Polymorphic Decision Diagrams Towards a Framework Generalizing Decision Diagrams
Alban Linard123 Emmanuel Paviot-Adet2 Fabrice Kordon2 Didier Buchs1 Samuel Charron3 1 University
of Geneva
2 LIP6 3 LRDE
24 june 2010
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Decision Diagrams?
1/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Decision Diagrams? r p
q
1/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Decision Diagrams? r p
q
State space ⇔ Function
p
,
p
×
q
,
q
×
r , r
→B
1/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Decision Diagrams? r p
q
State space ⇔ Function
p
,
p
×
q
,
q
×
r , r (
p q
×
(
r
×
(
7→
0
0
0
0
0
0
1
1
→
→B p q
, ,
p
)
q
)
r , r
)
B
1/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Decision Diagrams? (
p q
×
(
r
×
(
7→
0
0
0
0
0
0
1
1
→
p q
, ,
p
)
q
)
r , r
)
B
1/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Decision Diagrams? (
p q
×
(
r
×
(
7→
0
0
0
0
0
0
1
1
→
p q
, ,
p
)
q
)
r , r
)
B
DAG with maximal sharing q
r
1
q
r
0
p
1/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Decision Diagrams? (
p q
×
(
r
×
(
7→
0
0
0
0
0
0
1
1
→
p q
, ,
p
)
q
)
r , r
)
B
DAG with maximal sharing q
r
1
q
r
0
p
1/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Decision Diagrams? (
p q
×
(
r
×
(
7→
0
0
0
0
0
0
1
1
→
p q
, ,
p
)
q
)
r , r
)
B
DAG with maximal sharing q
r
1
q
r
0
p
1/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Decision Diagrams? (
p q
×
(
r
×
(
7→
0
0
0
0
0
0
1
1
→
p q
, ,
p
)
q
)
r , r
)
B
DAG with maximal sharing q
r
1
q
r
0
p
1/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Decision Diagrams? (
p q
×
(
r
×
(
7→
0
0
0
0
0
0
1
1
→
p q
, ,
p
)
q
)
r , r
)
B
DAG with maximal sharing q
r
1
q
r
0
p
1/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Decision Diagrams? (
p q
×
(
r
×
(
7→
0
0
0
0
0
0
1
1
→
p q
, ,
p
)
q
)
r , r
)
B
DAG with maximal sharing q
r
1
q
r
0
p
1/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Decision Diagrams? (
p q
×
(
r
×
(
7→
0
0
0
0
0
0
1
1
→
p q
, ,
p
)
q
)
r , r
)
B
DAG with maximal sharing q
r
1
q
r
0
p
1/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Decision Diagrams? (
p q
×
(
r
×
(
7→
0
0
0
0
0
0
1
1
→
p q
, ,
p
)
q
)
r , r
)
B
DAG with maximal sharing q
r
1
q
r
0
p
1/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Decision Diagrams? (
p q
×
(
r
×
(
7→
0
0
0
0
0
0
1
1
→
p q
, ,
p
)
q
)
r , r
)
B
DAG with maximal sharing q
r
1
q
r
0
p
1/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Decision Diagrams? (
p q
×
(
r
×
(
7→
0
0
0
0
0
0
1
1
→
p q
, ,
p
)
q
)
r , r
)
B
DAG with maximal sharing q
r
1
q
r
0
p
1/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Decision Diagrams? (
p q
×
(
r
×
(
7→
0
0
0
0
0
0
1
1
→
p q
, ,
p
)
q
)
r , r
)
B
DAG with maximal sharing q
r
1
q
r
0
p
1/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Decision Diagrams? (
p q
×
(
r
×
(
7→
0
0
0
0
0
0
1
1
→
p q
, ,
p
)
q
)
r , r
)
B
DAG with maximal sharing q
r
1
q
r
0
p
1/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Decision Diagrams? (
p q
×
(
r
×
(
7→
0
0
0
0
0
0
1
1
→
p q
, ,
p
)
q
)
r , r
)
B
DAG with maximal sharing q
r
1
q
r
0
p
1/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Decision Diagrams? (
p q
×
(
r
×
(
7→
0
0
0
0
0
0
1
1
→
p q
, ,
p
)
q
)
r , r
)
B
DAG with maximal sharing q
r
1
q
r
0
p
1/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
lat ion
tio n
ma nip u
nta rep res e
ing od enc Problem
Function
DD
Solution
2/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
lat ion
tio n
ma nip u
nta rep res e
ing od enc Problem
Function
DD
Solution
2/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
ma nip u tio
ma nip u
ent a rep res
Function
Solution
lat ion
n
DD
ing od enc
lat ion
tio n nta rep res e
ing od enc
Function
Problem
DD
cho
ice
Problem
Solution
DD category
2/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
ma nip u tio
ma nip u
ent a rep res
Function
Solution
lat ion
n
DD
ing od enc
lat ion
tio n nta rep res e
ing od enc
Function
Problem
DD
cho
ice
Problem
Solution
DD category
2/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
3/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Lack of unifying description for DD categories!
