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Artificial Intelligence

Artificial Intelligence. Predicate & Propositional Calculus. It is a representation language for AI Its advantages include: Well defined formal semantics Complete inference rules This part reviews propositional calculus and then describes predicate calculus. Propositional Calculus.

palmer-duncan
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Artificial Intelligence

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  1. Artificial Intelligence

  2. Predicate & Propositional Calculus • It is a representation language for AI • Its advantages include: • Well defined formal semantics • Complete inference rules This part reviews propositional calculus and then describes predicate calculus

  3. Propositional Calculus • Symbols: P, Q, R, S, … {Propositions or statements about the world that may be either true or false} • Truth symbols: true, false • Connectives:   ¬  

  4. Definition: Propositional Calculus Sentences (well formed formulas) • Every propositional symbol and truth symbol is a sentence example: true, P, Q, R are sentences • The negation of a sentence is a sentence example: ¬P , ¬false are sentences • The conjunction (and) of two sentences is a sentence example: P ¬P is a sentence

  5. Definition: Propositional Calculus Sentences (well formed formulas) • The disjunction (or) of two sentences is a sentence example: P ¬P is a sentence • The implication of one sentence from another is a sentence example: P  Q is a sentence • The equivalence of two sentences is a sentence example: P v Q  R is a sentence

  6. Defn: • In P  Q, P and Q are called conjuncts • In P  Q, P and Q are called disjuncts • In P  Q, P is called premise or antecedent and Q the conclusion or consequent

  7. Defn: • ( ) and [ ] are used to group symbols to control their order of evaluation and meaning: ( P  Q )  R is different from P  ( Q  R )

  8. Defn: • An expression is a sentence or well-defined formula of the propositional calculus if and only if it can be formed of legal symbols through some sequence of these rules. Example: ((P  Q)  R ) ¬ P ¬ Q  R

  9. ((P  Q)  R ) ¬ P ¬ Q  R Why correct? • P, Q and R are propositions and thus sentences • P  Q, conjunction is a sentence • (P  Q)  R , implication is a sentence • ¬ P and¬ Q, negation is a sentence • ¬ P ¬ Q, disjunction is a sentence • ¬ P ¬ Q  R disjunction is a sentence • ((P  Q)  R ) ¬ P ¬ Q  R, equivalence is a sentence

  10. Semantics of propositional calculus(Why important?) • Because AI programs must reason with their representational structures, it is important to demonstrate that the truth of their conclusion depends only on the truth of their initial knowledge, i.e. the logical errors are not introduced by the inference procedures.

  11. Interpretation: • It is the assignment of T or F to each propositional symbol • ,  and ¬ follow the normal truth tables •  gives F only if the precedent is T and the consequent is F. (Otherwise T) •  gives T only when both expressions have same truth assignment

  12. Example:

  13. Rules: • ¬(¬P)  P • (PQ)  (¬P  Q) • (PQ)  (¬Q  ¬P) • ¬(PQ)  (¬P  ¬Q) • ¬(PQ)  (¬P  ¬Q) • (P  Q)  (Q  P) • (P  Q)  (Q  P) • ((P  Q)  R)  (P  (Q  R)) • ((P  Q)  R)  (P  (Q  R)) • P  ( Q  R )  (P  Q )  (P  R) • P  ( Q  R )  (P  Q )  (P  R)

  14. (Example of Representing Knowledge in Propositional Logic) • Example: KB = { Mike comes to the party; If Cathy comes to the party then Becky comes; If Cathy doesn't come then Mike won't come to the party} • Let: M represent Mike comes to the party. C represent Cathy comes to the party. B represent Becky comes to the party. • KB = { M, C  B, ¬C  ¬M} (N.B. statements in the KB are implicitly connected by ^)

  15. QUIZ • Show that: (P  Q)  ( Q  P)  (PQ)  (PQ)

  16. Modus ponens and modus tollens • If the sentences P and P  Q are known to be true, then modus ponens lets us infer Q. • If the sentence P  Q is known to be true, and the sentence Q is known to be false, modus tollens lets us infer P.

  17. And elimination / and introduction • And elimination lets us infer the truth of either of the conjuncts from the truth of a conjunctive sentence. For instance, P  Q lets us conclude both P and Q are true. • And introduction allows us to infer the truth of a conjunction from the truth of its conjuncts. For instance, if P and Q are true, then P  Q is true.

  18. Universal instantiation • Universal instantion states that if any universally quantified variable in a true sentence is replaced by any appropriate term from the domain, the result is a true sentence. Thus, if a is from the domain of X, X P(X) lets us infer P(a).

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