1. Zoran Majkic Integration of Ontological Bayesian Logic Programs
in Deductive Knowledge Systems
Zoran Majkic
Computer Science Dept.
University of Maryland
College Park
September 2005

2. Motivation: Biological Ontologies
Data Bases and Ontologies
Information retrieval from unstructured Biomedical data
Genome databases and Annotation
Representation of sequence data and functional info
Biological Data Integration
Biological Data warehouse
Experimental facts and aggregations
Biological Datamining
Relational datamining

3. Motivation: Semantic Web for Genomics
Major chalenge for post genomic era
Formal Ontologies: provide consensus representation of bioinformatics data
Multi agent systems – modular middleware: Web services
Advanced relational data mining: generate probabilistic knowledge
Automated access to specific units of information

4. From Bayesian Networks to Relational Databases
Bayesian Clause
A A1 ,…. An
[Majkic 2004, 2005]
Ontological Bayesian Programs
A A1 ,…. An
Relational Data base

5. 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
Bayesian Network:
A qualitative component – acyclic influence graph among the random variables
A quantitative component that ecodes Probability density over these local influences

6. 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:

7. Example: Probability Density
apriory density
175
Aposteriory density
(m + f) / 2 = 168.5

8. Example: Dependency graph
f = 173
Inheritance of height :
(m+f)/2 = 168.5
m = 164

9. Bayesian Clause: A A1 ,….., An
Atoms: A = p(t1,…, t_k), with Dom(p) different from true values = (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 .

10. 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)

11. 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, , or higher types,
I(bt(dorothy)) in W = F(x,y) : x,y in a,b,0
Solution: Higher-order Herbrand model type Iabs: H W, with Iabs(A): W 2, and for any w in W,
Iabs (A)(w) = true if and only if I(A) = w

12. 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

13. Example: Flattening
Higher-order Herbrand model type Iabs: H T, where
T = ( 2 W2 )W1 with W1 =Dom(bt) = a, b, ab, 0 , W2 = [0,1].
m(ann, dorothy) ……………… bt(dorothy)
= …………… ( 2 W2 )W1
Transformation: bt(X) btF(X, w1, w2)

14. Ontological Bayesian Program
Two-valued logic program
Unique Herbrand model IF : HF
Example:
for the case when
we obtain

15. Advantages: full integration
More expressive Bayesian environment:
We can use negation:
We can use constraints:
Full integration with Relational Databases and Deductive Databases: Standard Query Language
Common Ontology DB

16. 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.

17. Thank you !
Any question ?