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Propositional Logic

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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

- 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

- And Elimination
1 2, ... n

1

- And Introduction
1, . . ., n

1 2, ... n

LOGIC

Inference Rules (cont’d)

- Or Introduction
i

1 2, ... i … n

- Double Negation Elimination

LOGIC

Inference Rules (cont’d)

- Modus Ponens
(Implication Elimination)

,

(Chaining)

,

LOGIC

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

Percepts:

[Stench, Breeze, Glitter, Bump, Scream]

Operators:

[Right 90, Left 90, Forward,

Grab, Shoot,Climb]

LOGIC

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 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 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 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

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)

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

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

Finding the Lion Form – R4

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

¬ L1,2

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

¬ L2,1

LOGIC

Finding the Lion Form – R4

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

Avoiding the Lion Form – R4

- 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 Form – R4

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 Form – R4

- Can’t express generalities
- Need new propositions for each time stamp

LOGIC