ARTIFICIAL INTELLIGENCE

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1 ARTIFICIAL INTELLIGENCE LECTURE # 04 Artificial Intelligence 2012 Lecture 04 Delivered By Zahid Iqbal 1

2 Review of Last Lecture Artificial Intelligence 2012 Lecture 04 Delivered By Zahid Iqbal 2

3 Review Reasoning Types of Reasoning Logic Propositional Logic Artificial Intelligence 2012 Lecture 04 Delivered By Zahid Iqbal 3

4 Today s Lecture Review of previous lecture Limitations of Propositional Logic Predicate Calculus First Order Predicate Logic First Order Predicate Logic Constructs Inference Rules Artificial Intelligence 2012 Lecture 04 Delivered By Zahid Iqbal 4

5 Limitations of Propositional Logic We Can t describe things in terms of their properties or relationships (very limited expressive power) Propositional logic is declarative Propositional logic is compositional. We can t express rules or generalizations If the train is late and there are no taxis, john is late for the meeting If trains are late and there are no taxis, anyone traveling by trains is late for the meeting Artificial Intelligence 2012 Lecture 04 Delivered By Zahid Iqbal 5

6 Limitations Propositions can only represent knowledge as complete sentences, e.g. a = the ball s color is blue. Cannot analyze the internal structure of the sentence. No quantifiers are available, e.g. for-all, there-exists Propositional logic provides no framework for proving statements such as: All humans are mortal All women are humans Therefore, all women are mortals This is a limitation in its representational power. Artificial Intelligence 2012 Lecture 03 Delivered By Zahid Iqbal 6

7 Predicate Calculus Provides a richer modeling language We have objects and properties We have relationships between objects We have quantification an ability to refer to all or some objects Retain connectives such as ~ = Instead of looking at sentences that are of interest merely for their truth values, predicate calculus is used to represent statements about specific objects or individuals. Examples of individuals: you, this page of lecture, the number 1, Socrates Artificial Intelligence 2012 Lecture 04 Delivered By Zahid Iqbal 7

8 Predicate Calculus (cont.) A predicate is that which says something about the subject. e. g., The book is red. subject color of the book represented as: is-red(book) or simply red(book) is-red: predicate book: argument A predicate statement takes the value true or false. Artificial Intelligence 2012 Lecture 04 Delivered By Zahid Iqbal 8

9 Predicate Calculus (cont.) red(book) is true if the book is red, false if it is not, then ~red( book) becomes false. Predicate with one argument is called a 1- place predicate. A predicate can have more than 1 argument: e. g., color( book, red) mother( john, mary) greater- than( 7, 4) The number of arguments of a predicate is called its arity. book, red, john, mary, 7, 4 are constants. We need variables and quantifiers to express sentences such as Everyone likes ice cream Ali has some friends Artificial Intelligence 2012 Lecture 04 Delivered By Zahid Iqbal 9

10 Predicate Calculus (cont.) " for all, for every (universal quantifier) $ there exists (some) (existential quantifier ) Examples: "X likes(x, ice_ cream) $Y friends(y, Ali) The quantifier specifies the extension of the variable (the total number of objects it applies, or the range of values it can take).. Universal and existential quantifiers allow expressing general rules with variables Universal quantification All cats are mammal It is equivalent to the conjunction of all the sentences obtained by substitution the name of an object for the variable x. Artificial Intelligence 2012 Lecture 04 Delivered By Zahid Iqbal 10

11 Predicate Calculus (cont.) Functions -- have a fixed number of arguments (arity) -- return (or evaluate to) objects instead of truth values. e. g., uncle- of( mary) = john plus( 4, 3) = 7 Arguments can be constants, variables, or functions. e. g., father- of( father- of( john)) Sometime we use something called a term, which is either a constant, variable, or function expression Artificial Intelligence 2012 Lecture 04 Delivered By Zahid Iqbal 11

