Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our User Agreement and Privacy Policy.

Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our Privacy Policy and User Agreement for details.

Successfully reported this slideshow.

Like this presentation? Why not share!

4,166 views

Published on

No Downloads

Total views

4,166

On SlideShare

0

From Embeds

0

Number of Embeds

2

Shares

0

Downloads

126

Comments

0

Likes

5

No embeds

No notes for slide

- 1. PropositionalLogicV.SaranyaAP/CSESri Vidya College of Engineering andTechnology,Virudhunagar
- 2. Definition• a branch of symbolic logic dealing withpropositions (proposal, scheme, plan) as unitsand with their combinations and theconnectives that relate them.
- 3. 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]
- 4. 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]
- 5. BNF (Backus Naur Form)• Grammar of sentences in propositional logicSentence Atomic Sentence | complex sentenceAtomic sentence True|False|SymbolSymbol P, Q,RComplex Sentence ¬ sentence|Sentence ˄ Sentence|Sentence ˄ Sentence|Sentence Sentence|Sentence Sentence
- 6. • Every sentence constructed with binaryconnectives must be enclosed in parenthesis((A ˄B) C) right formA ˄B C wrong oneMultiplication has higher precedence than additionOrder of precedence is, ˄,V, and (i) A ˄ B ˄ Cread as (A ˄B) ˄ C (or) A ˄(B ˄ C)(ii) ¬ P ˄Q˄ RS((¬ P) ˄(Q˄ R))S
- 7. Semantics• Defines the rules.• Model fixes truth vales true or false for everypropositional symbol.• Semantics specify how to compute thetruth of sentences formed with each of 5connectives.• Ex; (Wumpus World)M1= { P1,2 = False, P2,2 = False, P3,1= True}
- 8. • Atomic sentences are easy– True is true in every model– False is false in every model.• Complex Sentence– Using “ Truth Table”
- 9. 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 smartP => 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.
- 10. 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 FalseNowResult : TrueExample 3:• B1,1 (P1,2 ˄P2,1)• The result is true• But incomplete (violate the rules ofwumpus world)
- 11. 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]
- 12. KB1. There is no pit in [1,1] R1 : ¬P1,12. A square is breeze if and only if there is a pitin 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 wumpusworld. Now after visiting 2 squares R4 : ¬B1,1 R5 : B2,1
- 13. • KB consists of R1 to R5 Consider the allabove in 5 single sentences R1 ˄ R2 ˄ R3 ˄ R4 ˄ R5Concluded that all 5 sentences areTrue
- 14. Inference(conclusion, assumption..)• Used to decide whether α is true in everymodel in which KB is true.Example: Wumpus WorldB1,2 , B2,1 , P1,1 , P2,2 , P3,1, P1,2 , P2,1So totally 27=128 models are possible
- 15. Truth table for the given KB
- 16. 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]
- 17. Truth Table Enumeration Algorithm
- 18. • Here TT truth table• This enumeration algorithm is sound andcomplete because it works for any KB andalpha and always terminates.• Complexity:– Time complexity O(2 power n)– Space complexity O(n)n symbols
- 19. Equivalence• 2 sentences are logically true in the same set of models then P Q.• Also P ˄Q and Q ˄ P are logically equivalence
- 20. Validity• A sentence is valid if it is true in all the modelsExample:• P ˄ ¬P is valid.• Valid is also know as tautologies.
- 21. 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, CA sentence is unsatisfiable if it is true in no modelse.g., A A
- 22. • 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

No public clipboards found for this slide

×
### Save the most important slides with Clipping

Clipping is a handy way to collect and organize the most important slides from a presentation. You can keep your great finds in clipboards organized around topics.

Be the first to comment