1. CSE-291: Ontologies in Data Integration Department of Computer Science & Engineering University of California, San Diego CSE-291: Ontologies in Data Integration Spring 2003 Bertram Ludäscher LUDAESCH@SDSC.EDU Description Logics, Tableaux Calculus BREAK Finalizing assignment: Questions

2. CSE-291: Ontologies in Data Integration Description Logics Decidable Fragments of FO (aka terminological logics, member of concept languages)

3. CSE-291: Ontologies in Data Integration Formalism for Ontologies: Description Logic DL definition of “Happy Father” (Example from Ian Horrocks, U Manchester, UK)DL definition of “Happy Father” (Example from Ian Horrocks, U Manchester, UK)

4. CSE-291: Ontologies in Data Integration Description Logic Statements as Rules Another syntax: first-order logic in rule form (implicit quantifiers):Another syntax: first-order logic in rule form (implicit quantifiers): happyFather(X)  man(X), child(X,C1), child(X,C2), blue(C1), green(C2), not ( child(X,C3), poorunhappyChild(C3) ). poorunhappyChild(C)  not rich(C), not happy(C). Note:Note: –the direction “  ” is implicit here (*sigh*) –see, e.g., Clark’s completion in Logic Programming

5. CSE-291: Ontologies in Data Integration Description Logics Terminological Knowledge (TBox)Terminological Knowledge (TBox) –Concept Definition (naming of concepts): –Axiom (constraining of concepts): => a mediators “glue knowledge source” Assertional Knowledge (ABox)Assertional Knowledge (ABox) –the marked neuron in image 27 => the concrete instances/individuals of the concepts/classes that your sources export

6. CSE-291: Ontologies in Data Integration Formalizing Glue Knowledge: Domain Map for SYNAPSE and NCMIR Domain Map = labeled graph with concepts ("classes") and roles ("associations") additional semantics: expressed as logic rules Domain Map = labeled graph with concepts ("classes") and roles ("associations") additional semantics: expressed as logic rules Domain Map (DM) Purkinje cells and Pyramidal cells have dendrites that have higher-order branches that contain spines. Dendritic spines are ion (calcium) regulating components. Spines have ion binding proteins. Neurotransmission involves ionic activity (release). Ion-binding proteins control ion activity (propagation) in a cell. Ion-regulating components of cells affect ionic activity (release). Domain Expert Knowledge DM in Description Logic

7. CSE-291: Ontologies in Data Integration Source Contextualization & DM Refinement Source Contextualization & DM Refinement In addition to registering (“hanging off”) data relative to existing concepts, a source may also refine the mediator’s domain map...  sources can register new concepts at the mediator...

8. CSE-291: Ontologies in Data Integration Querying vs. Reasoning Querying:Querying: –given a DB instance I (= logic interpretation), evaluate a query expression (e.g. SQL, FO formula, Prolog program,...) –boolean query: check if I |=  (i.e., if I is a model of  ) –(ternary) query: { (X, Y, Z) | I |=  (X,Y,Z) } => check happyFathers in a given database Reasoning:Reasoning: –check if I |=  implies I |=  for all databases I, –i.e., if  =>  –undecidable for FO, F-logic, etc. –Descriptions Logics are decidable fragments  concept subsumption, concept hierarchy, classification  semantic tableaux, resolution, specialized algorithms

9. CSE-291: Ontologies in Data Integration Reasoning Example We want to show that (1) &... & (4) implies (5)We want to show that (1) &... & (4) implies (5) One approach: assume NEGATION of (5) and show that it leads to a contradiction.One approach: assume NEGATION of (5) and show that it leads to a contradiction. –Question: Why is this sound? Example from [Becker&Haehnle, Automatisches Beweisen, 2001]

10. CSE-291: Ontologies in Data Integration Tree Structure of the Proof  (5) W = contradiction (“Widerspruch”) [Becker&Haehnle, Automatisches Beweisen, 2001]

11. CSE-291: Ontologies in Data Integration (Semantic) Tableaux Rules A branch is closed if it contains complementary formulasA branch is closed if it contains complementary formulas A tableaux is closed if every branch is closedA tableaux is closed if every branch is closed t arbitrary c new (  ) rule for F = A  B(  ) rule for F = A  B (  ) rule for F = A  B(  ) rule for F = A  B (  ) rule for F =  x: A(X,...)(  ) rule for F =  x: A(X,...) –substitute a  -variable X with an arbitrary term t (  ) rules for F =  x: A(X,...)(  ) rules for F =  x: A(X,...) –substitute a  -variable X with a new constant c

12. CSE-291: Ontologies in Data Integration FO Tableaux Calculus Theorem (Soundness, Completeness of Tableaux calculus): Let A 1,..., A k and F be first-order logic sentences. (Recall: a sentence is a closed formula, i.e., has no free variables ) Then the following are equivalent: 1.A 1,..., A k |= F 2.A 1 ...  A k  F is unsatisfiable (inconsistent) 3.There is a closed tableaux for {A 1,..., A k,  F}

13. CSE-291: Ontologies in Data Integration Example Revisited Initial Example in FO logicInitial Example in FO logic How can we prove it in the Tableaux Calculus?How can we prove it in the Tableaux Calculus? (Assumption)

14. CSE-291: Ontologies in Data Integration Partially closed tableaux [Becker&Haehnle, Automatisches Beweisen, 2001]