# Lecture10HO - 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 Artificial Intelligence 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 How do we deal with quantifiers and variables? Solution 1: Propositionalization Ground all the variables. Solution 2: Lifted inference Ground ( skolemize ) all the existentially quantified variables. All remaining variables are universally quantified. Use unification . 3 CS440/ECE448: Intro AI Prerequisites for lifted inference: Skolemization and Unification

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Skolemization: remove existentially quantified variables Replace any existentially quantified variable ! x that is in the scope of universally quantified variables " y 1 " y n with a new function F(y 1 ,…,y n ) (a Skolem function ) Replace any existentially quantified variable ! x that is not in the scope of any universally quantified variables with a new constant c (a Skolem term ) 5 CS440/ECE448: Intro AI 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 Universal quantifiers: 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 A substitution " is a set of pairings of variables v i with terms t i : " = {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 " doesn ʼ t matter {x/y, y/f(a)} = {y/f(a), x/y} = {x/f(a), y/f(a)} Substitutions
Unification Two sentences # and \$ unify to % ( U NIFY ( # , \$ ) = % ) if % is a substitution such that S UBST ( % , # ) = S UBST ( % , \$ ) . Example: U NIFY (like(x, M’), like(C’,y)) ={x/C’, y/M’} Unification A set of sentences # 1, # n unify to % if for all i & j : S UBST ( % , # i ) = S UBST ( % , # j ). % is the unifier of # 1, # n S UBST ( % , # i ) is a unification instance. Standardizing apart Unification is not well-behaved if # and \$ contain the same variable: U NIFY (like( x , M’), like(C’, x )): fail. We need to standardize # and \$ apart (rename this variable in one term): U NIFY (like( x , M’), like(C’, y )) = {x/C’, y/M’} to yield like(C’,M’) 11 CS440/ECE448: Intro AI Do these unify?

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