First Order Logic CSL 302 ARTIFICIAL INTELLIGENCE SPRING 2014

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1 First Order Logic CSL 302 ARTIFICIAL INTELLIGENCE SPRING 2014

2 Propositional Logic Declarative - pieces of syntax correspond to facts Allows partial/disjunctive/negated information Compositional (B 11 P 12 is derived from B 11 and P 12 ) Context independent Unambiguous Very limited expressive power ocannot say pits cause breeze in adjacent squares, have to list this for each square. 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 2

3 Propositional vs. First Order Logic PL: Deals with facts and propositions that can be true or false op 21 - there is a pit in [2, 1] FOPL: closer to natural language, assumes the world contains objects and relations oobjects - Narayanan, AICourse, 302Student1 orelations - Person Narayanan, Person 302Student1, Teach Narayanan, AICourse, CSStudent 302Student1, Student AICourse, 302Student1, SleepsInTheClass( ) 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 3

4 Working Example 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 4

5 FOL - Syntax Definitions Constant Symbols: Name of a specific object John, Richard, Crown, Predicate Symbols: Properties of relationships between objects otakes the value true or false oargument includes one or more objects King, Sibling, Person, > Function Symbols: relations where there is only value for a given input omapping from objects to object Left_Leg_of, Brother_of, 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 5

6 FOL - Syntax Definitions Terms: logical expression that refers to an object f t 1, t 2,, t n, t i is a specific object. Brother_of(John), Left_Leg_of(John) Variables: Refer to an object without naming it x, y, z1, Connectives: Logical connectives to form complex sentences,,,,, = Quantifiers: express properties of entire collection of objects, instead of exhaustive enumeration, 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 6

7 FOL - Quantifiers - For all - Universal Quantifier ostatements about every object in the world without naming it o x P P is true for every object x oall kings are persons x King x Person(x) oall students in the AI class are smart x AIS x S(x) - There exists - Existential Quantifier ostatement about some object in the world without naming it o x P P is true for some object x ojohn has a crown on his head x Crown x OnHead x, John othere is smart student in the AI class x AIS x S x 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 7

8 Quantifier-Connective interaction x AIS x S x All AI students are smart x AIS x S(x) Everything is a smart AI student x AIS x S x There exists an object that is either smart, or not a AI student at all x AIS x S(x) There exists a smart AI student AIS(x) - x is an AI student S(x) - x is smart 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 8

9 Nested Quantifiers x y is same as y x x y is same as y x x y is not same as y x Every monkey has a tail m t has(m, t) Every body loves somebody x y loves(x, y) Every monkey shares a tail t m has(m, t) Somebody is loved by everybody y x loves(x, y) x Crown x x Brother Richard, x The rule is that the variable belongs to the innermost quantifier that mentions it, it will not then be subjected to any other quantification 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 9

10 Connections between and De Morgan s Laws x P x x P(x) x P(x) x P(x) x P(x) x P(x) x P x x P(x) 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 10

11 FOL - Semantics Semantics - What the arrangement of the symbols means in the world. Propositional Logic obasic elements are propositional variables e.g., P 11 opossible worlds - mappings from variables to T/F First order logic obasic elements are terms e.g., Richard, Father_of(Richard) ointerpretations: mappings from terms to real-world elements 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 11

12 Example Interpretations map syntactic tokens to model elements Constants: Richard, John Functions: LeftLeg(p) Relations: On(x, y), King(p) 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 12

13 Example Interpretations map syntactic tokens to model elements Constants: Richard, John Functions: LeftLeg(p) Relations: On(x, y), King(p) 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 13

14 How many interpretations? Two constants and 5 objects in the world orichard, John (objects: R, J, crown, RL, JL) o5 2 = 25 object mappings One unary relation oking(x): x - infinite number of values; infinite number of mappings osuppose x {R, J, Crown, RL, JL }, 2 5 = 32 truth values Two binary relations oleg(x, y); On(x, y) oif we restrict x and y to 5 objects each, yields mappings for each binary relation. 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 14

