Lecture 10: All men are mortal. Socrates is a man. Even more on - PowerPoint PPT Presentation

CS440/ECE448: Intro to Artificial Intelligence � Inference in predicate logic � Lecture 10: 
 All men are mortal. Socrates is a man. Even more on Socrates is mortal. predicate logic � We need a new version of modus ponens: � ∀ x P(x) ! Q(x) Prof. Julia Hockenmaier � P(s’) juliahmr@illinois.edu � !!!!!!!!!! � Q(s’) http://cs.illinois.edu/fa11/cs440 � � � CS440/ECE448: Intro AI � 2 � � How do we deal with 
 quantifiers and variables? � � Prerequisites 
 Solution 1: Propositionalization � Ground all the variables. � for lifted inference: 
 � Skolemization and Solution 2: Lifted inference � Ground (skolemize) all the existentially Unification � quantified variables. All remaining variables are universally quantified. � Use unification. � CS440/ECE448: Intro AI � 3 �

Skolemization: remove The effect of Skolemization � existentially quantified variables � " x " y ! w " z Q(x, y, w, z, G(w, x)) Replace any existentially quantified variable ! x that is in the scope of universally is equivalent to � quantified variables " y 1 … " y n with a new � " x " y " z Q(x, y, P(x, y), z, G(P(x, y), x )) � function F(y 1 ,…,y n ) (a Skolem function ) � where P is the Skolem function for w. Replace any existentially quantified variable ! x that is not in the scope of any universally NB: the Skolem function is a function, 
 quantified variables with a new constant c so this is not decidable anymore. (a Skolem term ) � � CS440/ECE448: Intro AI � 5 � CS440/ECE448: Intro AI � 6 � Universal quantifiers: 
 Substitutions � Modus ponens � A substitution " is a set of pairings of variables v i With propositionalization: � ∀ x human(x) ! mortal(x) human(s’) with terms t i : 
 " = {v 1 /t 1 , v 2 /t 2 , v 3 /t 3 , …, v n / t n } !!!!!!!!!!!!!!! (UI) � human(s’) ! mortal(s’) • Each variable v i is distinct � !!!!!!!!!!!!!!!!!!!!!!!! (MP) • t i can be any term (variable, constant, mortal(s’) function), as long as it does not contain v i � directly or indirectly � How can we match human(s’) and 
 � ∀ x human(x) ! mortal(x) directly? � 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)} � CS440/ECE448: Intro AI � 7 �

Unification � Unification � Two sentences # and $ unify to % 
 A set of sentences # 1, … # n unify to % � ( U NIFY ( # , $ ) = % ) 
 if for all i & j : S UBST ( % , # i ) = S UBST ( % , # j ). if % is a substitution such that 
 S UBST ( % , # ) = S UBST ( % , $ ). % is the unifier of # 1, … # n S UBST ( % , # i ) is a unification instance. Example: U NIFY (like(x, M’), like(C’,y)) ={x/C’, y/M’} Do these unify? 
 Standardizing apart � (Single lower case letters are variables) � Unification is not well-behaved if # and $ U NIFY (P(x,y,z), P(w, w, Fred)) contain the same variable: � � % = {x=Fred, y=Fred, z=Fred, w=Fred} U NIFY (like(x, M’), like(C’,x)): fail. We need to standardize # and $ apart 
 Equivalently: % ’ = {x=Fred, w=y, z=Fred , y=x} (rename this variable in one term): � � Both yield P(Fred,Fred,Fred) U NIFY (like(x, M’), like(C’,y)) = {x/C’, y/M’} to yield like(C’,M’) � CS440/ECE448: Intro AI � 11 �


 
 Most General Unifier (MGU) � Are there others? � % is the most general unifier (MGU) of # and $ 
 U NIFY (P(x,y,z) , P(w, w, Fred)) if it imposes the fewest constraints. 
 % = {x=Mary, y=Mary, z=Fred, w=Mary} The MGU of # and $ is unique. 
 (modulo alphabetic variants, i.e. different variable names) � Equivalently: % ’ = {x=Mary, w=y, z=Fred , y=x} Applying the MGU to an expression yields a Both yield P(Mary, Mary, Fred) most general unification instance . � We often define U NIFY ( # , $ ) to return MGU( # , $ ) � What is the MGU? � What is the MGU? � M GU ( m(Ann, x, Bob), m(Ann, x, Bob) ): MGU(P(x,y,z), P(w,w,Fred)) m(Ann, x, Bob) M GU ( m(Ann, x, Bob), m(y, x, Chuck) ): fail . % ={x=w, y=w, z=Fred} yields P(w,w,Fred) M GU ( m(Ann, x, Bob), m(y, x, Father-of(Chuck) ): fail. M GU ( p(w, w, Fred) , p(x, y, y) ): Equivalently, % ={x=u, y=u, w=u, z=Fred} p(Fred, Fred, Fred) yields the alphabetic variant P(u,u,Fred) M GU ( q(r, r), q(x, F(x)) ): fail M GU ( r(g(x,Bob),y, y), r(z,g(Fred,w),z) ): r(g(x,Bob), g(Fred,w), g(Fred(w)))

Generalized modus ponens � If p 1 ’…p n ’, p 1 …p n are atomic sentences with Lifted inference: 
 universally quantified variables, and there is a substitution " such that S UBST ( " , p i ’) = S UBST ( " , p i ) Generalized Modus p 1 ’ … p n ’ (p 1 ∧ … ∧ p n ) ' q Ponens 
 !!!!!!!!!!!!!!!!!!! (GMP) S UBST ( " , q) � � CS440/ECE448: Intro AI � 18 � Generalized modus ponens � With a slight abuse of notation…. � S UBST ( " , p 1 ’ ∧ … ∧ p n ’) S UBST ( " , p 1 ’ ∧ … ∧ p n ’) ' S UBST ( " ,q) Another way to look at GMP: � = # = # ' S UBST ( " ,q) !!!!!!!!!!!!!!!!!!!!!!!!!!! (MP) " makes p 1 ’ ∧ … ∧ p n ’ and p 1 ∧ … ∧ p n equal: � S UBST ( " , q) S UBST ( " , p 1 ’ ∧ … ∧ p n ’ ) = S UBST ( " , p 1 ∧ … ∧ p n ) # # � CS440/ECE448: Intro AI � 19 � CS440/ECE448: Intro AI � 20 �

Generalized modus ponens � Generalized modus ponens � Knowledge base: � ∀ x ∀ y[s(x,y) ∧ p(x) ∧ d(y) ! c(x)] p(s’) d(c’) ∀ zs(s’,z) A person that sells drugs is a criminal . � !!!!!!!!!!!!!!!!!!!!!!!!!!! (GMP) � ∀ x ∀ y [s(x,y) ∧ p(x) ∧ d(y) ! c(x)] � S UBST ({x/s’, y/c’, z/c’}, c(x)) Socrates is a person: p(s’) � ( c(s’) Socrates sells anything: ∀ z s(s’,z) � Cannabis is a drug: d(c’) S UBST ({x/s’,y/c’,z/c’}, ∀ x ∀ y[s(x,y) ∧ p(x) ∧ d(y) ! c(x)]) � ( s(s’,c’) ∧ p(s’) ∧ d(t’) ! c(s’) Query: � Is Socrates a criminal? c(s’) � CS440/ECE448: Intro AI � 21 � CS440/ECE448: Intro AI � 22 � Generalized modus ponens � This is a lifted version of modus ponens: 
 it raises modus ponens from ground Inference with GMP: 
 propositional logic to first-order logic. � Forward chaining for � Lifting is more efficient than definite clauses � propositionalization: only necessary substitutions are made. � CS440/ECE448: Intro AI � 23 �

