Sep 19, 2003
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- More concepts
	- satisfiability
	- validity

- Relating entailment to these
	- a entails b if ... is unsatisfiable
	- a entails b if ... is valid
		- fill up the blanks!

- a new island: 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
	- facts
	- rules
	- goal

- Using propositional logic
        - proving theorems by model-checking
        - proving theorems by rewrite rules

- Using propositional logic in software
	- forward chaining
		- careful book-keeping in arrays
	- backward chaining
		- goal-driven

- Recap algorithms for propositional logic
        - checking validity
                - resolution-refutation
        - checking satisfiability
                - GSAT

- But
        - GSAT is incomplete
        - it is sound though

- What is a complete and sound algorithm?
        - DPLL: Davis-Putnam-Logemann-Loveland

- Lots of nice tricks in DPLL
        - unit clause propagation
        - see if a given symbol appears with same parity
          throughout (AIMA calls this "pure symbol heuristic")
        - if all clauses have two or more literals
                - set something to true/false and branch
                - backtrack when unsatisfiable

- DPLL
        - is sound AND
        - complete
                - because it will can detect 
                  unsatisfiable formulas

- When to use GSAT versus DPLL
        - GSAT: when you know a solution exists
        - DPLL: when you don't know and might be interested
          in proving a formula unsatisfiable

- Practical applications of propositional reasoning
        - scheduling problems