15 Satisfiability, Validity and Entailment S is valid if is true in all interpretations S is satisfiable if it is true in some interpretation S is unsatisfiable if it is false in all interpretations S 1 S 2 for all interpretations where S 1 is true, S 2 is also true. 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 15

16 PL vs FOL PL FOL Ontology Facts (P, Q, ) objects, relations, functions Syntax Atomic sentences connectives variables, quantifications, terms, Semantics truth tables interpretations inference algorithm complexity WalkSAT, DPLL NP-Complete unification, forward/backward chaining, theorem proving semi-decidable, may run forever when no entailment 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 16

17 Examples All men are mortal Socrates is a man Socrates is mortal All purple mushrooms are poisonous A mushroom is poisonous only if it is purple No purple mushroom is poisonous There is exactly one mushroom No human enjoys Golf Some professor who is not a historian writes programs Every boy owns a dog 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 17

18 FOL for Wumpus World Objects osquares, Wumpus, agent ogold, pits, stench, breeze Relations osquare topology(adjacency) opits/breezes owumpus/stench 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 18

19 Wumpus World - Squares Each square as an object osquare 11, Square 12, Square 13,, Square 44 Adjacency relation oadjacent(square 11, Square 12 ), oadjacent(square 34, Square 44 ) Better: Squares as lists o[1, 1], [1, 2],,[4, 4] Square topology relations o x, y, a, b: Adjacent x, y, a, b a, b x + 1, y, x 1, y, x, y + 1, x, y 1 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 19

20 Wumpus World - Pits Each pit has an object opit 11, Pit 21,, Pit 44 oproblem? Not all squares have pits. List only the pits we have opit 31, Pit 33, Pit 44 ono reason to distinguish between the pits Pit as a unary predicate opit x, Pit 3, 1 will be true. 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 20

21 Wumpus World Breeze: Represent breezes like pits - unary predicate obreezy x, returns true for squares that are breezy squares adjacent to pits are breezy o a, b, c, d: Pit a, b Adjacent a, b, c, d Breezy c, d Agent s locations: changes over time oat Agent, s, t - agent is in square s at time t. If the agent is at a square and perceives a breeze, then the square is breezy o s, t At Agent, s, t Breeze t Breezy s 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 21

22 Wumpus World Breezy implies possible pits in adjacent squares o s Breezy s r Adjacent s, r Pit r Wumpus - Wumpus as an object owumpus The home square of the Wumpus as a unary predicate owumpusin x Better: Wumpus s home as a function ohome Wumpus references Wumpus s home square. 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 22

23 Reasoning and Inference in FOL Instantiation Techniques opropositionalization ounification oforward Chaining obackward Chaining oresolution 3/10/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 23

24 Universal Instantiation Universally quantified sentence oall students in the AI class are smart x AIS x S(x) Intuitively, x can be anything: oais Tom S Tom oais Chair S Chair oais LeftLeg John Formally: x Subst S LeftLeg John S x/p, S ox is replaced with p (ground term) in S, and the quantifier is removed. 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 24

25 Existential Instantiation Existentially Quantified sentence othere is smart student in the AI class x AIS x S x Intuitively, x must name something. But what? ocan we conclude:ais Tom S Tom??? ono! The sentence might not be true for Tom! Instead, Use a Skolem constant and draw the conclusion, Formally x S Subst x/k, S ok is called a Skolem constant and is a completely new symbol you created. 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 25

26 Inference I: Propositionalization Suppose KB contains just the following: x King x Greedy x Evil(x) King(John) Greedy(John) Brother(Richard, John) Instantiating will result in King John Greedy John Evil John King Richard Greedy Richard Evil Richard King John Greedy John Brother(Richard, John) The new KB is now propositionalized. What are the proposition symbols? 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 26

27 Inference I: Propositionalization Suppose KB contains just the following: x King x Greedy x Evil(x) King(John) Greedy(Richard) Brother(Richard, John) Instantiating will result in King John Greedy John Evil John King Richard Greedy Richard Evil Richard King John Greedy John Brother(Richard, John) The new KB is now propositionalized. What are the proposition symbols? King John, Greedy John, King Richard, 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 27

