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

• Description Logics, Tableaux Calculus
• BREAK
• Finalizing assignment Questions

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Description LogicsDecidable Fragments of FO

• (aka terminological logics,member of concept
  languages)

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Formalism for Ontologies Description Logic

• DL definition of Happy Father
  (Example from Ian Horrocks, U
  Manchester, UK)

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Description Logic Statements as Rules

• 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
• the direction ? is implicit here (sigh)
• see, e.g., Clarks completion in Logic
  Programming

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Description Logics

• Terminological Knowledge (TBox)
• Concept Definition (naming of concepts)
• Axiom (constraining of concepts)
• gt a mediators glue knowledge source
• Assertional Knowledge (ABox)
• the marked neuron in image 27
• gt the concrete instances/individuals of the
  concepts/classes that your sources export

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Formalizing Glue KnowledgeDomain Map for
SYNAPSE and NCMIR

• Domain Map
• labeled graph with
• concepts ("classes") and
• roles ("associations")
• additional semantics expressed as logic rules

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Source Contextualization DM Refinement

• sources can register new concepts at the
  mediator ...

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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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Reasoning Example
Example from BeckerHaehnle, Automatisches
Beweisen, 2001

• We want to show that (1) ... (4) implies (5)
• One approach assume NEGATION of (5) and show
  that it leads to a contradiction.
• Question Why is this sound?

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Tree Structure of the Proof
? (5)
W contradiction (Widerspruch)
BeckerHaehnle, Automatisches Beweisen, 2001
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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