Sep 18, 2006
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- Recap how to write puzzles as satisfiability problems
	- magic square puzzle
	- 8-queens puzzle
	- five letter words with four vowels

- Use consistent boolean form
	- conjunction of disjunctions
	- every clause is a disjunction
	- puzzle is a list of clauses

- After writing such expressions how do we solve them?
	- many approaches

- Trivial way: model checking
	- test out all exponential number of rows
	- brute force!

- Better algorithms
	- GSAT
	- DPLL 

- GSAT
	- Guess a value assignment for all variables
	- See if it works
	- Else, pick a variable and flip it
		- go back to step 2

- GSAT is
	- sound: if it finds an answer, that is indeed a solution
	- not complete: not guaranteed to find solution even when it exists

- DPLL
	- Named after inventors
	- Davis, Putnam, Logemann, and Loveland

- DPLL also has
	- exponential worst case time complexity
	- but has lots of tricks up its sleeve

- Example DPLL heuristic
	- "pure symbol" heuristic
	- if a variable always appears in negated, or non-negated form,
		- pick the value that will make it to be true
	- forward propagate the value
		- some new symbol might become a pure symbol
	- Example where this heuristic solves the puzzle!

- Another DPLL example
	- look for "unit clauses"
	- make them true
	- forward propagate and see what happens to rest

- Much more is needed in full fledged implementation of DPLL
	- modern implementations can solve problems with hundreds of
	  variables and thousands of clauses
	- example domain: hardware verification

- Next problem in propositional logic
	- Proving theorems, not model checking

- Simple rules
	- Modus Ponens
	- Modus Tollens
	- Resolution

- Using these rules to encode basic facts and prove results