CS 4341 Homework #5 Solutions

Problem 1:

Show the steps required to put the following axioms into clause form:
∀x ∀y [On(x,y) ⇒ Above(x,y)]
∀x ∀y ∀z [Above(x,y) ∧ Above(y,z) ⇒ Above(x,z)]

Remove the implications:
∀x ∀y [¬On(x,y) ∨ Above(x,y)]
∀x ∀y ∀z [¬(Above(x,y) ∧ Above(y,z)) ∨ Above(x,z)]

Move negations down to the atomic level:
∀x ∀y [¬On(x,y) ∨ Above(x,y)]
∀x ∀y ∀z [¬Above(x,y) ∨ ¬Above(y,z) ∨ Above(x,z)]

Rename variables (We won't deduct points if you forgot to do this):
∀x ∀y [¬On(u,v) ∨ Above(u,v)]
∀x ∀y ∀z [¬Above(x,y) ∨ ¬Above(y,z) ∨ Above(x,z)]

Drop the existential quantifiers
¬On(u,v) ∨ Above(u,v)
¬Above(x,y) ∨ ¬Above(y,z) ∨ Above(x,z)

Problem 2:

Example	Heuristic	Result
1	Initial model	Apple is red fragrant sphere with weight=4
2	Drop link	Apple is red sphere with weight=4
3	Forbid link	Apple is red sphere with weight=4, not inedible
4	Enlarge set	Apple is red or green sphere with weight=4, not inedible
5	Close interval	Apple is red or green sphere with 4≤weight≤7, not inedible
6	Climb tree	Apple is red or green rounded object with 4≤weight≤7, not inedible
7	none	Nothing learned, same model as above

Problem 3:

All the true successes involve not only George and Sally, but Frank as well. The false successes have nothing in common, except that George and Sally are involved. You should improve the rule to state that you only enjoy dinner parties attended by George, Sally, and Frank. Note that parties 5 and 7 have the same attendees, but different outcomes, so it is impossible to have a 100% accurate rule or rule set.

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