Department of Computer Science

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Department of Computer Science Engineering
University of California, San DiegoCSE-291
Ontologies in Data IntegrationSpring 2003

• Bertram Ludäscher
• LUDAESCH_at_SDSC.EDU

• Tableaux calculus II, introduction to the
  LeanTAP prover
• Example Reasoning about concepts with LeanTAP
• Definitorial terminologies, terminological
  cycles
• BREAK
• QA to Assignments

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(Semantic) Tableaux Rules

• (?) 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

• A branch is closed if it contains complementary
  formulas
• A tableaux is closed if every branch is closed

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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

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Example Revisited
(Assumption)

• Initial Example in FO logic
• How can we prove it in the Tableaux Calculus?

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Partially closed tableaux
BeckerHaehnle, Automatisches Beweisen, 2001
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Description Logic Revisited
Source F. Baader, W. Nutt. Basic Description
Logics. Description Logic Handbook, Cambridge
University Press, 2002.

• a whole family of DLs is obtained by adding
• full existential quantification ?R.C
• union
• ...

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... 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

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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?

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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

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Reasoning in the Tableaux calculus
From this
We want to show this
In First-order (LeanTap) syntax
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LeanTap Demo
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Computing the Negation Normal Form

• LeanTap Tableaux Prover
• Axioms ( Theorem )
• ? FO formula
• ? formula in NNF
• ? attempt to close tableaux

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The Sound and Complete LeanTap Tableaux Prover
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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

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The Sound and Complete LeanTap Tableaux Prover
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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, ..)

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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)
• gt check happyFathers in a given database
• Reasoning
• check if I ? implies I ? for all
  databases I,
• i.e., if ? gt ?
• undecidable for FO, F-logic, etc.
• Descriptions Logics are decidable fragments
• concept subsumption, concept hierarchy,
  classification
• semantic tableaux, resolution, specialized
  algorithms

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Mediator Demo query/view rewriting (aka
planning) is reasoning!
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Querying (a database) is formula evaluation (aka
running the query)