Semantic Web Techniques

1

• jDREW

BU
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Todays Topics

• Resolution
• top down and bottom up
• j-DREW BU procedure
• Subsumption
• change to procedure
• Infinite Loops
• RuleML input
• Prolog output

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Resolution

• Clausal Logic Syntax
• an atom in logic is (syntactically) a relation
  symbol followed by its arguments, each of which
  is a term
• a term is either a variable, a constant or a
  function symbol followed by its arguments (which
  are also terms)
• brother(Harry, future_king(England))
• passes(Fang, cs6999)
• each clause is a disjunction of positive or
  negative atoms
• attends(Student, Course) \/ passes(Student,
  Course)
• Rule Notation a \/ b \/ c \/ d becomes a \/ b
  ? c /\ d
• a literal is a positive or negative atom
• Definite clauses have
• one positive literal and any number of negative
  literals

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Resolution

• Given two clauses, to form the resolvent
• find two literals, one from each clause, that
• can be made to match (with assignments to the
  variables)
• differ in sign
• construct the disjunction of all literals except
  these two
• apply the substitution
• Resolve
• b(X) \/ c(X, 3) against c(2, Y) \/ d(Y, Z)
• gives b(2) \/ d(3, Z)
• In rule notation
• b(X) ? c(X, 3) against c(2, Y) ? d(Y, Z)
• gives b(2) ? d(3, Z)

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Bottom Up Reasoning

• Single literal definite clauses are called facts
• passes(Fang, cs6999)
• Resolving a fact against a condition in a rule is
  called forward reasoning
• attends(Student, Course) ? passes(Student,
  Course)
• Resolvent attends(Fang, cs6999)
• Usually a derivation tree is viewed with given
  facts at the bottom and derived facts at the top
• so forward reasoning also called bottom up
  reasoning
• Prolog does top-down, backward reasoning
• starting from a negative clause, called a query

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Theorem Provers Search Procedure
main loop select new fact for each
matching rule resolve process new
result add to old facts

• Priority queue
• new facts
• Discrimination trees
• used facts
• rules, indexed onfirst goal

process new result(C) if C is rule for
each old fact matching first goal
resolve process new result add C
to rules else add C to new facts
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Preventing Infinite Loops

• To prevent infinite loops in the search
  procedure,
• use subsumption (next slides) and either
• 1a) use ground facts and rules only, or
• 1b) use no function symbols
• Sometimes it helps if the positive literal has
  no new variables (that do not already occur in
  the body)

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Subsumption

• A clause C subsumes a clause D if there is a
  substitution ? such that C? ? D
• C is more general than D
• Given C you do not need D
• We use subsumption between facts (atoms)
• Atoms C subsumes atom D if there exists ? such
  that C? D
• Subsumption is checked by
• first grounding all variables in D to new
  constants
• then trying to unify C with grounded D

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Forward subsumption

• New result is checked against old results to see
  if it is more general
• We can prevent some infinte loops by checking
  subsumption of new fact against old facts
• Should this be done when the new fact is derived
  or when it is selected?
• experience shows that subsumption on selection is
  better
• What about subsumption of rules?

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Theorem Provers Search Procedure
main loop select new fact if it is not
forward subsumed for each matching rule
resolve process new result
add to old facts

• Priority queue
• new facts
• Discrimination trees
• used facts
• rules, indexed onfirst goal

process new result(C) if C is rule for
each old fact matching first goal
resolve process new result add C
to rules else add C to new facts
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RuleML
ltimpgt lt_headgt ltatomgt
lt_oprgtltrelgtdiscountlt/relgtlt/_oprgt
ltvargtcustomerlt/vargt ltvargtproductlt/vargt
ltindgt5.0 percentlt/indgt lt/atomgt lt/_headgt
lt_bodygt ltandgt ltatomgt
lt_oprgtltrelgtpremiumlt/relgtlt/_oprgt
ltvargtcustomerlt/vargt lt/atomgt ltatomgt
lt_oprgtltrelgtregularlt/relgtlt/_oprgt
ltvargtproductlt/vargt lt/atomgt lt/andgt
lt/_bodygt lt/impgt

• Input to j-DREW BU may also be RuleML
• RuleML queries are ignored currently

discount(customer,product1,'5.0 percent')
- premium(customer),regular(product).
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Related Literature

• Set of support prover for definite clauses
• Facts are supports
• Theorem Completeness preserved when definite
  clause resolutions are only between first
  negative literal and fact.
• Proof completeness of lock resolution (Boyers
  PhD)
• Use standard search procedure to reduce redundant
  checking
• Unlike OPS/Rete, returns proofs and uses first
  order syntax for atoms