Sep 13, 2006
-------------

- Recap knowledge representation and reasoning notions

- Propositional logic
	- can write formulas in many ways

- One standard notation: CNF 
	- conjunctive normal form
	- conjunction of disjunctions
		- really a form of CSPs

- Satisfying a given CNF formula
	- first problem to be shown to be NP complete

- How to solve CNF problems
	- truth table approach
	- trying out (search) values

- k-SAT	
	- also called k-CNF-SAT
	- 2-SAT is in P
	- 3-SAT is in NP-complete

- Can reduce higher SATs, like 4-SAT, 5-SAT
	- to 3-SAT
	- but not to 2-SAT (Gotcha!)

- Difficulty grows sharply
	- as we move from 2-SAT to 3-SAT
		- using (2+e)-SAT interpolation

- Critical parameter
	- # clauses to # variables ratio
	  (C/N)

- If C/N is very small
	- easy (to find an assignment)

- If C/N is very large
	- easy (to prove unsatisfiability)

- If C/N is in between
	- hard (takes most effort and time)

- Formulating puzzles as boolean satisfiability problem
	- formulate map-coloring as satisfiability (solved in class)
	- formulate magic square puzzle as satisfiability (do it yourself)