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Propositional Logic. An “adventure game” example Thinking?. PSSS. The Physical Symbol System Hypothesis: A physical symbol system has the necessary and sufficient means for general intelligent action.

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propositional logic
Propositional Logic
  • An “adventure game” example
  • Thinking?

LOGIC

slide2
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.

LOGIC

knowledge representation
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

LOGIC

inference rules
Inference Rules
  • And Elimination

1 2,  ...  n

1

  • And Introduction

1, . . ., n

1 2,  ...  n

LOGIC

inference rules cont d
Inference Rules (cont’d)
  • Or Introduction

i

1 2,  ...  i …  n

  • Double Negation Elimination



LOGIC

inference rules cont d1
Inference Rules (cont’d)
  • Modus Ponens

(Implication Elimination)

,

(Chaining)

, 



LOGIC

inference rules cont d2
Inference Rules (cont’d)
  • Unit Resolution:, 

(cf.Modus Ponens)

  • Resolution:, 



 is true or false. If  is true,  is true.

If  is false,  is true.

LOGIC

the lion world
The Lion World

Percepts:

[Stench, Breeze, Glitter, Bump, Scream]

Operators:

[Right 90, Left 90, Forward,

Grab, Shoot,Climb]

LOGIC

the lion world1
The Lion World

(1,1) [none,none,none,none,none]

ok

A

ok

ok

LOGIC

the lion world2
The Lion World

(1,1) [none,none,none,none,none]

(2,1) [none,breeze,none,none,none]

ok

P?

A

ok

B

P?

ok

LOGIC

the lion world3
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

ok

P?

ok

ok

B

P?

LOGIC

the lion world4
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

ok

B

P?

LOGIC

the lion world5
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

LOGIC

lion world implications
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

LOGIC

lion world implications transformed into conjunctive normal form r1 r3
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 )

LOGIC

lion world implications transformed into conjunctive normal form r4
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

LOGIC

the lion world6
The Lion World

(1,1) [none,none,none,none,none]

ok

A

ok

ok

LOGIC

finding the lion
Finding the Lion

¬ S1,1 , S1,1 \/ ¬L1,2 Unit Resolution

¬ L1,2

¬ S1,1 , S1,1 \/ ¬L2,1 Unit Resolution

¬ L2,1

LOGIC

the lion world7
The Lion World

(1,2) [stench,none,none,none,none]

L?

A

ok

P?

ok

ok

B

P?

LOGIC

slide20

Finding the Lion

S1,2 , ¬ S1,2 \/ L1,1 \/ L2,2 \/ L1,3

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

L1,1 \/ L2,2 \/ L1,3 ,¬ L1,1

L2,2 \/ L1,3 Unit Resolution

LOGIC

finding the lion1
Finding the Lion
  • How do we know ¬ L2,2 ?
  • L2,2 \/ L1,3 ,¬ L2,2
  • L1,3 Unit Resolution

LOGIC

avoiding the lion
Avoiding the Lion
  • Don’t go forward if the lion is in front –

A1,2 /\ NorthA /\ L1,3  ¬Forward

  • 64 rules (16 squares x 4 orientations)

LOGIC

avoiding the lion in the next move
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

LOGIC

limitations of propositional logic
Limitations of Propositional Logic
  • Can’t express generalities
  • Need new propositions for each time stamp

LOGIC

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