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 Tableaux calculus II, introduction to the LeanTAP prover Example: Reasoning about concepts with LeanTAP Definitorial terminologies, terminological cycles BREAK Q&A to Assignments

2. 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

3. 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}

4. 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)

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

6. CSE-291: Ontologies in Data Integration Description Logic Revisited a whole family of DLs is obtained by addinga whole family of DLs is obtained by adding –full existential quantification  R.C –union –... Source: [F. Baader, W. Nutt. Basic Description Logics. Description Logic Handbook, Cambridge University Press, 2002]. Basic description logic Basic description logic

7. CSE-291: Ontologies in Data Integration... Reasoning with the Family... concept definition: MyConcept  DL-formulaconcept definition: MyConcept  DL-formula concept inclusion: MyConcept  DL-formulaconcept inclusion: MyConcept  DL-formula finite set of definitions is a terminology or TBox if for every atomic concept A there is at most one axiom whose lhs is Afinite set of definitions is a terminology or TBox if for every atomic concept A there is at most one axiom whose lhs is A

8. CSE-291: Ontologies in Data Integration Definitorial Terminologies In a Tbox T we distinguish: primitive concepts (occurring only on rhs) and defined concepts (occurring on lhs)In a Tbox T we distinguish: primitive concepts (occurring only on rhs) and defined concepts (occurring on lhs) T is definitorial if every interpretation of primitive concepts yields exactly one model of T (and thus for the defined concepts)T is definitorial if every interpretation of primitive concepts yields exactly one model of T (and thus for the defined concepts)  meaning of defined concepts is fixed once the primitive concepts are interpreted ! A directly uses B in T if B appears in the rhs of the definition of AA directly uses B in T if B appears in the rhs of the definition of A A uses B is the transitive closure of ‘directly uses’A uses B is the transitive closure of ‘directly uses’ T is cyclic if A uses A for some A; else acyclicT is cyclic if A uses A for some A; else acyclic One can show: If T is acyclic then T is definitorial What about this one? What about this one?

9. CSE-291: Ontologies in Data Integration Expansion of Terminologies For acyclic T we can “unfold” concept definitions until every defined concepts is specified in terms of primitive concepts onlyFor acyclic T we can “unfold” concept definitions until every defined concepts is specified in terms of primitive concepts only  the expansion of a Tbox T Example:Example:

10. CSE-291: Ontologies in Data Integration Reasoning in the Tableaux calculus Tbox Expansion From this We want to show this In First-order (LeanTap) syntax

11. CSE-291: Ontologies in Data Integration LeanTap Demo

12. CSE-291: Ontologies in Data Integration Computing the Negation Normal Form LeanTap Tableaux Prover:LeanTap Tableaux Prover: –{Axioms} & –( Theorem ) –  FO formula –  formula in NNF –  attempt to close tableaux

13. CSE-291: Ontologies in Data Integration The Sound and Complete LeanTap Tableaux Prover

14. CSE-291: Ontologies in Data Integration How LeanTAP works (1) select A; put B in unexpanded list(1) select A; put B in unexpanded list (3) split branch; creates two new goals(3) split branch; creates two new goals (6) create new instance  (X1) from  (X) formula, add X1 to free vars; or backtrack if varlimit is reached(6) create new instance  (X1) from  (X) formula, add X1 to free vars; or backtrack if varlimit is reached (11) close branch for literals; recurse(11) close branch for literals; recurse

15. CSE-291: Ontologies in Data Integration The Sound and Complete LeanTap Tableaux Prover

16. CSE-291: Ontologies in Data Integration Reasoning in Database Mediation View expansion in Global-as-View mediation is similar to this concept expansionView expansion in Global-as-View mediation is similar to this concept expansion –uncle(X, Y) :- parent(X, Z), brother(Z, Y) ; parent(X, Z), brother_in_law(Z, Y). –aunt(X, Y) :- parent(X, Z), sister(Z, Y) ; parent(X, Z), sister_in_law(Z, Y). –parent(X, Y) :- father(X, Y) ; mother(X, Y). –brother_in_law(X, Y) :- sister(X, Z), spouse(Z, Y) ; spouse(X, Z), brother(Z, Y).... Goal: find a “query plan” that expresses the derived relation uncle/2 in terms of only base relations (father/2, mother/2,..)Goal: find a “query plan” that expresses the derived relation uncle/2 in terms of only base relations (father/2, mother/2,..)

17. 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

18. CSE-291: Ontologies in Data Integration Mediator Demo: query/view rewriting (aka planning) is reasoning!

19. CSE-291: Ontologies in Data Integration Querying (a database) is formula evaluation (aka running the query)