fs1 ,s2 :
{(s, n) | 1 ≤ n ≤ σ1s1 (s)} →
s∈S1
[
{(s, n) | 1 ≤ n ≤ σ2s2 (s)},
s∈S2
such that s R t whenever fs1 ,s2 (s, n) = (t, m).
The largest r -bisimulation on multiframes, also written -r , is also
called r -bisimilarity.
32/53
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Multi-frame r -bisimulation that is a function
A function g : S1 → S2 is a multiframe r -bisimulation between
M1 = (S1 , Σ1 ) and M2 = (S2 , Σ2 ) precisely when for all
(s1 , s2 ) ∈ S1 × S2
if g (s1 ) = s2 ,
then for each y ∈ S2 ,
σ2s2 (y ) =
X
{x|g (x)=y }
33/53
σ1s1 (x) = σ1s1 (g −1 [y ])
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Relationships between r -bisimilarities
Given pointed multiframes (M1 , s1 ) and (M2 , s2 )
(M1 , s1 ) -r (M2 , s2 ) iff (K(M1 ), (s1 , 1)) -r (K(M2 ), (s2 , 1))
Given pointed Krikpe frames (F1 , s1 ) and (F2 , s2 )
(F1 , s1 ) -r (F2 , s2 ) iff (M(F1 ), s1 ) -r (M(F2 ), s2 )
34/53
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Progress so far
The following are equivalent
(F , x) -r (K , y ).
(M(F ), x) -r (M(K ), y )
Direct argument
The following are equivalent
L(CM(F ), x) = L(CM(K ), y )
L(M(F ), x) = L(M(K ), y )
Basic Induction
L(F , x) = L(K , y ).
Basic Induction
(F , x) -g (K , y ).
35/53
HM property
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Coalgebraic homomorphism
Definition (homomorphism)
A map f from (A, α) to (B, β) is a homomorphism if the following
diagram commutes
X
α
f
FX
/Y
Ff
β
/ FY
If F = P, then coalgebraic homomorphism coincides with bounded
morphism on frames (a function that is a bisimulation).
36/53
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Equivalent multiframe homomorphism
If F = B, then we examine point by point:
x_ f
α
σx / f (x)
_
Bf
β
/ Bf (σ x ) = σ f (x)
Lifting σ x additively to sets, f is a coalgebraic homomorphism iff
for each x ∈ X , β(f (x)) = σ f (x) is given by
σ f (x) : y 7→ σ x (f −1 (y )).
The coalgebraic morphism is a function that is a multiframe
resource bisimulation.
37/53
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Coalgebraic bisimulation
Let F : Set → Set.
Definition (Coalgebraic bisimulation)
A relation R ⊆ A × B is a coalgebraic bisimulation between
F -coalgebras (A, α) and (B, β) if there is a morphism δ : R → FR,
such that the following commutes:
Ao
α
FA o
π1
F π1
R
π2
δ
FR
F π2
/B
β
/ FB
where π1 : R → A and π2 : R → B are projections that are
F -coalgebra morphisms.
States x ∈ A and y ∈ B are bisimilar, x -c y , if there exists a
bisimulation R between (A, α) and (B, β), such that (x, y ) ∈ R.
38/53
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Relationship between bisimulation and coalg. bisimulation
A coalgabraic bisimulation is a bisimulation
If F = P <∞ or F = B, then F -morphisms are bisimulations.
Hence R = π1 ◦ π2−1 is the composition of bisimulations and thus a
bisimulation.
A bisimulation is a coalgebraic bisimulation
If F = P and R is a bisimulation, then δ exists:
δ : (x, y ) 7→ (α(x) × β(y )) ∩ R.
If F = B and R is a multiframe resource bisimulation, then R
is a coalgebraic bisimulation via a matrix property
(for each (a, b) ∈ R, δ(a, b)(x, y ) is given by the (x, y )
coordinate of a |X | × |Y | matrix.)
39/53
Graded ML
40/53
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Coalgebraic bisimilarity and resource bisimilarity
Definition (Matrix property of a relation)
Let (A, Σ) and (B, T ) be multiframes. A relation R ⊆ A × B
satisfies the matrix property if for every (a, b) ∈ R, there exists an
|A| × |B|-matrix (mx,y ) with entries from N such that
1
all but finitely many mx,y are 0,
2
mx,y 6= 0 implies (x, y ) ∈ R,
P
for each x, σ a (x) = {mx,y | y ∈ B}, and
P
for each y , τ b (y ) = {mx,y | x ∈ A}.
3
4
R is a multiframe r -bisimulation iff matrix property holds
mx,y > 0 is the number of pairs ((x, n), (y , m)), such that
fa,b : (x, n) 7→ (y , m).
R is a coalgebraic bisimulation iff matrix property holds
δ(a, b) : (x, y ) 7→ mx,y
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Progress so far
The following are equivalent
(F , x) -r (K , y ).
(M(F ), x) -r (M(K ), y )
(CM(F ), x) -c (CM(K ), y )
Direct argument
Matrix property
The following are equivalent
L(CM(F ), x) = L(CM(K ), y )
L(M(F ), x) = L(M(K ), y )
Basic Induction
L(F , x) = L(K , y ).
Basic Induction
(F , x) -g (K , y ).
41/53
HM property
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Coalgebraic behavioral equivalence
Let F : Set → Set.
Definition (Behavioural equivalence)
Pointed F -coalgebras ((A, α), x) and ((B, β), y ) are behaviorally
equivalent, written ((A, α), x) -b ((B, β), y ), if there exist a
coalgebra (C , γ), and coalgebra homomorphisms f : A → C and
g : B → C , such that f (x) = g (y ).
