Propositional Logic

1
Propositional Logic

• An adventure game example
• Thinking?

2
PSSS

• The Physical Symbol System Hypothesis A physical
  symbol system has the necessary and sufficient
  means for general intelligent action.
  
• Where a symbol is a designating pattern that can
  be combined with others to form another
  designating pattern.

3
Knowledge Representation

• Key is problem formulation
• What happens when an n-dimensional array is
  insufficient?
  
• Need a language that is
• Expressive and concise
• Unambiguous and independent of context
• Has an inference procedure for new sentences

4
Inference Rules

• And Elimination
• ?1? ?2, ? ... ? ?n
• ?1
• And Introduction
• ?1, . . .,
  ?n
  
• ?1? ?2, ? ... ? ?n

5
Inference Rules (contd)

• Or Introduction
• 
  ?i
  
• ?1? ?2, ? ... ?
  ?i ? ? ?n
  
• Double Negation Elimination
• ???
• 
  ?
  

6
Inference Rules (contd)

• Modus Ponens
• (Implication Elimination)
• ???,?
• ?
• (Chaining)
• ???, ???
• ???

7
Inference Rules (contd)

• Unit Resolution ????, ?
• ?
• (cf.Modus Ponens)
• Resolution ????, ????
• ????
• ? is true or false. If ? is true, ? is
  true.
  
• If ? is false,
  ?? is true.

8
The Lion World
Percepts Stench, Breeze, Glitter, Bump, Scream
Operators Right 90, Left 90, Forward, Gra
b, Shoot,Climb
9
The Lion World
(1,1) none,none,none,none,none
ok
A
ok
ok
10
The Lion World
(1,1) none,none,none,none,none
(2,1) none,breeze,none,none,none
ok
P?
A
ok
B
P?
ok
11
The Lion World
(1,1) none,none,none,none,none
(2,1) none,breeze,none,none,none
L?
A
(1,2) stench,none,none,none,none
ok
P?
ok
ok
B
ok
P?
12
The Lion World
(1,1) none,none,none,none,none
(2,1) none,breeze,none,none,none
(1,2) stench,none,none,none,none
A
L?
ok
ok
(2,2) none,none,none,none,none
(2,3)Stench,none,Glitter,none,none
ok
B
ok
P?
13
The Lion World

• The Knowledge Base
• S1,1 , B1,1
  P3,1 , B4,1
  
• S2,1 , B2,1
  S3,2 , B3,2
  
• S1,2 , B1,2
  S2,2 , B2,2 S3,3
  , B3,3 Gl 1,4
  , S1,4 , B1,4
• S2,4 , B2,4 , G 2.4
  B3,4 , Gl 3,4
  
• S1,3 , B1,3 , L 1,3
  B4,3
  
• S 2.3 , B 2.3 , Gl 2.3
  P4,4
  

14
Lion World Implications

• R1 S1,1 ? L1,2 /\ L2,1
• R2 S2,1 ? L1,1 /\ L2,2 /\ L3,1
  
  
• R3 S1,2 ? L1,1 /\ L2,2 /\ L1,3
• R4 S1,2 ? L1,1 \/ L2,2 \/
  L1,3
  

15
Lion World Implications transformed into
Conjunctive Normal Form (R1-R3)

• R1 S1,1 ? L1,2 /\ L2,1
• R1 S1,1 \/ ( L1,2 /\ L2,1)
• R1 S1,1 \/ ( L1,2 /\ L2,1)
• R1 (S1,1 \/ L1,2 )/\ (S1,1 \/ L2,1 )
• R1 (S1,1 \/ L1,2 ), (S1,1 \/ L2,1 )

16
Lion World Implications transformed into
Conjunctive Normal Form R4

• R4 S1,2 ? L1,1 \/ L2,2 \/
  L1,3
  
• R4 S1,2 \/ (L1,1 \/ L2,2 \/
  L1,3)
  
• R4 S1,2 \/ L1,1 \/ L2,2 \/
  L1,3
  

17
The Lion World
(1,1) none,none,none,none,none
ok
A
ok
ok
18
Finding the Lion

• S1,1 , S1,1 \/ L1,2 Unit Resolution
• L1,2
• S1,1 , S1,1 \/ L2,1 Unit Resolution
• L2,1

19
The Lion World
(1,2) stench,none,none,none,none
L?
A
ok
P?
ok
B
ok
P?
20
Finding the Lion

• S1,2 , S1,2 \/ L1,1 \/ L2,2 \/ L1,3
  
  
• L1,1 \/ L2,2 \/ L1,3
  Unit Resolution
  
• L1,1 \/ L2,2 \/ L1,3 , L1,1
• L2,2 \/ L1,3
  Unit Resolution
  

21
Finding the Lion

• How do we know L2,2 ?

L2,2 \/ L1,3 , L2,2 L1,3
Unit
Resolution

22
Avoiding the Lion

• Dont go forward if the lion is in front
• A1,2 /\ NorthA /\ L1,3 ? Forward
• 64 rules (16 squares x 4 orientations)

23
Avoiding the Lion in the next move
After the Agent moves, A1,2 is no longer true,
now A2,3 is true. A2,3 /\ WestA /\ L1,3 ?
Forward
24
Limitations of Propositional Logic

• Cant express generalities
• Need new propositions for each time stamp