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**Zoran Majkic Integration of Ontological Bayesian Logic Programs**

in Deductive Knowledge Systems Zoran Majkic Computer Science Dept. University of Maryland College Park September 2005

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**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

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**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

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**From Bayesian Networks to Relational Databases**

Bayesian Clause A A1 ,…. An [Majkic 2004, 2005] Ontological Bayesian Programs A A1 ,…. An Relational Data base

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**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

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**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:

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**Example: Probability Density**

apriory density 175 Aposteriory density (m + f) / 2 = 168.5

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**Example: Dependency graph**

f = 173 Inheritance of height : (m+f)/2 = 168.5 m = 164

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**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 .

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**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)

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**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

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**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

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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)

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**Ontological Bayesian Program**

Two-valued logic program Unique Herbrand model IF : HF Example: for the case when we obtain

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**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

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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.

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