In this talk: only DDs with results in terminals (BDD, ZDD, ADD, MDD, DDD, etc.) not Edge-Valued DDs
3/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Function’s characteristics Fixed number of variables Finite variable domains
{x:B; y :B; z:B}→ B
Finite codomain
4/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Function’s characteristics Fixed number of variables {x:B; y :B; z:B}→ B
Finite variable domains Finite codomain Example 0
y
1
z
0, 1
0
0
x
1 1
y 0, 1
z
0
1
4/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Example 0
y
1
z
0, 1
0
0
x
1 1
y 0, 1
z
0
1
Category description x
hx, Bi 7→ y
hy , Bi 7→ z
hz, Bi 7→ τ B
4/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Example 0
1
y
z
0, 1
0
0
x
1 1
y
z
0, 1
0
1
Category description x
hy , Bi 7→ z
hz, Bi 7→ τ B
hx, Bi 7→ y
4/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Example 0
y
1
z
0, 1
0
0
x
1 1
y 0, 1
z
0
1
Category description x
hz, Bi 7→ τ B
(
hx, Bi 7→ y hy , Bi 7→ z
4/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Example 0
y
1
z
0, 1
0
0
x
1 1
y 0, 1
z
0
1
Category description x
hx, Bi 7→ y hy , Bi 7→ z hz, Bi 7→ τ 4/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Example 0
1
y
z
0, 1
0
0
x
1 1
y
z
0, 1
0
1
Category description first
(
hi, B, succ(i)i hlast, B, τ i
when i 6= last
i ∈ [x, y , z] Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
4/11
Function’s characteristics Infinite variable domains
{x:B; y :N} → B
5/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Function’s characteristics {x:B; y :N} → B
Infinite variable domains Example 0
y
x 1
{n ∈ N | n is even } {n ∈ N|n is o d d}
y
0
1 N
5/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Function’s characteristics {x:B; y :N} → B
Infinite variable domains Example y
0 x 1
{n ∈ N | n is even } {n ∈ N|n is o d d}
y
0
1 N
Category description hx, Bi 7→ y
hy , Ni 7→ z B
5/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Function’s characteristics | A of {x1 :B}
Discriminant
| B of {x2 :{0, 1, 2}; y :{0, 1, 2}}
→B
6/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Function’s characteristics | A of {x1 :B}
Discriminant
| B of {x2 :{0, 1, 2}; y :{0, 1, 2}}
→B
Example à faire
Category description
d
hd , {A, B}i 7→ x1
hx1 , Bi 7→ x2
hx2 , {0, 1, 2}i 7→ y hy , {0, 1, 2}i 7→ τ B 6/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Function’s characteristics | A of {x1 :B}
Discriminant
| B of {x2 :{0, 1, 2}; y :{0, 1, 2}}
→B
Example à faire
Category description
d
hd , {A, B}i 7→ x1
lehxs,!{0, 1, 2}i 7→ y b a i var
hx1 , Bi 7→ x2
ed
s Unu
2
hy , {0, 1, 2}i 7→ τ B 6/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Function’s characteristics | A of {x1 :B}
Discriminant
| B of {x2 :{0, 1, 2}; y :{0, 1, 2}}
→B
Example à faire
Category description
d
hd , {A, B}i 7→ x
hx, {0, 1, 2}i 7→ y
hy , {0, 1, 2}i 7→ τ B
6/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Function’s characteristics | A of {x1 :B}
Discriminant
| B of {x2 :{0, 1, 2}; y :{0, 1, 2}}
→B
