Bertram Lud ä scher [email protected]

CSE-291: Ontologies in Data Integration

Department of Computer Science & Engineering Department of Computer Science & Engineering University of California, San DiegoUniversity of California, San Diego

CSE-291: Ontologies in Data IntegrationCSE-291: Ontologies in Data IntegrationSpring 2003Spring 2003

Bertram LudBertram Ludää[email protected]@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


(Semantic) Tableaux Rules(Semantic) Tableaux Rules

• A A branch is closedbranch is closed if it contains complementary formulas if it contains complementary formulas

• A A tableaux is closed tableaux is closed if every branch is closedif every branch is closed

t arbitraryc new

• (() rule for F = ) rule for F = A A B B• (() rule for F = ) rule for F = A A B B • (() rule for ) rule for F = F = x: A(X,...)x: A(X,...)

– substitute a -variable X with an arbitrary term t

• (() rules for F = ) rules for F = x: A(X,...) x: A(X,...) – substitute a -variable X with a new

constant c


FO Tableaux CalculusFO Tableaux Calculus

TheoremTheorem (Soundness, Completeness of Tableaux (Soundness, Completeness of Tableaux calculus):calculus):

Let ALet A11,..., A,..., Akk and F be first-order logic and F be first-order logic sentences.sentences.

(Recall: a sentence is a closed formula, i.e., has no free variables)

Then the following are equivalent:Then the following are equivalent:

1.1. AA11, ..., A, ..., Akk |= F |= F

2.2. AA11 ... ... A Akk F is unsatisfiable (inconsistent) F is unsatisfiable (inconsistent)

3.3. There is a closed tableaux for {AThere is a closed tableaux for {A11, ..., A, ..., Akk , , F} F}


Example RevisitedExample 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)


Partially closed Partially closed tableauxtableaux

[Becker&Haehnle, Automatisches Beweisen, 2001]


Description Logic RevisitedDescription Logic Revisited

• a whole family of DLs is obtained by adding 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].

Basic description logicBasic description logic


... Reasoning with the Family ...... Reasoning with the Family ...

• concept concept definition: definition: MyConcept MyConcept DL-formulaDL-formula• concept concept inclusion: inclusion: MyConcept MyConcept DL-formulaDL-formula• finite set of definitions is a finite set of definitions is a terminologyterminology or or TBoxTBox if for every if for every

atomic concept atomic concept A A there is at most one axiom whose lhs is there is at most one axiom whose lhs is AA


Definitorial TerminologiesDefinitorial Terminologies

• In a Tbox In a Tbox T T we distinguish: we distinguish: primitive concepts primitive concepts (occurring only on (occurring only on rhs) and rhs) and defined concepts defined concepts (occurring on lhs)(occurring on lhs)

• T T is is definitorialdefinitorial if every interpretation of primitive concepts yields if every interpretation of primitive concepts yields exactly one model of exactly one model of T T (and thus for the defined concepts) (and thus for the defined concepts)

meaning of defined concepts is fixed once the primitive concepts are meaning of defined concepts is fixed once the primitive concepts are interpreted !interpreted !

• A directly uses B A directly uses B in in TT if if B B appears in the rhs of the definition of appears in the rhs of the definition of AA• A uses BA uses B is the transitive closure of ‘ is the transitive closure of ‘directly uses’directly uses’• TT is is cyclic cyclic if if A uses A A uses A for some for some A; A; else else acyclicacyclic

One can show: If One can show: If TT is acyclic then is acyclic then T T is definitorialis definitorial

What about this one?What about this one?


Expansion of TerminologiesExpansion of Terminologies• For acyclic For acyclic T T we can “unfold” concept definitions until we can “unfold” concept definitions until

every defined concepts is specified in terms of every defined concepts is specified in terms of primitive concepts onlyprimitive concepts only

the the expansion expansion of a Tbox of a Tbox TT• Example: Example:


Reasoning in the Tableaux calculusReasoning in the Tableaux calculus

Tbox

Expansion

From this

We want to show this

In First-order (LeanTap) syntax


LeanTap DemoLeanTap Demo


Computing the Negation Normal FormComputing the Negation Normal Form

• LeanTap Tableaux Prover:LeanTap Tableaux Prover:– {Axioms} & –( Theorem ) FO formula formula in NNF attempt to close tableaux


The Sound and The Sound and Complete LeanTap Complete LeanTap Tableaux ProverTableaux Prover


How LeanTAP worksHow LeanTAP works

• (1) select A; put B in (1) select A; put B in unexpanded listunexpanded list

• (3) split branch; (3) split branch; creates two new goalscreates two new goals

• (6) create new (6) create new instance instance (X1) from (X1) from (X) formula, add X1 (X) formula, add X1 to free vars; or to free vars; or backtrack if varlimit backtrack if varlimit is reachedis reached

• (11) close branch for (11) close branch for literals; recurseliterals; recurse


The Sound and The Sound and Complete LeanTap Complete LeanTap Tableaux ProverTableaux Prover


Reasoning in Database Mediation Reasoning in Database Mediation • View expansion in View expansion in Global-as-ViewGlobal-as-View mediation is similar to this mediation is similar to this

concept expansionconcept 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 Goal: find a “query plan” that expresses the derived relation uncle/2 in terms of only base relations (father/2, mother/2, ..)uncle/2 in terms of only base relations (father/2, mother/2, ..)


Querying vs. ReasoningQuerying 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


Mediator Demo: Mediator Demo: query/view rewritingquery/view rewriting (aka (aka planningplanning) is ) is reasoningreasoning!!


QueryingQuerying (a database) is (a database) is formula evaluationformula evaluation (aka (aka runningrunning the query) the query)

Bertram Lud ä scher [email protected]

Documents

Transcript of Bertram Lud ä scher [email protected]