Reasoning in Propositional Logic

1
Reasoning in Propositional Logic
2
Learning Objectives

• Introduction to knowledge bases
• Wumpus world
• Introduction to logic
• Propositional logic
• Normal forms
• Inference rules

3
Acknowledgements

• These slides have been adapted from Stuart
  Russell and Peter Norvig.

4
(No Transcript)
5
(No Transcript)
6
(No Transcript)
7
(No Transcript)
8
Where is the Wumpus?
9
A
10
S1,2 so W1,3 or W2,2 No S2,1 so No W2,2
No B1,2 so No P2,2 No P2,2 so P1,3
S
A
11
OK
S
A
12
(No Transcript)
13
(No Transcript)
14
(No Transcript)
15
(No Transcript)
16
(No Transcript)
17
(No Transcript)
18
(No Transcript)
19
(No Transcript)
20
(No Transcript)
21
(No Transcript)
22
(No Transcript)
23
(No Transcript)
24
(No Transcript)
25
(No Transcript)
26
Natural Deduction
27
(No Transcript)
28
(No Transcript)
29
(No Transcript)
30
(No Transcript)
31
(No Transcript)
32
(No Transcript)
33
(No Transcript)
34
(No Transcript)
35
(No Transcript)
36
(No Transcript)
37
(No Transcript)
38

• Let Si,jj be the proposition There is a stench
  at cell i,j and Bi,j be the proposition There
  is a breeze in cell i,j.
• Assume that in the Wumpus World, the KB contains
  the following
• S1,1 B1,1
• S2,1 B2,1
• S1,2 B1,2
• The KB also contains among other rules, the
  following
• R1 S1,1 -gt W1,1 W1,2 W2,1
• R2 S2,1 -gt W1,1 W2,1 W2,2 W3,1
• R3 S1,2 -gt W1,1 W1,2 W2,2 W1,3
• Also, for each cell in the grid the following is
  a rule
• R4 S1,2 -gt W1,3 v W1,2 v W2,2 v W1,1

39

• Show that W1,3 is entailed by KB.
• 1. W1,1 W1,2 W2,1 by MP,R1 S1,1
• 2. W1,1 W1,2 W2,1 by -elim.
• 3. W1,1 W2,1 W2,2 W3,1 by MP,R2 S2,1
• 4. W1,1 W2,1 W2,2 W3,1 by -elim.
• 5. W1,3 v W1,2 v W2,2 v W1,1 by MP,R4 S1,2
• 6. W1,3 v W2,2 resolve (2),(5) for W1,1 W1,2
• 7. W1,3 resolve (6),(4) for W2,2

40

• Solution 2. Show that W1,3 is entailed by KB. Our
  goal is to show that KB W1,3 is unsatisfiable.
• First, translate the KB into CNF then show
  unsatisfiability by resolution.

(S1,1 v W1,1) R11 (S1,1 v W1,2) R12 (S1,1 v
W2,1) R13 (S2,1 v W1,1) R21 (S2,1 v W2,1)
R22 (S2,1 v W2,2) R23 (S2,1 v W3,1) R24 (S1,2 v
W1,1) R31 (S1,2 v W1,2) R32 (S1,2 v W2,2)
R33 (S1,2 v W1,3) R34
(S1,2 v W1,3 v W1,2 v W2,2 v W1,1) W1,3
(negation of goal) S1,1 S2,1 S1,2 B1,1
B2,1 B1,2
41
(No Transcript)
42
(No Transcript)
43
(No Transcript)
44
(No Transcript)
45
(No Transcript)
46
Resolution is refutation-complete

• Resolution can not be used to find all the
  logical consequences of a given set of sentences.
• But, it can be used to find all answers to a
  given question using the negated-goal method.
• If a set of sentences is unsatisfiable, then
  resolution will always derive a contradiction.