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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) -> brother(X1,X2). Query: brother(Frank,X).

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RDF Graph URI: T(def:Frank,gd:son,def:Guido).

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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) -> brother(X1,X2). Proof of “brother(Frank,Wim).”: {(son(Frank, Guido), son(Wim, Guido)), son(Frank,Guido), son(Frank,Wim) -> 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 ==> 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

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