Normal Symbolic Form
Marc CSERNEL
Paris IX Dauphine
Inria(Axis)
Supported by the French embassy in Brazil with the help of
the cultural and technical services in RECIFE
Recife Nov 2004
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outlines
•
•
•
•
•
•
•
•
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Symbolic description with rules
Comparison of S.D
Notion of coherence
Description Potential
Normal Symbolic Form (NSF)
Computation using N.S.F.
Problem of memory growth
The effective Growth
Perspectives et Conclusion
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Symbolic descriptions
Classical
individuals
Symbolic
Descriptions
•
•
•
•
•
Color
Size
Beatles 1
Blue
8.0
Beatles 2
Red
10.0
color
size
specie1
{Blue,Red}
[7:12]
specie2
{Yellow}
[8:9]
S.D are well adapted to describes species, classes, collections .
S.D. allows to take into account variability.,uncertainty….
A Symbolic Description represent an intention
Usually the extension of a S.D. is made from usual individual
S.D are a good instrument to summarize information
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Background Knowledge
Background Knowledge can be introduced as rules
• Two kinds of rules
– hierarchical (mother-daughter)
if wings  {absent} then wings_colour = NA
– logical:
if wings_colour  {red} then Thorax_colour  {blue}
• Dependencies reduce the description space
– they introduces holes in the description space
• But hierarchical dependencies reduce also the number of
dimensions.
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NA semantic
• NA means that the variable is not applicable, (or
non-existent or meaningless) if the premise is true.
• NA should not be considered as a value, but for
conveniences we denote: var = NA.
– remark 1: hierarchical rules induce a kind of
inheritance.
– remark 2: As NA is not a possible value of a variable,
if a variable is Not Applicable, it can’t have any value.
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The Importance of the rules
Similarity problem
if we consider two symbolic descriptions:
x = [a {a1, a2, a3, a4}] ^ [b {b1, b2, b3, b4}]
y = [a {a3, a4}] ^ [b {b2, b3, b4, b5}]
• if a  {a1,a2} then b = NA
a4
a4
a3
a3
a2
a2
a1
a1
b1 b2 b3 b4 b5
b1 b2 b3 b4 b5
the objects look more similar
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Problem of Identification
If we considers two objects
x = [a { a3, a4}] ^ [b { b2, b3 }]
y = [a {a2, a3}] ^ [b { b3, b4 }]
x full white
y grey shade
if a  {a3} then b  {b1,b2, b4 }
a4
a3
a2
a1
a4
a3
a2
a1
b1 b2 b3 b4
Recife Nov 2004
b1 b2 b3 b4
These objects cannot always be discriminated
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Our needs
•
•
•
•
Be able to compare S.D. contrained by rules
Make then, data analysis, data mining…..,
Specially distance computation
d(a,b)=  (ab)0.5( (a) (b))
 (ab)
• where  is the join operator
• (a) the description potential
a
b
of a
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The Coherence
• An individual is coherent if it's description respect the rules.
• The coherent part of an S.D. is the part of the description which
respect the rules.
• An S.D. is coherent if a coherent part exist.
• An S.D. is fully coherent when all the H-volume it describe is
coherent.
– If wing {Absent} then wings_colour = NA
wings
Wings-color
d1
{absent}
{blue,yellow,red}
d2
{absent,present}
{blue,yellow,red}
d3
{present}
{blue,yellow,red}
• d1 is not coherent, d2 is coherent, d3 is fully coherent
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The Description Potential
• Description Potential: the measure of the COHERENT part of the
volume described by a symbolic description.
• The introduction of dependence rules changes the potential of an
description.
• The Computation of the Description Potential is combinatorial.
r1
r2
D
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combinatorial aspect of D.P
x : {a1,a2}{b1,b2}{c1,c2}{d1,d2}
Potential without rules = 2x2x2x2 = 16
if a  {a1} then b  { b1 } ;(r1) if c  {c1 } then d  { d1 } ;(r2)
a1 b1 c1 d1
a1 b1 c1 d2
a1 b1 c2 d1
a1 b1 c2 d2
a1 b2 c1 d1
a1 b2 c1 d2
a1 b2 c2 d2
a1 b2 c2 d2
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Y
N(r2)
Y
Y
N (r1)
N(r1,r2)
N(r1)
N (r1)
a2 b1 c1 d1
a2 b1 c1 d2
a2 b1 c2 d1
a2 b1 c2 d2
a2 b2 c1 d1
a2 b2 c1 d2
a2 b2 c2 d1
a2 b2 c2 d2
Normal Symbolic Form
Y
N(r2)
Y
Y
Y
N(r2)
Y
Y
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computation of D.P.
without dependencies
p
 a   Ai 
i1
  Ai