12 Quantification: Existential Existential quantification : an existentially quantified sentence is true in case one of the disjunct is true $ x Sister(x, Ibrahem) // $ : There exists Equivalent to disjunction: Sister(fatima, ibrahem) V Sister(zainab, ibrahem) V Sister(kalsoom, ibrahem) We can mix existential and universal quantification. Artificial Intelligence 2012 Lecture 04 Delivered By Zahid Iqbal 12

13 Predicate Calculus Syntax Every atomic sentence is a sentence. If s is a sentence, so is ~s. If s 1 and s 2 are sentences, so is s 1 s 2 ;. so is s 1 s 2 ;. so is s 1 s 2 ;. so is s 1 = s 2 ; If X is a variable and s a sentence, then "X s is a sentence.. then $X s is a sentence. For example: "X "Y father( X, Y) mother( X, Y) parent( X, Y) is a well- formed predicate calculus sentence. Artificial Intelligence 2012 Lecture 04 Delivered By Zahid Iqbal 13

14 First-order logic Simplest form of predicate logic, Propositional logic assumes the world contains facts First-order logic (like natural language) assumes the world contains Objects: people, houses, numbers, colors, baseball games, wars, centuries properties: blue, small, tall, ugly,... Relations: prime, brother of, bigger than, part of, comes between, bogus Functions: father of, best friend, third inning of, one more than, plus, Artificial Intelligence 2012 Lecture 04 Delivered By Zahid Iqbal 14

15 FOPC The main types of symbols used are Constant: name specific object Predicates: a fact or predicate is divided into 2 parts. Predicate: the assertion of the proposition Argument: the object of the proposition. E.g. Ali likes bananas in predicate logic will b, Likes( Ali, bananas), Variables: use for general representation of objects Likes ( X, Y), Formula: combine predicate and quantifiers to represent information. Connectives: ^, v, ~, Quantifiers: $, " Function: father(x) = y: A function that specifies the unique element, that is the father of Ali Artificial Intelligence 2012 Lecture 04 Delivered By Zahid Iqbal 15

16 FOL: Basic Elements Predicates, variables, functions, constants, connectives, quantifiers Constants: (first letter small) blue a color santro a car crow a bird Variables: (first letter capital) Represent general class of objects/properties Dog: an element that is a dog, but unspecified Color: an unspecified color

17 We can have functions with arity > 1, e.g. student(amar, mcs) FOL: Basic Elements Function (evaluates to a constant / variable)maps Sentences to Objects Denote a mapping 4m the domain of function to range of function. Ali is father of Babar Babar is son of Ali father(babar) = ali son(ali) = babar If you write father(baber), the answer should be ali For the above functions the arity is 1 (number of arguments to the function)

18 FOL: Basic Elements Functions: 1) ali likes akram likes(ali) = akram 2) atif likes abid likes(atif) = abid 3) Constants to Variables likes(x) = Y {X,Y} have two possible BINDINGS {X, Y} could be {ali, akram} Or {X,Y} could be {atif, abid}

19 FOL: Basic Elements Predicate Maps Sentences to Truth Values (True/False) Predicate name a relationship b/w zero or more objects in the world. (arity) 1) Shahid is student student(shahid) 2) Sana is a girl girl(sana) 3) Father of baber is elder than Hamza elder(father(babar), hamza) For 1 and 2 arity is 1 and for 3 the arity is 2

20 FOL: Basic Elements Predicate 1) Shahid is a good student student(shahid,good) or good_student(shahid) or is_good(shahid,student) 2) Sana is a friend of Saima, Sana and Saima both are girls friend_of(sana,saima)^girl(sana)^girl(saima) 3) Bill helps Fred helps(bill,fred)

21 FOL: Basic Elements Connectives: ^ and v or ~ not Implication Quantification To express sentences like: All persons can see There is a person who cannot see Use: Universal quantifiers " (ALL) Existential quantifiers $ (There exists)

