Download Integration of Ontological Bayesian logic programs in

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
Genome databases and
Annotation



Biological Data Integration

Biological Data warehouse

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Biological Datamining

Information retrieval from
unstructured Biomedical data
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:
3.
Multi agent systems – modular middleware:
4.
Advanced relational data mining:
5.
Automated access to specific units of information
provide consensus representation of bioinformatics data
Web services
generate probabilistic knowledge
From Bayesian Networks to
Relational Databases
Bayesian Network
Bayesian Clause
A A1 ,…. An
[Majkic 2004, 2005]
Ontological Bayesian Programs
Relational Data base
A A1 ,…. An
Bayesian Network:
Genetics and Probability
Genetics:


Has a probabilistic nature given by the biological laws
of inheritance
Requires the representation of the relational familiar
structure of the objects under study
Bayesian Network:

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A qualitative component – acyclic influence graph
among the random variables
A quantitative component that ecodes Probability
density over these local influences
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
Aposteriory density
175
(m + f) / 2 = 168.5
apriory density
Example: Dependency graph
Inheritance of height :
f = 173
(m+f)/2 = 168.5
m = 164
Bayesian Clause:

Atoms: A = p(t1,…, t_k),
2 = (true, false).
A A1 ,….., An
with Dom(p)
different from true values
Ground atoms = random variables in Bayesian network

Symbol:
Example:
complex probability distribution operation
is not logic implication
“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)
With Dom(bt) =
mc(X, pc(X)
a, b, ab, 0 , Dom(mc) = Dom(pc) = a, b, 0 .
Bayesian Clause: probabilistic model
Conditional probability distribution of a clause c
Bayesian program:
m(ann, dorothy) , f(brian, dorothy), pc(ann), pc(brian), mc(brian), mc(ann)
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
Example:

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2,
2
= (true, false).
I(m(ann, dorothy)) = true, I(m(dorothy, ann))= false
Problem:
Example:
I: H
Bayesian model
I: H
W , W is not 2.
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
with
Iabs(A): W
2,
and
Iabs (A)(w) = true
if and only if
type
for any
Iabs: H
w in W,
I(A) = w
2W,
Program transformation: flattening

Higher-order Herbrand interpretation
A type T denotes a functional space

Hidden parameters

Transformation of Atoms

Flattened interpretation
for any
Example : for
with
Example: Flattening

Iabs: H
Higher-order Herbrand model type
T =(2
W )W
2
1
with W1 =Dom(bt) =
m(ann, dorothy)
=

Transformation:
2
bt(X)
+
T,
where
a, b, ab, 0 , W2 = [0,1].
……………….
bt(dorothy)
……………. + ( 2
btF(X, w1, w2)
W )W
2
1
Ontological Bayesian Program
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Two-valued logic program
Unique Herbrand model
Example:
for the case when
we obtain
IF : HF
2
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
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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 ?