Lecture-11

Lecture-11 - Artificial Intelligence CS 165A 165A Tuesday,...

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Artificial Intelligence S 165A CS 165A Tuesday, Feb 8, 2011 ference (Ch 9) Inference (Ch. 9) 1
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Notes • Midterm, this Thursday –O n e H o u r – Close Book 2
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What we’ve been talking about Complete and sound inference procedures ew inference rules: New inference rules: – Universal Instantiation (gets rid of ) – Existential Instantiation (gets rid of ) – Existential Introduction (adds ) • Generalized Modus Ponens • Generalized (First-Order) Resolution – Complete but semidecidable nification Unification – Finds the substitution(s) necessary to make two sentences match •C o n junctive normal form (CNF) 3 j( )
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Generalized Modus Ponens • For atomic sentences p i , p i ', and q , where there is a substitution such that S UBST ( , p ' ) = S UBST ( , p ) for all i i i , then Specific facts General rule ) , ( ) ( , , , , 2 1 2 1 q SUBST q p p p p p p n n • How to prove this? Instantiated conclusion 4
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Unification Unification takes two atomic sentences p and q and returns a substitution that would make p and q look the same (or else it fails) U NIFY ( p, q ) = where S UBST ( , p ) = S UBST ( , q ) is the unifier of the sentences p and q •E x a m ples p = Knows(John, x ) q = Knows(John, Jane) = { x /Jane} p = Knows(John, x ) q = Knows( y , Fred) = { x /Fred, y /John} p = Knows(John, x ) q = Knows( y , MotherOf(y)) = { x /MotherOf(John), y /John} p = Knows(John, x ) q = Knows( x , Mary) = fail p = Knows(John, x ) q = Knows( x , Mary) 5 1 2 = { x 1 /Mary, x 2 /John}
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Incompleteness of GMP x P(x) Q(x) x P(x) R(x) Q(x) (x) x Q(x) S(x) x R(x) S(x) Want to conclude S(A) S(A) is true if Q(A) or R(A) is true, and one of those must be true because either P(A) or P(A) Incomplete: there are entailed sentences that the procedure cannot infer. 6
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Inference (search) strategies Forward chaining Generalized Modus Ponens ackward chaining Backward chaining 7
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Generalized Modus Ponens in Horn FOL • A Horn clause is a sentence of the form: x (P1(x) ˄ P2(x) ˄ ... ˄ Pn(x)) Q(x) – where there are 0 or more Pi 's, and the Pi 's and Q are positive (i.e., n egated) literals un-negated) literals • Horn clauses represent a subset of sentences in FOL (First Order Logic). For example, P(a) ˅ Q(a) is not a Horn clause. • Generalized Modus Ponens (GMP) is complete for KBs containing only Horn clauses – Proofs start with the given axioms/premises in KB, deriving new ntences using GMP until the goal is derived. This defines a sentences using GMP until the goal is derived. This defines a forward chaining inference procedure because it moves “forward” from the KB to the goal. 8
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Example of forward chaining All cats like fish. Cats eat everything they like. Ziggy is a cat. Data-driven KB = at(x) ikes(x Fish) 1. x cat(x) likes(x, Fish) 2. x y (cat(x) ˄ likes(x,y)) eats(x,y) 3. cat(Ziggy) Goal : Does Ziggy eat fish?
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Lecture-11 - Artificial Intelligence CS 165A 165A Tuesday,...

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