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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

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Motivation: Biological Ontologies Data Bases and Ontologies Data Bases and Ontologies Information retrieval from unstructured Biomedical data Information retrieval from unstructured Biomedical data Genome databases and Annotation Genome databases and Annotation Biological Data Integration Biological Data Integration Biological Data warehouse Biological Data warehouse Biological Datamining Biological Datamining Representation of sequence data and functional info Representation of sequence data and functional info Experimental facts and aggregations Experimental facts and aggregations Relational 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 Network Bayesian Clause Ontological Bayesian Programs Relational Data base A A 1,…. A n [Majkic 2004, 2005]

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Bayesian Network: Genetics and Probability Has a probabilistic nature given by the biological laws of inheritance Has a probabilistic nature given by the biological laws of inheritance Requires the representation of the relational familiar structure of the objects under study Requires the representation of the relational familiar structure of the objects under study A qualitative component – acyclic influence graph among the random variables A qualitative component – acyclic influence graph among the random variables A quantitative component that ecodes Probability density over these local influences A quantitative component that ecodes Probability density over these local influences Genetics: Bayesian Network:

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Example: Individuals 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 Aposteriory density (m + f) / 2 =

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

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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.

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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)

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Herbrand models ? Logic: Herbrand model I: H 2, 2 = (true, false). Logic: Herbrand model I: H 2, 2 = (true, false). Example: I(m(ann, dorothy)) = true, I(m(dorothy, ann))= false Example: I(m(ann, dorothy)) = true, I(m(dorothy, ann))= false Problem: Bayesian model I: H W, W is not 2. Problem: Bayesian model I: H W, W is not 2. Example: I(mc(ann)) in W = a, b, 0, or higher types, 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 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, 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 I abs (A)( w ) = true if and only if I(A) = w

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Program transformation: flattening Higher-order Herbrand interpretation Higher-order Herbrand interpretation A type T denotes a functional space A type T denotes a functional space Hidden parameters Hidden parameters Transformation of Atoms Transformation of Atoms Flattened interpretation with Flattened interpretation with for any for any Example : for

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Example: Flattening Higher-order Herbrand model type I abs : H T, where 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]. T = ( 2 W 2 ) W 1 with W 1 =Dom(bt) = a, b, ab, 0, W 2 = [0,1]. m(ann, dorothy) ………………. bt(dorothy) m(ann, dorothy) ………………. bt(dorothy) = 2 + ……………. + ( 2 W 2 ) W 1 = 2 + ……………. + ( 2 W 2 ) W 1 Transformation: bt(X) bt F (X, w 1, w 2 ) Transformation: bt(X) bt F (X, w 1, w 2 )

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Ontological Bayesian Program Two-valued logic program Two-valued logic program Unique Herbrand model I F : H F 2 Unique Herbrand model I F : H F 2 Example: Example: for the case when for the case when we obtain we obtain

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Advantages: full integration More expressive Bayesian environment: 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 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, Ontological encapsulation of many-valued logic, 19th Italian Symposium of Computational Logic (CILC04), June 16-17, Parma, Italy, 2004Ontological encapsulation of many-valued logic,Ontological encapsulation of many-valued logic, 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, 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.Constraint Logic Programming and Logic Modality for Event's Valid-time Approximation,IICAI-05Constraint Logic Programming and Logic Modality for Event's Valid-time Approximation,IICAI-05 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, 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.Beyond Fuzzy: Parameterized approximations of Heyting algebras for uncertain knowledgeIICAI-05)Beyond Fuzzy: Parameterized approximations of Heyting algebras for uncertain knowledgeIICAI-05) Z. Majkic, Kripke Semantics for Higher-order Herbrand Model Types, Technical Report, 2005, College Park, University of Maryland. Z. Majkic, Kripke Semantics for Higher-order Herbrand Model Types, Technical Report, 2005, College Park, University of Maryland.Kripke Semantics for Higher-order Herbrand Model Types,Kripke Semantics for Higher-order Herbrand Model Types,

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Thank you ! Any question ?

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