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,S b,S n )  (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: (1) F is tautology  M=(I,S), S is the set of all interpretations. (2) F is not tautology but a logical consequence of G  no model (3) 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 Σ if:  (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 = =

13. ICs  Example 1

14. ICs  Example 2  IC: Each known employee has known social security number, which is known to be valid Σ = =