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Integration of Ontological Bayesian Logic Programs

in Deductive Knowledge Systems


Zoran Majkic


Computer Science Dept.

University of Maryland

College Park

September 2005

Motivation:

Biological Ontologies


Data Bases and Ontologies



Information retrieval from
unstructured

Biomedical data



Genome databases and
Annotation



Biological Data Integration


Biological Data warehouse


Biological Datamining



Representation of
sequence

data and
functional
info




Experimental facts and
aggregations





Relational
datamining





Motivation:

Semantic Web for Genomics

1.
Major chalenge for post genomic era


2.
Formal Ontologies
:

provide consensus representation of bioinformatics data


3.
Multi agent systems


modular middleware:

Web services


4.
Advanced relational data mining:

generate probabilistic knowledge



5.
Automated access to specific units of information


From Bayesian Networks to
Relational Databases

Bayesian Network

Bayesian Clause

Ontological Bayesian Programs

Relational Data base


A A
1
,….

A
n

A A
1
,….

A
n

[Majkic 2004, 2005]

Bayesian Network:


Genetics and Probability


Has a probabilistic nature given by the biological

laws
of inheritance


Requires the representation of the
relational

familiar

structure

of the objects under study


A
qualitative

component


acyclic influence graph
among the random variables


A
quantitative

component that ecodes Probability
density over these local influences

Genetics:

Bayesian Network:

Example
:
Individual’s phenotype

“each individual ha a polygenic value, or polygenotape, which in the population is

normally

(Gaussian density)

distributed


“each gene independently effects additive changes of the
phenotype


(ex. height of a person)

Values of phenotype when the number of underlying genes increases:

Example
: Probability Density

apriory density

Aposteriory density

(m + f) / 2 = 168.5

175

Example
: Dependency graph

Inheritance of height :

f = 173


m = 164

(m+f)/2 = 168.5

Bayesian Clause:
A A1 ,….., An


Atoms:

A = p(t1,…, t_k),


with

Dom(p)

different from true values

2 =
(true, false).


Ground atoms

=
random variables

in Bayesian network



Symbol:

complex probability distribution

operation

is not

logic

implication

Example:

“blood
-
type

bt

of a person

X
depends on the inherited information
of

X”

Each person

X
has two copies of the chromosome containing gene,

mc(Y), pc(Z),
inherited from her mother

m(Y,X)
and father


f(Z,X)
:


bt(X) mc(X, pc(X)

With

Dom(bt) = a, b, ab, 0
,

Dom(mc) = Dom(pc) = a, b, 0 .

Bayesian Clause
:
probabilistic model

Conditional probability distribution of a clause c

m(ann, dorothy) , f(brian, dorothy), pc(ann), pc(brian), mc(brian), mc(ann)

Bayesian program:

mc(X) m(Y,X) , mc(Y) , pc(Y)

pc(X) f(Y,X) , mc(Y) , pc(Y)

bt(X) mc(X) , pc(X)


Herbrand models ?


Logic:

Herbrand model

I: H 2 , 2
=
(true, false).


Example:
I(m(ann, dorothy)) = true, I(m(dorothy, ann))= false




Problem:

Bayesian model

I: H W , W
is not

2.


Example:
I(mc(ann)) in
W

= a, b, 0 , or higher types,


I(bt(dorothy)) in W = F(x,y) : x,y in a,b,0




Solution:

Higher
-
order

Herbrand model


type

I
abs
: H 2
W
,
with
I
abs
(A): W 2,
and for

any

w

in

W,




I
abs
(A)(
w
) =
true

if and only if

I(A) =
w



Program transformation:
flattening


Higher
-
order

Herbrand interpretation



A type T denotes a functional space



Hidden

parameters





Transformation

of Atoms





Flattened

interpretation with


for any








Example

: for

Example:

Flattening


Higher
-
order

Herbrand model


type

I
abs
: H T,

where



T = ( 2
W
2
)
W
1

with
W
1

=Dom(bt) =

a, b,

ab, 0 ,
W
2

= [0,1].



m(ann, dorothy) ………………. bt(dorothy)




=
2 + ……………. + ( 2
W
2
)
W
1



Transformation:
bt(X) bt
F
(X, w
1
, w
2
)



Ontological

Bayesian Program


Two
-
valued

logic program


Unique

Herbrand model
I
F

: H
F

2


Example:



for the case when



we obtain

Advantages:
full integration


More expressive

Bayesian environment:

1.
We can use
negation:


2.
We can use

constraints:


Full integration with
Relational Databases

and Deductive
Databases: Standard Query Language

Common Ontology DB

References


Z. Majkic,

Ontological encapsulation of many
-
valued logic,

19th Italian Symposium
of Computational Logic (CILC04), June 16
-
17, Parma, Italy, 2004



Z.Majkic,

Constraint Logic Programming and Logic Modality for Event's Valid
-
time
Approximation,



2nd Indian
International Conference on Artificial Intelligence
(
IICAI
-
05
),


December 20
-
22, 2005, Pune, India.


Z.Majkic,


Beyond Fuzzy: Parameterized approximations of Heyting algebras for
uncertain knowledge
,


2nd Indian
International Conference on Artificial
Intelligence (
IICAI
-
05)
,


December 20
-
22, 2005, Pune, India.



Z. Majkic,

Kripke Semantics for Higher
-
order Herbrand Model Types
,


Technical
Report, 2005

, College Park, University of Maryland.

Thank you !

Any question ?