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Minimal Knowledge and Negation as Failure

Minimal Knowledge and Negation as Failure. Ming Fang 7/24/2009. Outlines. Propositional MBNF Positive MKNF General MKNF Extended MBNF with First-order Quantification Description Logics of MKNF ICs. Propositional MKNF .

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Minimal Knowledge and Negation as Failure

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  1. Minimal Knowledge and Negation as Failure Ming Fang 7/24/2009

  2. Outlines • Propositional MBNF • Positive MKNF • General MKNF • Extended MBNF with First-order Quantification • Description Logics of MKNF • ICs

  3. Propositional MKNF • Built from propositional symbols (atoms) using standard propositional connectives and two modal operators B and not. B: “knowledge operator”K not : “assumption operator”A • Positive: if a formula or a theory (set of formulas) does not contain the negation as failure operator not.

  4. Propositional MKNF • Define when a positive formula F is true in a structure (I,S): • (I,S) is a model of positive theory T if: • (i) the axioms of T are true in (I,S) • (ii) there is no (I’,S ’) such that S’ is a proper superset of S and the axioms of T are true in (I ’,S ’) • S is maximized, so the believed propositions are minimized.

  5. Propositional MKNF • General MKNF: truth will be defined by a triple (I,Sb,Sn) • (I,S) is a model of positive theory T if: • (i) the axioms of T are true in (I,S,S) • (ii) there is no (I’,S’) such that S ’ is a proper superset of S and the axioms of T are true in (I,S’,S)

  6. Propositional MKNF • An example: • It is true in (I,S’S) when:  Then a model must satisfy: (i) (ii) Three cases: • F is tautology  M=(I,S), S is the set of all interpretations. • F is not tautology but a logical consequence of G  no model • F is not a logical consequence of G  M=(I,Mod(G))

  7. Quantification • Names: object constants representing all elements of |I | • (I,S) is a model of positive theory T if: • (i) the axioms of T are true in (I,S,S) • (ii) there is no (I’,S’) such that S ’ is a proper superset of S and the axioms of T are true in (I,S’,S)

  8. Quantification • An example: • Which courses are taught? • Which courses are taught by known individuals?

  9. MKNF-DL • Goal: • represent non-first-order features of frame systems

  10. MKNF-DL • A set of interpretations M is a model of Σif: • (i) the structure (M,M) satisfies Σ • (ii) for each set of interpretations M’, if M’M, then (M’,M) does not satisfy Σ

  11. MKNF-DL • An ideal rational agent trying to decide which set of propositions to believe. • Set of prior beliefs + set of rules  new beliefs • “logical closure” • Deduced set of beliefs coincides with the assumed believe  assumed set is justified  candidate for the agent to believe in • Two kinds of beliefs: • Beliefs that the agent assumed (A operator) • New beliefs that derived (K operator)

  12. ICs • Example 1 • IC: Each known employee must be known to be either male or female. Σ = <T,A> = <{},{employee(bob)}>

  13. ICs • Example 1

  14. ICs • Example 2 • IC: Each known employee has known social security number, which is known to be valid Σ = <T,A> = <{},{employee(bob)}>

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