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Understanding Minimal Knowledge and Negation as Failure: Propositional MKNF

This document outlines the concepts of Minimal Knowledge and Negation as Failure, specifically focusing on Propositional MKNF frameworks. It explains positive and general MKNF theories, extended MBNF with first-order quantification, and description logics related to MKNF. Key definitions are provided, including the role of beliefs, knowledge operators, and assumption operators. Practical examples are given to illustrate how a model can be satisfied under various conditions. This overview is essential for understanding knowledge representation in logical systems.

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Understanding Minimal Knowledge and Negation as Failure: Propositional MKNF

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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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