22 Complex sentences Complex sentences are made from atomic sentences using connectives S, S 1 S 2, S 1 S 2, S 1 S 2, S 1 S 2, sibling(ali,hamza) sibling(hamza,ali) >(1,2) (1,2) (1 is greater than 2 or less than equal to 2) >(1,2) >(1,2) (1 is greater than 2 and is not greater than equal to 2)

23 Universal quantification "<variables> <sentence> Every girl at UOG is smart: "X at(x, uog) smart(x) Roughly speaking, equivalent to the conjunction of instantiations of P at(rabia,uog) smart(rabia) at(safia,uog) smart(safia) at(amna,uog) smart(amna) Typically, is the main connective with ". Common MISTAKE: ^ is main connective with ". "X at(x, uog) smart(x) Every girl is at UOG and every girl is smart.

24 Existential quantification $<variables> <sentence> Some boys at UOG are smart: $X at(x,uog) smart(x) Roughly speaking, equivalent to the disjunction of instantiations of P at(amir,uog) smart(amir) at(bashir,uog) smart(bashir) at(asim,uog) smart(asim)... Typically, is the main connective with $.

25 Properties of quantifiers Quantifiers can be nested E.g. "X $Y or $X "Y "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 $X "Y loves(x,y) There is a person who loves everyone in the world "Y $X loves(x,y) Everyone in the world is loved by at least one person

26 Properties of quantifiers (Contd.) Quantifier duality: each can be expressed using the other "X likes(x,car) $X likes(x,car) $X likes(x,bread) "X likes(x,bread) "X likes(x,car) $X likes(x,car) $X likes(x,bread) "X likes(x,bread)

27 FOPC Inference Rules The ability.to infer new correct expressions from a set of true assertion is an important feature of the predicate calculus Allow the deduction of new sentences from previously given sentences. If we know that [all humans are mortal] is true, and that [Socrates is a human] is true than we can conclude that, [Socrates is mortal] In FOPC. e. g., "X human( X) mortal( X) human( Socrates) It should logically follow that: mortal( Socrates) Artificial Intelligence 2012 Lecture 04 Delivered By Zahid Iqbal 27

28 Some rules of inference Here are some examples of sound rules of inference A rule is sound if its conclusion is true whenever the premise is true Each can be shown to be sound using a truth table RULE PREMISE CONCLUSION Modus Ponens A, A B B And Introduction A, B A B And Elimination A B A Double Negation A A Unit Resolution A B, B A Resolution A B, B C A C

29 Some Useful Inference Rules modus ponens (MP) If P is true and P Q is true then Q is true e. g., If we know that John is an uncle is true and that If John is an uncle then John is male is true. Then we can conclude that John is male is true. Let P = John is an uncle Q = John is male Hence if P is true and P Q is true then Q is true. This is known as the modus ponens rule, or the implication elimination rule. 29

30 inference rules Modus Ponens (MP) P, P Q Q Modus pones can also be applied to expression containing variables.?????????????????? "X (man(x) mortal( X) man(waleed) It should logically follow that: mortal(waleed)

31 inference rules Modus Tolens If X is true then Y is true. ( X Y) Y is false. ~Y Therefore X is false. ~A Example If there is smoke, there is fire. There is not fire, so there is no smoke. If I am happy, then I smile. I am not smiling, therefore I am not happy. if P Q is true and Q is false or ~Q is true then ~P is true e. g., sick( student) not_ attend_ lecture( student) ~not_ attend_ lecture( student) produces: ~sick( student) 31

32 Inference Rules And-Introduction (AI) P Q P ^ Q Intelligent( Saira) CSMajor (Saira) Intelligent(Saira) ^ CSMajor( Saira)

33 Inference Rules And-Elimination (AE) P ^Q P Intelligent(Saira) ^ CSMajor( Saira) Intelligent( Saira)

34 Inference Rules Universal Elimination (UE) Any universal quantified variable in a true sentence is replaced by any appropriate term form the domain, the result is a true sentence. "x Takes( x, AI) & Intelligent(x) Takes( Pat, AI ) & Intelligent (Pat) The substitution has to be done by a Ground Term.