28 Propositionalization contd. Every FOL KB and query can be propositionalized in such a way that entailment is preserved. opropositionalize the KB and query oapply resolution! Problems can occur with Function Symbols. oe.g., Father Father John Herbrand Theorem: If a sentence α is entailed by a FOL KB, it is entailed by a finite subset of the propositionalized KB. Idea: For n = 0 to do ocreate a propositional KB by instantiating with depth nterms. osee if α is entailed by this KB. 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 28

29 Propositionalization contd. Every FOL KB and query can be propositionalized in such a way that entailment is preserved. opropositionalize the KB and query oapply resolution! Problems can occur with Function Symbols. oe.g., Father Father John Herbrand Theorem: If a sentence α is entailed by an FOL KB, it is entailed by a finite subset of the propositional KB. Idea: For n = 0 to do ocreate a propositional KB by instantiating with depth nterms. osee if α is entailed by this KB. Works fine if α is entailed, but loops otherwise. Similar to halting problem in Turning machines - Semidecidable. 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 29

30 Problems with Propositionalization Can generate lot of irrelevant sentences e.g., suppose we have x King x Greedy x Evil x King John y Greedy y Brother(Richard, John) It seems obvious that Evil John, but propositionalization produces lots of facts such as Greedy Richard that are irrelevant With p k-ary predicates and n constants there are p n k instantiations With function symbols, it gets much much worse!! 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 30

31 Propositionalization contd. What if we want to use modus ponens from propositional logic? α β, α β γ γ In FOL? x AIS x S(x) AIS Ram???? Must unify x with Ram: oneed to substitute x/ram in AIS x S(x)to infer S(Ram) 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 31

32 Unification Going back, suppose we have x King x Greedy x Evil x King John Greedy John Brother(Richard, John) We can get the inference immediately if we can find a substitution θ such that oking(x) and Greedy(x) match with King(John) and Greedy John oθ = x/john works 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 32

33 Unification Going back, suppose we have x King x Greedy x Evil x King John y Greedy y Brother(Richard, John) We can still find a substitution θ such that King(x) and Greedy(x) match King(John) and Greedy y θ = x/john, y/john works UNIFY α, β = θ if αθ = βθ 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 33

34 Unification - Most General Unifier Match up expressions by finding variable values that make the expressions identical Unify AIS x and AIS(Ram) using {x/ram} Unify(a, b) returns the most general unifier. othat places fewest restrictions on values of variables Unify Knows Ram, x, Knows y, z returns { y Ram, x/z} or y Ram, x Ram, z Ram MGU Unification vs Substitution? 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 34

35 Unification Examples Unify Knows John, x, Knows John, Jane Unify Brother x, John, Brother Richard, y Unify Brother x, x, Brother(John, Richard) 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 35

36 Unification Examples Unify Knows John, x, Knows John, Jane Unify Brother x, John, Brother Richard, y Unify Brother x, x, Brother(John, Richard) otwo sentences happen to use the same variable ostandardizing apart - renaming the variables in one of the sentences to avoid name clashes. 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 36

37 Unification Examples Unify Knows John, x, Knows John, Jane Unify Brother x, John, Brother Richard, y Unify Brother x, x, Brother(John, Richard) Unify f g x, dog, y, f g cat, y, dog Unify f g x, f(x) 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 37

38 Unification Examples Unify Knows John, x, Knows John, Jane Unify Brother x, John, Brother Richard, y Unify Brother x, x, Brother(John, Richard) Unify f g x, dog, y, f g cat, y, dog Unify f g x, f(x) oa variable may not contain itself in a substitution ooccur-check 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 38

39 Unification Examples Unify Knows John, x, Knows John, Jane Unify Brother x, John, Brother Richard, y Unify Brother x, x, Brother(John, Richard) Unify f g x, dog, y, f g cat, y, dog Unify f g x, f(x) Unify f g cat, y, y, f x, dog Unify f g y, f x 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 39