 

 


cardinal A , if y is discrete
i
 i
Range A , if y is continu
i
 i
where Range(Ai) is the absolute value of
the difference between the upper bound
and the lower bound of interval Ai.
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computation of D.P.
with dependencies
d =  [yi  Ai] be a S.D. and rj  {r1, , rt} a rule
t
  (d rj)
j1
   ((d  r j )  r ) 
k
jk
p
 (d / r  rt )    (Di )
1
i1
(1)t 1 ((d  r )  r )  rt )
1
2
complexity:- Exponential according to the number of rules
- Linear according to the number of variables
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Our Aim
• Our aim is to represent only the valid part of a
S.D. (fully coherent)
– We have to split the description space in different
subspaces.
– Each subspace will correspond to one premise
variable, and all conclusion Variables linked with.
– In subspace will be cut in different slices where all
the values of the premise variable lead to the same
conclusion
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Normal Symbolic Form
• If Hand {Absent} then Hand_Size= N.A.
• If Hand {Absent} then Finger = N.A.
• If Finger  {Absent} then Finger_size  N.A.
d1
d2
hand
{absent, present}
{absent, present}
hand_size
{big,midle}
{big,small}
finger
finger_size …
{absent, present} {big, small} …
.
{ present}
{small}
Hand
Finger
Hand_size
Finger_Size
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N.S.F.
Hand
Hand_size
d1 {absent, present} {big,midlle}
d2 {absent, present} {big,small}
d1
d2
Main table
N°
1
2
3
Hand
Hand_size
{present } {big,middle}
{present} {big,small}
{absent}
N.A.
Hand
{ 1,3}
{2,3}
finger
{absent, present}
{ present}
..
….
….
finger_size …
{big, small} …
.
{small}
Secondary tables
finger_table
{1,3}
{2}
N.A.
N°
1
2
3
finger
{ present}
{present}
{absent}
finger_size
{big,small}
{ small}
{NA}
3 tables but NO MORE RULES
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Normal Symbolic Form
• First NSF condition:
– If no dependency occurs between the variables, or if a dependency
occurs between the first variable V1 and the others
• Second NSF condition
– If all values expressed for one object by the premise variable V1
leads to the same conclusion
• Valid Only if the rules form a tree or a set of tree (first
condition)
• inspired by Codd's Normal Form in Databases
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Consequences
Two consequences:
• Cut the data tables into different tables according to the
dependence tree
– possible only if the dependence form a tree or a forest
• Cut each symbolic description in two parts (CQ2)
– One where the premise is true
– One where the premise is false
If Finger {Absent} then Finger_Size  N.A.
Finger
Finger_Size
{absent,present}
{big,small}
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Normal Symbolic Form
finger
Finger_size
{present}
{big,small}
{absent}
N.A.
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The dependence tree
• For each rule we draw an edge from the
premise variable to the conclusion
variable
• Each node correspond to one variable
• Each node can be linked to more than a
rule
• Each node and his son correspond to a
secondary table
• Y1 form a table with Y2,Y3,Y4
• Y2 form a table with Y5 and Y6
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Normal Symbolic Form
y1
y2
y5
y3
y4
y6
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Potential computation with N.S.F
(d1) = (ta(1) +ta(3))*…= (6+1)*….
ta(1) = 1*2*(tc(1) +tc(3)) = 1*2*(1 +2) = 6
ta(3) = 1*1 = 1
(a1) = 2 * (ta(1) +ta(2)) = 2*(3+2) = 10
7*..
d1
d2
Hand
{ 1,3}
{2,3}
N°
Hand
Hand_Size Finger_Table
6 1 {present } {big,med}
{1,3}
2 {present} {big,small}
{2}
{absent}
N.A.
N.A.
1 3
ta
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..
….
….
2
1
N°
Finger
1 { present}
2 {present}
3
{absent}
Finger_size
{big,small}
{ small}
{NA}
tc
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Memory Growth
(1st approach)
• CQ2=> size double for each node of the dependence tree
N
T: nb of premise variables, N: nb of descriptions, S: size of the biggest
secondary table
• S = < N*2D (D depth of the tree)
• tree well balanced then D = Log2 (T)
N*22
N*23
S= N*2 Log2 (T) = N*T => POLYNOMIAL
N*24
• tree not balanced (worst case) then S = N*2T
N*25
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Normal Symbolic Form
N*2T
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Data Factorisation
wings
d1 {absent, present}
d2 {absent, present}
d3
{absent}
d1
d2
d3
wings
Color
1
2
{pres }
{abs}
{1,2}
{4}
3
{pres}
{1,3}
wings_color
Thor_col
Thorax_size
{red, blue} {blue, yellow} {big, small}
{red, green}
{blue, red}
{small}
NA
{ blue,yellow } {med,small}
Wings...
Thorax_Size
{ 1, 2}
{2,3}
{2}
{big, small}
{small}
{med,small}
color
Wings_co
Thorax_Col
1
2
3
4
{ red}
{blue}
{green}
NA
{blue}
{ blue, yell}
{blue,red}
{ blue, yell }
if wings  {Absent} then wings_colour = NA.
if wings_colour {red} then Thorax_colour {blue}.
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usual operations with N.S.F.
•
•
no changes BUT recursion due to numerous
tables (following table's tree)
Two kinds of operation
–
–
Creating a new Volume (join, union..)
Restriction of a existing volume (intersection..)
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The Join (without N.S.F.)
An Operation Creating a new volume
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operations with N.S.F.
(creating a new volume)
OK
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Suicide ??
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The Bounds
We will first consider nominal variables and hierarchical
rules.
2 Cases:
- Locally : between mother and daughter
- Globally : between the root and a leaf
Sn: size according to CQ2
Sv : size according the number of
possible different descriptions
Ndaughter, Nmother, Fl = Nd/Nm
Flocal 
N daughter
N mother
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min S n , Sv 

