Gremlins and Goblins

1
Gremlins and Goblins

• Gremlins always tell the truth. Goblins sometimes
  lie. Unfortunately, their appearance is
  identical.
• You are in a dungeon with two gremlins and a
  goblin (but you dont know which is which). There
  is a red door and a blue door. One door leads to
  the exit. The other door leads to a pit of deadly
  venomous snakes.

2
The Query

• Your objective is to find the exit. Your
  companions give you some advice
• Gorbnitz (G1) The red door is the way out. I am
  telling the truth!
• Grovnack (G2) The blue door is the way out.
  Gorbnitz is lying!!
• Glidnoph (G3) I dont know which door is the way
  out, but Gorbnitz is telling the truth.

3
Solving the Problem

• Express the problem in first-order logic.
• Express the negated goal in first-order logic and
  add it to the KB.
• Translate all of the knowledge into standard form
  (clausal normal form).
• Apply resolution to derive a contradiction.
• The answer is in the bindings.

4
Describing the Problem

• Goblins /gremlins mutually exclusive
• ?x Gob(x) ? ?Grem(x)
• ?x Grem(x) ? ?Gob(x)
• One goblin
• Gob(G1) ? Gob(G2) ? Gob(G3)
• Two gremlins
• (Grem(G1) ? Grem(G2)) ? (Grem(G1) ? Grem(G3)) ?
  (Grem(G1) ? Grem(G3))

5
The Problem II

• Exactly one door is the exit
• Exit(Red) ? Exit(Blue)
• ?(Exit(Red) ? Exit(Blue)
• The creatures statements
• Grem(G1) ? Exit(Red) ? Grem(G1)
• Grem(G2) ? Exit(Blue) ? Grem(G2)
• Grem(G3) ? Grem(G1)

6
Describing the Negated Goal

• Goal ?e Exit(e)
• Negated goal ?e ?Exit(e)

7
Clausal Normal Form

• ?Gob(x) ? ?Grem(x)
• ?Grem(y) ? ?Gob(y) redundant
• Gob(G1) ? Gob(G2) ? Gob(G3)
• Grem(G1) ? Grem(G2)
• Grem(G2) ? Grem(G3)
• Grem(G1) ? Grem(G3)
• ?Grem(G1) ? Exit(Red)
• ?Grem(G1) ? Grem(G1) tautology
• ?Grem(G2) ? Exit(Blue)
• ?Grem(G2) ? Grem(G2) tautology
• ?Grem(G3) ? Grem(G1)
• ?Exit(e) negated goal

8
Think about the proof!

• Both G1 and G3 say that G1 is telling the truth,
  so this must be true (since they cant both be
  lying)
• If G1 is telling the truth, then the red door is
  the exit!
• Now all we have to do is construct the resolution
  proof tree that demonstrates this line of
  reasoning

9
Resolution Proof Tree
?Grem(G3) ? Grem(G1)
Grem(G3) ? Grem(G1)
?Grem(G1) ? Exit(Red)
Grem(G1)
Exit(Red)
?Exit(e)
e/Red
FALSE
10
The Answer

• The exit is red!