Propositional logic
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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

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Propositional logic

Propositional Logic

  • An “adventure game” example

  • Thinking?

LOGIC


Propositional logic

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


Propositional logic

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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