Sep 20, 2006
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- Recap propositional logic rules
	- Modus Ponens
	- Modus Tollens
	- Resolution

- What is the difference between "entails" and "if ... then ..."?
	- one is an expression, the other is a sentence!
	- one is true in some models, the other is a statement
	  about general models

- Using these rules to prove simple facts
	- From KB, derive goal
	- First, convert the given KB into clauses
		- each clause is a disjunction
	- Try to apply these rules in order to
	  arrive at the desired conclusion

- Two 'tricks' that we use
	- can rearrange literals in a clause (disjunction)
	- can eliminate duplicate literals

- Worked out example with resolution applied three times
	- Given: p->q, ^p->r, q->s, r->s
	- Prove: s

- Doesn't always work (why?)
	- Might need to negate goal: ^s
	- Add it back to the KB
	- And prove a contradiction

- This approach is called
	- refutation

- Resolution is complete if
	- we use the refutation mode of proving
	
- Can also say
	- "Resolution-refutation is complete", or
	- "Resolution is refutation-complete" (better)

- What this means is:
	- you do not need any other rule, just resolution is sufficient

- Another worked out example from your book

- How to search for a systematic proof
	- breadth first search (forward chaining)
	- set of support strategy
		- both are complete!

- Set of support strategy
	- Initially "set of support" contains only negated goal
	- All resolutions must include one argument from set of support
	- Accumulate results back into set of support

- Horn clauses
	- a representation for which Modus Ponens
          is complete
	- note that Modus Ponens is really a 
          specialization of Resolution

- Definition of Horn Clause
	- a disjunction with at most one positive literal

- Types of Horn Clauses
	- exactly 1 positive literal and no negative literals
		- called "facts"
	- exactly 1 positive literal and at least one negative literal
		- called "rules"
	- 0 positive literals
		- serves as the "goal"

- Predicate logic
	- generalization of propositional logic

- Simple example
	- CS 4804 is a CS course.
	- All CS courses are easy.
	- Therefore CS 4804 is easy.

- Possible predicates
	- CScourse(X): true when X is a CScourse.
	- easy(X): true when X is easy.

- Predicates are also called
	- tables/relations in database literature
	- schema: predicate name, columns; instances: rows