35 Methods of Inference Forward Chaining: Starts with the available data and uses inference rules to extract more data until a goal is reached. Backward Chaining: Starts with a goal and works backwards from the goal to the facts to see if there is data available that will support the goal

36 Forward Chaining From "x[man(x) mortal(x)], we infer that man(talha) mortal(talha) using UE. From man(talha) and man(talha) mortal(talha), we infer mortal(talha) using Modus Ponens. We started with the available data and used inference rules to extract more data until a goal is reached.

37 Backward Chaining We start with goal. man(talha) mortal(talha) using UE. To show that mortal(talha), we have to show man(talha). Man(Talha) is given in KB. Therefore, mortal(talha). We Started with a goal and worked backwards from the goal to the facts to see if there is data available that will support the goal

38 Resolution Rule Deduction mechanism we discussed above, using the four rules of inference may be used in practical systems, but is not feasible. It uses a lot of inference rules that introduce a large branch factor in the search for a proof. An alternative is approach is called resolution, a strategy used to determine the truth of an assertion, using only one resolution rule: A B ~B C A C Artificial Intelligence 2012 Lecture 04 Delivered By Zahid Iqbal 38

39 Resolution-Truth Table Artificial Intelligence 2012 Lecture 04 Delivered By Zahid Iqbal 39

40 Conjunctive Normal Form (CNF) Resolution requires all sentences to be converted into a special form called conjunctive normal form (CNF). A statement in conjunctive normal form (CNF) consists of ANDs of ORs. The outermost structure is made up of conjunctions. Inner units called clauses are made up of disjunctions. The components of a statement in CNF are clauses and literals. A clause is the disjunction of many units. The units that make up a clause are called literals. And a literal is either a variable or the negation of a variable. Artificial Intelligence 2012 Lecture 04 Delivered By Zahid Iqbal 40

41 Conversion to CNF Eliminate arrows (implications) A B = ~A B 2. Drive in negations using De Morgan s Laws, which are given below ~(A B) = ~A ~B 3. Distribute OR over AND A (B C) = (A B) (A C) Artificial Intelligence 2012 Lecture 04 Delivered By Zahid Iqbal 41

42 Example: CNF (A B) (C D) 1. ~(A B) (~C D) 2. (~A ~B) (~C D) 3. (~A ~C D) (~b ~C D) Artificial Intelligence 2012 Lecture 04 Delivered By Zahid Iqbal 42

43 Resolution by Refutation Now, we will look at a proof strategy called resolution refutation. The steps for proving a statement using resolution refutation are: 1. Write all sentences in CNF 2. Negate the desired conclusion 3. Apply the resolution rule until you derive a contradiction or cannot apply the rule anymore. 4. If we derive a contradiction, then the conclusion follows from the given axioms 5. If we cannot apply anymore, then the conclusion cannot be proved from the given axioms Artificial Intelligence 2012 Lecture 04 Delivered By Zahid Iqbal 43

44 Resolution-Refutation Example 1 Artificial Intelligence 2012 Lecture 04 Delivered By Zahid Iqbal 44

45 Resolution-Refutation Example 2 Artificial Intelligence 2012 Lecture 04 Delivered By Zahid Iqbal 45

46 Resolution-Refutation Example 2 Artificial Intelligence 2012 Lecture 04 Delivered By Zahid Iqbal 46

47 Resolution-Refutation Example 2 Artificial Intelligence 2012 Lecture 04 Delivered By Zahid Iqbal 47

48 References Artificial Intelligence, A modern approach by Russell: (Chapter # 8,9) Artificial Intelligence: Structures and Strategies for Complex Problem Solving (Chapter # 2) Internet Artificial Intelligence 2012 Lecture 04 Delivered By Zahid Iqbal 48

49 End of Lecture Artificial Intelligence 2012 Lecture 04 Delivered By Zahid Iqbal 49