40 Generalized Modus Ponens (GMP) AIS Ram x AIS x S(x)???? Unify x with Ram - x/ram in AIS x S(x) Apply MP to infer S(Ram) Generalized Modus Ponens (GMP) o Lifted version of MP (lifts MP from ground Pl to FOL) p 1, p 2,, p n, (p 1 p 2 p n q) q where p i θ = p i θ for all i GMP used with KB of definite clauses, all variables assumed universally quantified. Prove GMP is sound! 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 40

41 Inference II: Forward Chaining The algorithm: Start with the KB Add any fact you can generate with GMP (unify + GMP) Repeat until goal is reached or generation halts. 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 41

42 Forward Chaining Example It is a crime for an Indian to sell weapons to hostile nations. The country Nono, an enemy of India, has some missiles. All of its missiles were sold to it by Traitor, who is an Indian. Is Traitor a criminal? 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 42

43 Forward Chaining Example It is a crime for an Indian to sell weapons to hostile nations. The country Nono, an enemy of India, has some missiles. All of its missiles were sold to it by Traitor, who is an Indian. Is Traitor a criminal? Criminal Traitor? KB of definite clauses (exactly one positive literal) o Indian x Weapon y Sells x, y, z Hostile z Criminal(x) o Enemy(Nono, India) o Owns Nono, M 1 M 1 is a Skolem constant o Missile M 1 o Missile x Owns Nono, x Sells(Traitor, x, Nono) o Indian Traitor o Missle x Weapon x o Enemy x, India Hostile(x) 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 43

44 Forward Chaining Example Missile x Owns Nono, x Sells Traitor, x, Nono Missle x Weapon x Enemy x, India Hostile(x) Indian Traitor Missile M 1 Owns Nono, M 1 Enemy(Nono, India) Initial Facts in the KB 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 44

45 Forward Chaining Example Missile x Owns Nono, x Sells Traitor, x, Nono Missle x Weapon x Enemy x, India Hostile(x) Weapon M 1 Sells Traitor, M 1, Nono Hostile Nono 3 x/m x/nono x/m 1 Indian Traitor Missile M 1 Owns Nono, M 1 Enemy(Nono, India) Initial Facts in the KB 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 45

46 Forward Chaining Example Indian x Weapon y Sells x, y, z Hostile z Criminal(x) Therefore, Traitor is a criminal Criminal Traitor x Traitor, y M 1, z Nono Weapon M 1 Sells Traitor, M 1, Nono Hostile Nono 3 x/m x/nono x/m 1 Indian Traitor Missile M 1 Owns Nono, M 1 Enemy(Nono, India) Facts after 1 st iteration 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 46

47 Inference II: Forward Chaining Sound oyes, Because GMP is sound Complete oyes, if KB contains only definite clauses Check pg. 331 in textbook Efficiency ounification via exhaustive pattern matching orule rechecking for premise satisfaction at every iteration oirrelevant fact generation Check section for possible strategies 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 47

48 Inference III: Backward Chaining Start with KB and goal Find all rules whose results unify with goal: Add the premises of these rules to the goal list Remove the corresponding result from the goal list Stop when goal list is empty(success) or progress halts (failure) 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 48

49 Backward Chaining Example Goal Criminal Traitor 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 49

50 Backward Chaining Example Indian x Weapon y Sells x, y, z Hostile z Criminal(x) Criminal Traitor x Traitor 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 50

51 Backward Chaining Example Depth-First Traversal Criminal Traitor x Traitor Indian x Weapon y Sells x, y, z Hostile z 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 51

52 Backward Chaining Example Depth-First Traversal Criminal Traitor x Traitor Indian Traitor Weapon y Sells x, y, z Hostile z 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 52

53 Backward Chaining Example Depth-First Traversal Criminal Traitor x Traitor Indian Traitor Weapon y Sells x, y, z Hostile z New Subgoal 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 53

54 Backward Chaining Example Depth-First Traversal KB: Missile y Weapon y ; Missile M 1 Criminal Traitor x Traitor, y M 1 Indian Traitor Weapon y Sells x, y, z Hostile z Missile y y M 1 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 54

