Minimal knowledge and negation as failure
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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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Outlines
Outlines

  • Propositional MBNF

    • Positive MKNF

    • General MKNF

  • Extended MBNF with First-order Quantification

  • Description Logics of MKNF

  • ICs


Propositional mknf
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.


Propositional mknf1
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.


Propositional mknf2
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)


Propositional mknf3
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))


Quantification
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)


Quantification1
Quantification

  • An example:

  • Which courses are taught?

  • Which courses are taught by known individuals?


Mknf dl
MKNF-DL

  • Goal:

    • represent non-first-order features of frame systems


Mknf dl1
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


Mknf dl2
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)


Minimal knowledge and negation as failure
ICs

  • Example 1

    • IC: Each known employee must be known to be either male or female.

      Σ = <T,A> = <{},{employee(bob)}>


Minimal knowledge and negation as failure
ICs

  • Example 1


Minimal knowledge and negation as failure
ICs

  • Example 2

    • IC: Each known employee has known social security number, which is known to be valid

      Σ = <T,A> = <{},{employee(bob)}>