propositional logic & inference

Propositional Logic

V.SaranyaAP/CSESri Vidya College of Engineering and Technology,Virudhunagar

Definition

• a branch of symbolic logic dealing with propositions (proposal, scheme, plan) as units and with their combinations and the connectives that relate them.

Syntax• Defines the allowable sentences.• Atomic Sentence:– Consist of single proposition symbol.– Either TRUE or FALSE

• Rules:– Uppercase names used for symbols P,O,R– Names are arbitrary (uninformed or random)

»Example:»W[1,3] Wumpus in [1,3]

Complex sentences• Constructed from simple sentences.• Using logical connectives.

...and [conjunction]

...or [disjunction]

...implies [implication / conditional]

..(if & only if)is equivalent [bi-conditional]

...not [negation]

• Every sentence constructed with binary connectives must be enclosed in parenthesis((A ˄B) C) right formA ˄B C wrong one

Multiplication has higher precedence than additionOrder of precedence isØ, ˄,V, and

(i) A ˄ B ˄ C read as (A ˄B) ˄ C (or) A ˄(B ˄ C)(ii) ¬ P ˅Q ˄ R S

((¬ P) ˅(Q˄ R)) S

Semantics• Defines the rules.• Model fixes truth vales true or false for every

propositional symbol.• Semantics specify how to compute the

truth of sentences formed with each of 5 connectives.

• Ex; (Wumpus World)M1= { P1,2 = False, P2,2 = False, P3,1= True}

• Atomic sentences are easy– True is true in every model– False is false in every model.

• Complex Sentence– Using “ Truth Table”

Example 1:• Evaluate the sentence

¬ P1,2 ˄(P2,2 ˅ P3,1) (True ˄ (False ˅ True)

Result= TrueExample 2:5 is even implies sam is smartThis sentence will be true if sam is smart

P => Q is only FALSE when the Premise(p) is TRUE AND Consequence(Q) is FALSE.

P => Q is always TRUE when the Premise(P) is FALSE OR the Consequence(Q) is TRUE.

Example 3:• B1,1 (P1,2 ˅ P2,1)– B1,1 means breeze in [1,1]

– P1,2 means pit in [1,2]

– P2,1 means pit in [2,1]– So False False

Now Result : True

Example 3:• B1,1 (P1,2 ˅ P2,1)• The result is true• But incomplete (violate the rules of

wumpus world)

A Simple Knowledge Base

• Take Pits alone

• i,j values

• Let Pi,j be true if there is a pit in [i,j]

• Let Bi,j be true if there is a breeze in [i,j]

KB

1. There is no pit in [1,1] R1 : ¬P1,1

2. A square is breeze if and only if there is a pit in a neighboring square. R2 : B1,1 (P1,2 ˅ P2,1) R3 : B2,1 (P1,1 ˅ P1,2 ˅ P3,1)

3. The above 2 sentences are true in all wumpus world. Now after visiting 2 squares R4 : ¬B1,1

R5 : B2,1

• KB consists of R1 to R5 Consider the all above in 5 single sentences R1 ˄ R2 ˄ R3 ˄ R4 ˄ R5

Concluded that all 5 sentences are True

Inference(conclusion, assumption..)

• Used to decide whether α is true in every model in which KB is true.

Example: Wumpus WorldB1,2 , B2,1 , P1,1 , P2,2 , P3,1, P1,2 , P2,1

So totally 27=128 models are possible

Truth table for the given KB

From the table KB is true if R1 through R5 is true in all 3 rows P1,2 is false so there is no pit in

[1,2].There may be or may not be pit in [2,2]

Truth Table Enumeration Algorithm

• Here TT truth table• This enumeration algorithm is sound and

complete because it works for any KB and alpha and always terminates.

• Complexity:– Time complexity O(2 power n)– Space complexity O(n)n symbols

Equivalence• 2 sentences are logically true in the same set of models then P Q.• Also P ˄Q and Q ˄ P are logically equivalence

Validity • A sentence is valid if it is true in all the modelsExample:• P ˅ ¬P is valid.• Valid is also know as tautologies.

Satisfiability

• A sentence is true if it is true in some model.

A sentence is satisfiable if it is true in some modele.g., A B, C

A sentence is unsatisfiable if it is true in no modelse.g., A A

• Validity and satisfiability are connected.

• α is valid if α is satisfiable.• α is valid if ¬α is unsatisfiable.• ¬α is satisfiable if ¬α is not valid

propositional logic & inference

Education