Advanced Topics in Tableaux Calculus: Ontologies & Data Integration

1. Department of Computer Science & Engineering University of California, San DiegoCSE-291: Ontologies in Data IntegrationSpring 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. c new t arbitrary (Semantic) Tableaux Rules • A branch is closed if it contains complementary formulas • A tableaux is closed if every branch is closed • () rule for F = A  B • () rule for F = A  B • () rule for F = x: A(X,...) • substitute a -variable X with an arbitrary term t • () rules for F = x: A(X,...) • substitute a -variable X with a new constant c

3. FO Tableaux Calculus Theorem (Soundness, Completeness of Tableaux calculus): Let A1,..., Ak 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: • A1, ..., Ak |= F • A1  ...  Ak  F is unsatisfiable (inconsistent) • There is a closed tableaux for {A1, ..., Ak ,  F}

4. Example Revisited • Initial Example in FO logic • How can we prove it in the Tableaux Calculus? (Assumption)

5. Partially closed tableaux [Becker&Haehnle, Automatisches Beweisen, 2001]

6. Basic description logic Description Logic Revisited • a 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].

7. ... Reasoning with the Family ... • concept definition: MyConcept  DL-formula • concept 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 A

8. Definitorial Terminologies • 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)  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 A • A uses B is the transitive closure of ‘directly uses’ • T 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?

9. Expansion of Terminologies • For 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:

10. Tbox Expansion Reasoning in the Tableaux calculus From this We want to show this In First-order (LeanTap) syntax

11. LeanTap Demo

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

13. The Sound and Complete LeanTap Tableaux Prover

14. How LeanTAP works • (1) select A; put B in unexpanded list • (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 • (11) close branch for literals; recurse

15. The Sound and Complete LeanTap Tableaux Prover

16. Reasoning in Database Mediation • View 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, ..)

17. Querying vs. Reasoning • 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 happyFathersin a given database • Reasoning: • check if I |=  impliesI |=  for all databases I, • i.e., if =>  • undecidable for FO, F-logic, etc. • Descriptions Logics aredecidable fragments • concept subsumption, concept hierarchy, classification • semantic tableaux, resolution, specialized algorithms

18. Mediator Demo: query/view rewriting(aka planning) is reasoning!

19. Querying (a database) is formula evaluation (aka running the query)