Date Topic Readings Due 11/18 First-order logic Ch. 9 HW 6 11/25 First-order inference, Ch. 10 HW7 (Othello) 12/2 Planning. Chs. 11 & 12 HW8.

Transcription

1 Today Lecture 8 Administrivia Discuss test 1 Discuss projects First order logic CS /543 Artificial Intelligence Fall 2008 Administrivia Readings for today: Nov. 18: Chs. 8 & 9 Readings for next week: Ch. 10 New semester average formula: Homework: 20% Test 1 25% Test 2 25% 25% Project 25% Class participation 5% This week s clear & muddy reviewer: Snehal Schedule Date Topic Readings Due 11/18 First-order logic Ch. 9 HW 6 11/25 First-order inference, Ch. 10 HW7 (Othello) 12/2 Planning Chs. 11 & 12 HW8 Test 2 12/9 Probabilistic Chs. 13 & 14 HW9 (Wumpus- reasoning world) 12/16 Project & paper review presentations (finals week) 6:00 8:30 Paper reviews Projects Presentations 4 > 90%: %: 11 < 80%: 6 Average: 83% Test 1 Correct your mistakes: ½ back Omit the emptystack method from problems 4, 5, & 6. Due in 1 week (11/25) No extensions, except for exceptional circumstances Put corrections on separate paper, stapled to your test Class Project I need project proposals from: Megan, Ben, and Chris Project writeups and paper reviews will be due on the last day of class I ll provide instructions later 1

2 Partners Review Othello Code Andrew & Phil Anthony & Robert Haiyang & Joel Alan, Ken, & Uma Ian & Nat Snehal & Jim Mark & Yan Ben & Chris &? Beibei & Zheng James & Siva Where Are We? Expressiveness of propositional logic Heuristic search general AI problem solving engine Adversarial search and zero-sum games Logic and reasoning Propositional logic Simple, but solveable First-order logic Complex, insolveable Knowledge representations systems Logic + bells & whistles Planning Probability A full-size KB for a wumpus-world world agent using propositional logic: P 1,1 W 1,1 B x,y (P x,y+1 P x,y-1 P x+1,y P x-1,y 1,y) S x,y (W x,y+1 W x,y-1 W x+1,y W x-1,y 1,y) W 1,1 W 1,2 W 4,4 W 1,1 W 1,2 W 1,1 W 1,3 64 distinct proposition symbols, 155 sentences Not very expressive! PL summary General inference with propositional logic is NP- complete Resolution is complete for propositional logic Forward, backward chaining are linear-time and complete for Horn clauses Propositional logic lacks expressive power Too many sentences to represent a simple world such as the Wumpus world Have to write a separate rule for every situation, no matter how similar Need variables to be able to write general rules about the world E.g.,., All squares have four corners Basic Elements of FOL Objects: names of objects Functions: Function(termterm 1,, term n ) Refer to objects Can nest functions Variables: x, y, a, b,... Style: capitalize constants, predicates and functions. Variables are lower-case Predicates: Relationships among objects Evaluate to TRUE or FALSE Connectives: Same as for propositional logic ~,,,, Quantifiers:, 2

3 FOL Sentences Atomic sentence: evaluates to TRUE or FALSE predicate (term 1,...,term n ) or (term 1 = term 2 ) Term: referse to an object constant, variable, or function (term 1,..., term n ) Bush is Honest Honest(Bush) King John s brother is Richard the Lionheart Brother(KingJohn, RichardTheLionheart) The left leg of Richard is longer than the left leg of King John >(Length(LeftLegOf(Richard)), Length(LeftLegOf(KingJohn))) The president of the USA in 2003 is honest Honest(President(USA, 2003) Complex sentences Complex sentences are made from atomic sentences using connectives S, S1 S2, S1 S2, S1 S2, S1 S2, If King John is a sibling of Richard, then Richard is a sibling of King John Sibling(KingJohn,Richard) Sibling(Richard,KingJohn) Either 1 is greater than 2 or 1 is less than or equal to 2: >(1, 2) (1, 2) Practice Anyone with two or more spouses is a bigamist All coats in Fred s closet belong to Sarah (use a function for closet) Fred did something to annoy Wilma Blondes have more fun More Practice (with partners) Some students took French in Spring 2001 Every student who takes French passes it Only one student took Greek in Spring 2001 The best score in Greek is always higher than the best score in French Every person who buys a policy is smart No person buys an expensive policy There is an agent who sells policies only to people who are not insured There is a barber who shaves all men in town who do not shave themselves A person born in the UK, each of whose parents is a UK citizen or a UK resident, is a UK citizen by birth A person born outside the UK, one of whose parents is a UK citizen by birth, is a UK citizen by descent Politicians can fool some of the people all of the time, and they can fool all of the people some of the time, but they can t fool all of the people all of the time FOL Inference First-order logic is much more expressive than propositional logic However Complexity of ffoli is much greater than for propositional logic General inference in FOL is semi-decidable: If KB a, the inference procedure will halt; otherwise, the inference procedure may not halt FOL Inference Propositionalization Rewrite FOL KB into PL KB and then make inference Inefficient Horn clauses Forward chaining Backward chaining Resolution 3