2
N mother
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One set of premise Value
• One premise variable y divided in two set of values
A and A
• Locally
– m conclusion variables x1…xm
– Nmother,Ndaugther : tables sizes
– according to CQ2:
Sn = 2* Nm
– according to the variables domain


m
Xj



Sv  2  1   2  1  2  1


 j 1 
A
A
– Flocal <= 2
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Globally
-Line with NA can not refer
to secondary tables
N
A
A
-Fglobal <= 2
-There is no following
element for A
-No global growing
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Normal Symbolic Form
A1
A1
N
N
N
N
28
More than 1set of premise values
• Until now 1 set of premise value
• But more than one (n) can exist
if travel  {plane} then car = N.A.
if travel  {plane} then train = N.A.
if travel  {car} then plane = N.A ….
N

F N
D
l
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M

min T n,T d 
N
n
M
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Tables Occupation
N Items to
Put
I31,I37,I70
I10,I20
m possible
descriptions
(places)
I15,I27
• how many busy places ?
• Statistic Maxwell Boltzman in physic
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Tables Occupation (without rules)
N initial Individuals, m possible descriptions
if d1, , dj, , dpvariables
m = P(X1) -   P(Xj) -   P(Xp) - 
• With independent variables and equirepatition
p
p ( d1 ,...d n )  
i 1
1

P ( Xi)  
1
 2
p
i 1
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Normal Symbolic Form
Xi

1
31
Tables occupation with rules
• 2 tables are generated
– one where the rules did NOT apply
– one where the rules did apply
(T1)
(T2)
• For T2 preceding problem
• T1 size :
– small compare with N=> T1 mostly full
– => factorisation
• the size will be growing according to
• greater
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P( A)
P(A)
A is greater the factorisation will be
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Application
• Phlebotomies (Shadflies)
– 73 species (descriptions)
– 53 nominal variables
– 5 rules in 3 different trees with 8 variables
• Fg = Fl
• Secondary tables
– 32 lines (56)
– 18 lines (39)
– 16 lines (30)
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 Fl = 32/73 = 0.438
 Fl = 18/73 = 0.246
 Fl = 16/73 = 0.219
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About complexity
•
Size
–
–
•
Over cost with references variable
Over cost induced by the possible growing factor F
Computation
–
–
–
–
the related to the rules disappeared
On over cost due to the NSF transformation
appears (N2 like a sort) but only Once
A minor over cost (linear) appears with some
operation (like the join)
Over cost due to the recursion
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perspectives and conclusions
• Our work arrive to a Mature point
• But still uncompleted
– Accept dependence graph instead of
dependence tree
– Adapt algorithm to N.S.F
– First distances and comparisons
– Clustering, factorial analysis …..
• Made simulation studies
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