Sep 17, 2003
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- Recap propositional logic
        - difference between -> and entails
        - one is an operator, yielding an expression
          the other is a statement of validity

- Simple propositional rewrite rules

- Two important rules
        - Modus Ponens
        - Resolution

- Proving validity of sentences
        - using truth table approach
        - using inference rules

- To prove that A entails B
        - show that all models of A are
          also models of B

- Simple inference rules
        - AND elimination
        - AND introduction
        - OR introduction
        - Double negation elimination

- Statements that are true but 
  which cannot be proved by Modes Ponens
        - {P=>Q, Q=>R} entails P=>R
                - because output is an implication
        - {P=>Q,~P=>R,Q=>S,R=>S} entails S
                - even though output is an atom,
                   Modus Ponens cannot find it

- Statements that are true but 
  which cannot be proved by Resolution
        - (nothing) entails (P OR ~P)
        - (P AND Q) entails (P OR Q)

- Moral of the story
        - both these rules are not complete

- With a small trick, resolution can be made complete
        - use refutation

- To prove that KB entails P
        - show that {KB U ~P} is not satisfiable

- We say
        - resolution-refutation is complete, or
        - resolution is refutation complete
          (better)