Lecture10HO - CS440/ECE448 Intro to Articial Intelligence...

Info icon This preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 2
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?
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern