Resolution


Do you ever get stuck with a proof by deduction
and not know what rule or law to apply next?

Would you like a way to do deductive proofs
where you never get stuck?

Then you need the Resolution inference rule.
(This rule is often used for automated proofs.)

	(P or A) and (not P or B) => (A or B)

Why is Resolution a valid rule?

	(P or A) and (not P or B) => (A or B)

	both (P or A) and (not P or B) must be true
	if P is false, A must be true
	if P is true, B must be true
	so either A or B must be true

Why is Resolution such a great rule?

	resolution by itself is all you need
	(no more modus ponens, modus tollens, etc)

What happens if A or B (or both) are false?

	(P or A) and (not P or B) => (A or B)

	(P or false) and (not P or B) => (false or B)
	(P or A) and (not P or false) => (A or false)
	(P or false) and (not P or false) => (false or false)


















How do you do proofs using resolution?

	1. negate the conclusion, add it to the premises
	2. convert the premises to clauses (CNF, product of sums)
	3. repeatedly apply resolution (don't use any other rules)
	4. the goal is to arrive at false

How do you convert a formula to CNF?

	1. remove -> and <->         (implication law)
	2. push NOTs                 (DeMorgans law)
	3. distribute ORs over ANDs  (distributive law)

Show how to convert the expression to clauses.

	(A or B) -> C

	not (A or B) or C		implication
	(not A and not B) or C		demorgans
	(not A or C) and (not B or C)	distributive


















Prove the argument using resolution and contradiction.

	There is an undeclared variable or
	there is a syntax error.
	If there is a syntax error,
	there is a missing semicolon or
	a variable name is misspelled.
	There is not a missing semicolon.
	There is not a misspelled variable name.
	Therefore there is an undeclared variable.

	U  undeclared variable
	E  syntax error
	S  missing semicolon
	M  misspelled variable name

	premises:
		U or E
		E -> (S or M)
		not S
		not M

	conclusion:
		U


















Classwork
You may work with a partner.

Prove the argument using resolution and contradiction.

	If the house is next to a lake,
	the treasure is not in the kitchen.
	If the tree in the yard is an elm,
	the treasure is in the kitchen.
	The house is next to a lake.
	The tree in the yard is an elm or
	the treasure is buried under the flagpole.
	Therefore the treasure is under the flagpole.

	L  house is next to lake
	K  treasure is in kitchen
	E  tree is an elm
	P  treasure is under flagpole

	premises:
		L -> not K
		E -> K
		L
		E or P

	conclusion:
		P