(B, β)
(A, α)
f
$
z
(C , γ)
42/53
g
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Bisimulation and Behavioral Equivalence
Definition (Final coalgebra)
A final coalgebra is a coalgebra (A, α), such that for every
coalgebra (B, β) there is a unique coalgebraic homomorphism ϕ
from (B, β) to (A, α).
The finite multiset functor B has a final coalgebra.
Theorem
If two pointed coalgebras are bisimilar, then they are behaviorally
equivalent, so long as a final coalgebra exists.
For the functor B, if two two models are coalgebraically bisimilar
then they are behavioral equivalent.
43/53
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Preserving weak pullbacks
Definition (Weak pullback)
Given functions f : B → D and g : C → D, a weak pullback is a
pair of functions h : A → B and k : A → C , such that
g ◦ k = h ◦ f and whenever f (b) = g (c) for some b ∈ B and
c ∈ C , there exists an a ∈ A such that h(a) = b and k(a) = c.
This is depicted by a diagram called a weak pullback square:
A
h
k
B
/C
f
g
/D
Definition (Preserving weak pullbacks)
A functor F : Set → Set preserves weak pullbacks if the image of a
weak pullback square under F is also a weak pullback square.
44/53
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Finite multiset functor preserves weak pullbacks
The key step in proving that the Finite multiset functor preserves
weak pullbacks is to apply the following:
Theorem (Row-column theorem (integer version))
if p1 , . . . , pm , qP
1 , . . . , qn ∈ N are such that
P
p
=
1≤i≤m i
1≤j≤n qj , then for each 1 ≤ i ≤ m and 1 ≤ j ≤ n,
there exists rij ∈ N, such that
X
X
rij = pi , for 1 ≤ i ≤ m, and
rij = qj , for 1 ≤ j ≤ n.
1≤j≤n
45/53
1≤i≤m
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Coalgebraic bisimilarity and behavioral equivalence
Theorem
Behavioral equivalence implies bisimilarity when the functor
preserves weak pullbacks.
46/53
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Progress so far
The following are equivalent
(F , x) -r (K , y ).
(M(F ), x) -r (M(K ), y )
(CM(F ), x) -c (CM(K ), y )
(CM(F ), x) -b (CM(K ), y )
Direct argument
Matrix property
⇑ weak pullbacks; ⇓ final coalg
The following are equivalent
L(CM(F ), x) = L(CM(K ), y )
L(M(F ), x) = L(M(K ), y )
Basic Induction
L(F , x) = L(K , y ).
Basic Induction
(F , x) -g (K , y ).
47/53
HM property
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Behavioral equivalence and formulas
Definition (Separating)
A set {λnX }n∈N of predicate liftings is separating if, for every set X ,
any multiset σ ∈ B(X ) can be uniquely determined by the set
S(σ) = {(λnX , A) | n ∈ N, A ⊆ X , σ ∈ λnX (A)}.
Our set {λnX }n∈N is separating, since given x ∈ X and σ : X → N,
X = {(λnX , {x}) | σ ∈ λnX ({x})} = {(λnX , {x}) | n ≤ σ(x)} ⊆ S(σ)
is enough to recover σ (by σ = x 7→ max{n | (λnX , {x}) ∈ X}).
Theorem
If L is a B-coalgebraic modal logic using a separating set of
predicate liftings, then the logic is expressive (that is, if
L(X , s) = L(Y , t), then (X , s) -b (Y , t)).
48/53
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
homomorphisms preserving semantics
Definition
Homomorphisms preserve the semantics if for every homomorphism
f : X → Y (with X = (A, α), Y = (B, β)), formula ϕ, and a ∈ A,
(X , a) |= ϕ if and only if (Y , f (a)) |= ϕ.
A simple induction shows that homomorphisms preserve our
semantics.
49/53
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
behavioral equivalence preserves semantics
Suppose ((A, α), a) -b ((B, β), b). Then the following diagram
(A, α)
(B, β)
f
$
z
g
(C , γ)
commutes, and f (a) = g (b).
As homomophisms preserve the semantics,
L((A, α), a) = L((C , γ), f (a)) = L((B, β), b).
50/53
Graded ML
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Outline of proof
The following are equivalent
(F , x) -r (K , y ).
(M(F ), x) -r (M(K ), y )
(CM(F ), x) -c (CM(K ), y )
Direct argument
Matrix property
(CM(F ), x) -b (CM(K ), y ) ⇑ Weak pullbacks; ⇓ Final coalg
L(CM(F ), x) = L(CM(K ), y ) ⇑ Separating; ⇓ Hom pres sem
L(M(F ), x) = L(M(K ), y )
Basic Induction
L(F , x) = L(K , y ).
Basic Induction
(F , x) -g (K , y ).
51/53
HM property
Graded ML
52/53
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
Coalg bisim
Behavioral equiv
Related References
L. Goble. Grades of Modality. Logique Et Analyse, 1970.
Earliest work on graded modal logic.
K. Fine. In so many possible worlds. Notre Dame Journal of Formal
Logic, 1972.
Early work on graded modal logic.
M. de Rijke. A note on graded modal logic. Studia Logica, 2000
Graded bisimulation
F. Corradini, R. De Nicola, A. Labella. Models of nondeterministic
regular expressions Journal of Computer and System Sciences, 1999.
Research bisimulation
L. Aceto, A. Infgolfsdottir, J. Sack. Research Bisimilarity and
graded bisimilarity coincide. Information Processing Letters, 2010.
Gives the result presented in this talk.
K. Sano, M. Ma. Goldblatt-Thomason-style Theorems for Graded
Modal Language. AiML 2010.
Related work on graded modal logic
Graded ML
53/53
Coalgebraic ML
Bisimulations
Goal
HM-property
MF r-bisim
THANK YOU!
Coalg bisim
Behavioral equiv