Example à faire
Category description
d
ns! i a hd , {A, B}i 7→ x hx, {0, 1, 2}i 7→m o y hy , {0, 1, 2}i 7→ τ d rge a l To o
B
6/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Function’s characteristics | A of {x1 :B}
Discriminant
| B of {x2 :{0, 1, 2}; y :{0, 1, 2}}
→B
Example à faire
Category description ( d
hd , {A}i 7→ x1 hd , {B}i 7→ x2
hx1 , Bi 7→ τ hx2 , {0, 1, 2}i 7→ y
hy , {0, 1, 2}i 7→ τ B
6/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Function’s characteristics | A of {x1 :B}
Discriminant
| B of {x2 :{0, 1, 2}; y :{0, 1, 2}}
→B
Example à faire
Category description
d
hd
hx1 , Bi 7→ τ
x1 }i 7→ , {A
B
hd , { B }i
7→ x
2
hx2 , {0, 1, 2}i 7→ y
τ }i 7→ 2 , 1 0, hy , { 6/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Function’s characteristics →B
Several codomains
| A of {x1 :B}
Infinite codomains
| B of {x2 :{0, 1, 2}; y :{0, 1, 2}}
→N
7/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Function’s characteristics →B
Several codomains
| A of {x1 :B}
Infinite codomains
| B of {x2 :{0, 1, 2}; y :{0, 1, 2}}
→N
Example à faire
7/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Function’s characteristics →B
Several codomains
| A of {x1 :B}
Infinite codomains
| B of {x2 :{0, 1, 2}; y :{0, 1, 2}}
→N
Example à faire
Category description
d
hd
x1 }i 7→ , {A
hd , {
B }i
hx1 , Bi 7→ τ B
7→ x
2
hx2 , {0, 1, 2}i 7→ y
hy , {0, 1, 2}i 7→ τ
N 7/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Function’s characteristics l → B where Recursive domain
type l = Nil | Cons of {x:B; n:l }
8/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Function’s characteristics l → B where Recursive domain
type l = Nil | Cons of {x:B; n:l }
Example Representation of { [1], [1;1], [1;0], [0;0], [0;1] }
Nil 1 l
Cons Nil
l
Cons
x
1 {0, 1}
x
l 0
l
Cons
x
Nil Cons
x
{0, 1}
...
{0, 1} 0
Nil
8/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Example Representation of { [1], [1;1], [1;0], [0;0], [0;1] }
Nil 1 l
Cons Nil
l
Cons
1
x
{0, 1}
x
l 0
l
Cons
Nil Cons
x
{0, 1}
...
{0, 1}
x
0
Nil
Category description l B
hx, Bi 7→ l
hl , {Nil }i 7→ τ hl , {Cons}i 7→ x 8/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Example Representation of { [1], [1;1], [1;0], [0;0], [0;1] }
Nil 1 l
Cons Nil
l
Cons
1
x
{0, 1}
x
l 0
l
Cons
Nil Cons
x
{0, 1}
...
?
{0, 1}
x
0
Nil
Category description l B
hx, Bi 7→ l
hl , {Nil }i 7→ τ hl , {Cons}i 7→ x 8/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Example Representation of { [1], [1;1], [1;0], [0;0], [0;1] }
Nil 1 l
Cons Nil
l
x 0
l
Cons
1
x
{0, 1}
Redx
Cons
s!
ion t c u
l
Nil Cons
x
{0, 1}
...
?
{0, 1}
0
Nil
Category description l B
hx, Bi 7→ l
hl , {Nil }i 7→ τ hl , {Cons}i 7→ x 8/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
9/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
10/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
11/11
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
Bibliography
Linard, Paviot-Adet, Kordon, Buchs, Charron
Polymorphic Decision Diagrams
