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Problems

• Title
• cfr.Prolog database of facts, rules, query and
  solutions

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Example 1

• Database
• son(Frank, Guido).
• son(Wim, Guido).
• son(X1,Y), son(X2,Y) -gt brother(X1,X2). Query
• brother(Frank,X).

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RDF Graph

• URI
• T(defFrank,gdson,defGuido).

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Definitions

• A rule generates a triple
• A query is a graph
• Closure graph
• A solution is a subgraph of a graph

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Graph theory of resolution

• not based on logic
• proof of
• 1) completeness
• 2) monotonicity
• deduction of proof based on forwards reasoning

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Proof format

• son(Frank, Guido).
• son(Wim, Guido).
• son(X1,Y), son(X2,Y) -gt brother(X1,X2).
• Proof of brother(Frank,Wim).
• (son(Frank, Guido), son(Wim, Guido)),
• son(Frank,Guido), son(Frank,Wim) -gt
  brother(Frank,Wim).

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Constructive logic

• BHK-interpretation(Brouwer, Heyting, Kolmogorov)
• A proof of A and B is given by presenting a proof
  of A and a proof of B.
• A proof of A or B is given by presenting either a
  proof of A or a proof of B or both.
• A proof of A ? B is a procedure which permits us
  to transform a proof of A into a proof of B.
• The constant false has no proof.

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Open World Consequences

• No complement of set
• No general negation
• No universal quantifier
• a or b if no proof of a and no proof of b gt no
  proof of a or b

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Open versus closed

• Set of all large internet sites?
• Complement of this set?
• x in complement if x is small
• Only members that can be constructed are in the
  set

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Negation and disjunction

• RDF has no negation and disjunction
• Nevertheless needed
• Introduction by ontology
• Proposal for constructive negation and
  disjunction
• capable, not_capable/not(capable)

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Conclusions 1

• Graph theory of RDF resolution inferencing
  permits
• 1) clear definition of rules, queries, solutions
  and proofs
• 2) proof of completeness and monotonicity
• 3) simple proof format based on forwards
  reasoning
• 4) constructive logic avoids problems with
  closed /open world.

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Conclusions 2

• RDFEngine is a constructive resolution engine in
  Haskell
• Proposal for constructive negation and disjunction