55 Backward Chaining Example Depth-First Traversal Criminal Traitor x Traitor, y M 1 Indian Traitor Weapon y Sells x, y, z Hostile z New Subgoal Missile y z? y M 1 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 55

56 Backward Chaining Example Depth-First Traversal Missile y Owns Nono, y Sells Traitor, y, Nono Missile M 1 Owns Nono, M 1 Criminal Traitor x Traitor, y M 1, z Nono Indian Traitor Weapon y Sells x, y, z Hostile z z Nono Missile y Missile M 1 Owns Nono, M 1 y M 1 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 56

57 Backward Chaining Example Depth-First Traversal Criminal Traitor x Traitor, y M 1, z Nono Indian Traitor Weapon y Sells x, y, z Hostile z z Nono New Subgoal Missile y Missile M 1 Owns Nono, M 1 y M 1 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 57

58 Backward Chaining Example Depth-First Traversal KB: Enemy z, India Hostile z ; Enemy(Nono, India) Criminal Traitor x Traitor, y M 1, z Nono Indian Traitor Weapon y Sells x, y, z Hostile z z Nono Missile y Missile M 1 Owns Nono, M 1 Enemy(Nono, India) y M 1 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 58

59 Backward Chaining Example Depth-First Traversal Criminal Traitor x Traitor, y M 1, z Nono Indian Traitor Weapon y Sells x, y, z Hostile z z Nono Missile y Missile M 1 Owns Nono, M 1 Enemy(Nono, India) y M 1 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 59

60 Inference III: Backward Chaining Depth-First recursive search: space is linear in size of proof. Incomplete due to infinite loops(e.g., repeated states) ofix by checking current goal against all goals on stack ocannot fix infinite paths though Inefficient due to repeated computations ofix using caching of previous results (extra space) Widely used for Logic Programming oprolog (Check Section 9.4 for more information). 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 60

61 Inference IV: Resolution Recall from propositional logic p q, p r s R q r s Literal in one clause Its negation in the other clause Result is the disjunction of the remaining literals. 3/9/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 61

62 Inference IV: Resolution In general p x q A, p B r x s y R q A r B s y l 1 l k, m 1 m n (l 1 l i 1 l i+1 l k m 1 m j 1 m j+1 m n )θ Where Unify l i, m j = θ The two clauses are assumed to be standardized apart so that they share no variables. Apply resolution steps to CNF(KB α). Substitute MGU x/b in all literals 3/10/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 62

63 FOL: Conversion to CNF Everyone who loves all animals is loved by someone x y Animal y Loves x, y y Loves(y, x) Steps to convert to CNF oeliminate biconditionals and implications omove inwards x, p x p, x p x p ostandardize the variables: each quantifier should use a different variable oskolemize: A more general form of existential instantiation Each existential variable is replaced by a Skolem function of the enclosing universally quantified variables odrop universal quantifiers odistribute over 3/10/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 63

64 FOL: Conversion to CNF It is a crime for an Indian to sell weapons to hostile nations. The country Nono, an enemy of India, has some missiles. All of its missiles were sold to it by Traitor, who is an Indian. o Indian x Weapon y Sells x, y, z Hostile z Criminal(x) o Enemy(Nono, India) o Owns Nono, M 1 o Missile M 1 o Missile x Owns Nono, x Sells(Traitor, x, Nono) o Indian Traitor o Missle x Weapon x o Enemy x, India Hostile(x) Resolution uses proof by contradiction o Show KB α by showing KB α is unsatisfiable. Variables are not standardized here. To prove, Traitor is a criminal. Add Criminal Traitor to KB and derive empty clause. 3/10/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 64

65 FOL Resolution Let s try it 3/10/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 65

66 Logic gone crazy 3/10/2014 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 66

First-order logic. Chapter 8, Sections 1 3

First-order logic. Chapter 8, Sections 1 3 First-order logic Chapter 8, Sections 1 3 Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides c Stuart Russel and Peter Norvig, 2004 Chapter 8, Sections 1 3 1 Outline Why FOL? Syntax