4 Substitution Practice θ = {x/jack, y/john}, S = Likes(x, y)), Sθ??? θ = {x/z, y/john}, S = Attached(Head(x), Body(y))), Sθ??? θ = {x/y, y/john), S = Likes(x, y)), Sθ??? Unification Unification is finding substitutions that will make two FOL statements the same Unification allows inference methods to avoid explicitly representing all possible propositionalizations Confusing, normally not done this way The substitution is called the unifier of the two sentences Standardizing Apart Standardizing apart eliminates overlap of variables, e.g., Knows(z17,OJ) X can t be bound to two values Standardizing apart replace all variables in one of the sentence with brand new variables if the variables overlap Unification Practice Unify(Knows(John, x), Knows(John, Jane))??? Unify(Knows(John, x), Knows(y, OJ))??? Unify(Knows(John, x), Knows(y, Mother(y)))??? Unify(Knows(John, x), Knows(x, OJ))??? Unify(Knows(John, x), ) Knows(x, Elizabeth))??? Unify(Knows(John, x),knows( ),Knows(x,, Elizabeth)) Unify(Knows(John, x), Knows(x0001, Elizabeth) θ = {x0001/john, x/elizabeth} Most General Unifier The Forward-chaining Algorithm The unifier that makes the least commitment to bindings is the most general unifier FOL inference requires the most general unifier There is an efficient algorithm for calculating the most general unifier We won t cover it; you nearly always copy the code from someone else if you need it 4

5 FOL Forward Chaining Example KB in FOL Horn form: (American(x) Weapon(y) Hostile(z) Sells(x, y, z) Criminal(x)) Owns(Nono, M1) Missile(M1) (Missile(x) Owns(Nono, x) Sells(West, x,, Nono)) (Missiles(x) Weapons(x)) (Enemy(x) Hostile(x)) American(West) Enemy(Nono, America) Prove that West is a criminal Backward Chaining If the query is a literal, then return the null substitution If you can find a literal that unifies with the query, then return the substitution If you can find a rule whose conclusion unifies with the query: Apply the substitution to the antecedents For each literal in the antecedent Find a proof for it (by calling BC recursively), which results in a substitution being returned Compose that substitution into the original substitution and the substitutions of the remaining antecedents Go on to the next literal in the antecedent. Once all antecedents are satisfied, return the final substitution If an antecedent cannot be satisfied, return failure FOL Backward Chaining Example KB in FOL Horn form: (American(x) Weapon(y) Hostile(z) Sells(x, y, z) Criminal(x)) Owns(Nono, M1) Missile(M1) (Missile( Missile(a) Owns(Nono Nono, a) Sells(West, a, Nono)) (Missiles( Missiles(b) Weapons(b)) (Enemy( Enemy(c) Hostile(c)) American(West) Enemy(Nono, America) Review Othello Code Partners share ideas on where and how to modify code. Is there a criminal: Ask(KB, Criminal(x)). 5

6 CS /543 Lecture 8 Nov. 18 Learning objectives: Students should be able to: Define, discuss, and provide and explain an example of the following first-order logic (FOL) concepts: first-order logic, propositional logic, constants, predicates, functions, variables, quantifiers, atomic sentences, terms, complex sentences, models, interpretation, universal quantification ( ), and existential quantification ( ) Convert a universally quantified sentence into an equivalent existentially quantified sentence. Convert an existentially quantified sentence into an equivalent universally quantified sentence. Convert a set of facts into FOL sentences, if they exist Universal quantification <variables> <sentence> Use it to say something about a group of objects E.g., Everyone at UML is smart: x At(x, UML) Smart(x) x P is true iff P is true for all x Roughly speaking, x P is equivalent to the conjunction of instantiations of P At(KingJohn, UML) Smart(KingJohn) At(Richard, UML) Smart(Richard) At(UML, UML) Smart(UML)... If the premise is FALSE, then the implication is automatically true Thus, the premise describes the types of objects we want to make a statement about A common mistake to avoid Typically, is the main connective of Common mistake: using as the main connective of : Everyone is at UML and everyone is smart If you are using as the main connective with, you are making an error Existential quantification <variables> <sentence>

7 Used to state that at least one object with certain properties exists E.g., Someone at UML is smart: x At(x, UML) Smart(x) x P is true iff P is true for some possible object x Roughly speaking, equivalent to the disjunction of instantiations of P At(KingJohn, UML) Smart(KingJohn) At(Richard,UML) Smart(Richard) At(UML,UML) Smart(UML)... Another common mistake to avoid Typically, is the main connective with Common mistake: using as the main connective with : There is somebody who is either no at UML or is smart If you are using as the main connective with, you are making an error Properties of quantifiers x y is the same as y x x y is the same as y x x y is not the same as y x E.g., x y Loves(x,y)? y x Loves(x,y)? Quantifier duality: each can be expressed using the other E.g., x Likes(x, IceCream) <-->? x Likes(x, Broccoli) <-->? Equality term1 = term2 is true under a given interpretation if and only if term1 and term2 refer to the same object Equality is commonly to constrain two variables to either (1) refer to the same object, or (2) refer to different objects E.g., definition of Sibling in terms of Parent:

8 x, y Sibling(x, y) [ (x = y) m, f (m = f) Parent(m, x) Parent(f, x) Parent(m, y) Parent(f, y)] FOL Writing Practice In class Exercise Break Introduction to FOL inference see PPT slides Propositionalization see PPT slides Substitution and Inference with FOL KBs Suppose a wumpus-world agent is using an FOL KB and wants to know what is the best action Ask(KB, BestAction(a)) I.e., does the KB entail some best action? Suppose the best action is to Shoot (the arrow), then ASK(KB, BestAction(a)) will return {a/shoot} substitution (binding list) Ask(KB,S) returns some/all θ such that KB Sθ Given a sentence S and a substitution θ, Sθ denotes the result of plugging θ into S E.g., θ = {x/k, y/john} x gets k, y gets John or x is replaced by y, y is replaced by John Can t let same variable get two substitutions {x/gary, x/y} is wrong Substitution practice on PPT slide Unification, standardizing apart, and the most general unifier: on PPT slides Practice: substitution and unification examples on slides