Lecture10 - CS440/ECE448 Intro to Articial Intelligence...

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Lecture 10: Even more on predicate logic Prof. Julia Hockenmaier [email protected] http://cs.illinois.edu/fa11/cs440 CS440/ECE448: Intro to ArtiFcial Intelligence
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Inference in predicate logic All men are mortal. Socrates is a man. Socrates is mortal. We need a new version of modus ponens: x P(x) Q(x) P(s’) ────────── Q(s’) 2 CS440/ECE448: Intro AI
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How do we deal with quantifers and variables? Solution 1: Propositionalization Ground all the variables. Solution 2: LiFted inFerence Ground ( skolemize ) all the existentially quantifed variables. All remaining variables are universally quantifed. Use unifcation . 3 CS440/ECE448: Intro AI
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Prerequisites for lifted inference: Skolemization and UniFcation
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Skolemization: remove existentially quantifed variables Replace any existentially quantifed variable x that is in the scope oF universally quantifed variables y 1 y n with a new Function F(y 1 ,…,y n ) (a Skolem function ) Replace any existentially quantifed variable x that is not in the scope oF any universally quantifed variables with a new constant c (a Skolem term ) 5 CS440/ECE448: Intro AI
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The effect of Skolemization x y w z Q(x, y, w , z, G( w , x)) is equivalent to x y z Q(x, y, P(x, y) , z, G( P(x, y) , x )) where P is the Skolem function for w. NB: the Skolem function is a function, so this is not decidable anymore. 6 CS440/ECE448: Intro AI
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Universal quantifers: Modus ponens With propositionalization: x human(x) mortal(x) human(s’) ─────────────── (UI) human(s’) mortal(s’) ──────────────────────── (MP) mortal(s’) How can we match human(s’) and x human(x) mortal(x) directly? 7 CS440/ECE448: Intro AI
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A substitution θ is a set of pairings of variables v i with terms t i : θ = {v 1 /t 1 , v 2 /t 2 , v 3 /t 3 , …, v n / t n } Each variable v i is distinct t i can be any term (variable, constant, function), as long as it does not contain v i directly or indirectly NB: the order of variables in θ doesn ʼ t matter {x/y, y/f(a)} = {y/f(a), x/y} = {x/f(a), y/f(a)} Substitutions
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Unifcation Two sentences φ and ψ uniFy to σ ( U NIFY ( φ , ψ ) = σ ) if σ is a substitution such that S UBST ( σ , φ ) = S UBST ( σ , ψ ) . Example: U NIFY (like(x, M’), like(C’,y)) ={x/C’, y/M’}
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Unifcation A set of sentences φ 1, φ n uniFy to σ if for all i j : S UBST ( σ , φ i ) = S UBST ( σ , φ j ). σ is the uniFer of φ 1, φ n S UBST ( σ , φ i ) is a uniFcation instance.
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Standardizing apart Unifcation is not well-behaved iF φ and ψ contain the same variable: U NIFY (like( x , M’), like(C’, x )): fail. We need to
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Lecture10 - CS440/ECE448 Intro to Articial Intelligence...

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