� � � Generalized Modus Ponens 
 First order definite clauses � in definite clause form � Given (p 1 ∧ … ∧ p n ) ' q and p 1 ’ , … , p n ’ Definite clauses have exactly one positive literal. � with U NIFY (p 1 ∧ … ∧ p n, p 1 ’ ∧…∧ p n ’) = " , � � prove q. Implications: � [p 1 (x 1 ,…,x n ) ∧ … ∧ p m (x 1 ,…,x n )] ' q(x 1 ,…,x n ) As def. clause: (p 1 ∧ … ∧ p n ) ' q ( ¬ (p 1 ∧ … ∧ p n ) ∨ q � premise consequent � ( ¬ p 1 ∨…∨ ¬ p n ∨ q ≡ ¬ [p 1 (x 1 ,…,x n ) ∧ … ∧ p m (x 1 ,…,x n )] ∨ q(x 1 ,…,x n ) ≡ ¬ p 1 (x 1 ,…,x n ) ∨ … ∨ ¬ p m (x 1 ,…,x n ) ∨ q(x 1 ,…,x n ) p 1 ’ … p n ’ ¬ p 1 ∨…∨ ¬ p n ∨ q !!!!!!!!!!!!!!!!!!!!!!!! (MP) Facts: q(x 1 ,…,x n ). S UBST ( " , q) CS440/ECE448: Intro AI � 26 � 1. Americans who sell weapons to enemies are In definite clause form � criminals � ∀ x ∀ y ∀ z [(a(x) ∧ w(y) ∧ e(z) ∧ sell(x,y,z)) ' c(x)] 1. ∀ x ∀ y ∀ z [(a(x) ∧ w(y) ∧ e(z) ∧ sell(x,y,z)) ' c(x)] ¬ a(x) ∨ ¬ w(y) ∨ ¬ e (z) ∨ ¬ sell(x,y,z) ∨ c(x) 2. Nono has some weapons. � ∃ x[owns(N’, x) ∧ w(x)]. 2. ∃ x[owns(N’, x) ∧ w(x)]. 2a) owns(N’, M’) 2b) w(M’) 3. Its weapons were sold by West. � ∀ x[(owns(N’, x) ∧ w(x)) ' sell(W’, N’,x)]. 3. ∀ x[(owns(N’, x) ∧ w(x)) ' sell(W’, N’,x)]. ¬ owns(N’, x) ∨ ¬ w(x) ∨ sell(W’, N’,x). 4. West is an American. 5. Nono is an enemy � a(W’) e(N’) 4. a(W’) a(W’) Query: is West a criminal? c(W’) 5. e(N’) e(N’)

Forward chaining: 
 apply GMP, starting from premises � Inference with 
 owns(N,M) w(M) ¬ owns(N,x) ∨¬ w(x) ∨ sell(W,N,x). ──────────────────────────────────── (GMP 2a, 2b, 3) � definite clauses: 
 6. sell(W,N,M). � a(W) e(N) w(M) sell(W,N,M) ¬ a(x) ∨¬ w(y) ∨¬ e(z) ∨¬ sell(x,y,z) ∨ c(x) backward chaining � ──────────────────────────────────── (GMP 4, 5, 2b, 6, 1) 7. c(W). Yes, West is a criminal. � CS440/ECE448: Intro AI � 29 � Backward chaining � Two ways to use modus ponens � Goal: prove that the literal q’ is true. � # ! $ # � !!!! (MP) 1. Find an implication clause ¬ p 1 ∨…∨ ¬ p n ∨ q $ � such that goal q’ unifies with consequent q. Forward: I know that # implies $ . I also know # . U NIFY (q’, q) = " ’ � Hence, I can conclude that $ is true as well. � 2. Apply " ’ to ¬ p 1 ∨…∨ ¬ p n ∨ q. � S UBST ( " ’, ¬ p 1 ∨…∨ ¬ p n ∨ q) = ¬ p’’ 1 ∨…∨ ¬ p’’ n ∨ q’’ � Backward: I want to know whether $ is true. � � I know that # implies $ . Hence, if I can prove # , 3. Find a unifier " ’’ that allows you to prove that I can conclude that $ is true as well. � each literal p’’ i is true. (Recursion!) � CS440/ECE448: Intro AI � 32 