Agent that reason logically

1. Agent that reason logically 지식표현

2. Knowledge Base • A set of representations of facts about the world • Knowledge representation language • tell : what has been told to the knowledge base previously • ask : a question and the answer • Inference : what follows from what the KB has been Telled • Background knowledge : a knowledge base which may initially contained • Sentence : individual representation of a fact Agent that reason logically

3. Knowledge base • The knowledge level :: saying what it knows to KB  “Golden Gates Bridge links San Francisco and Marin Country • The logical level :: the knowledge is encoding into sentences  Links(GGBridge, SF, Marin) • The implementation level :: the level that runs on the agent architecture (data structures to represent knowledge or facts) Agent that reason logically

4. Knowledge • declarative/procedural • love(john, mary). • can_fly(X) :- bird(X), not(can_fly(X)), !. • learning : general knowledge about the environment given a series of percepts • Commonsense knowledge Agent that reason logically

5. Figure 6.2 A typical wumpus world Specifying the environment Agent that reason logically

6. Domain specific knowledge • Domain specific knowledge • In the squares directly adjacent to a pit, the agent will perceive a breeze • Commonsense knowledge • logical reasoning • stench(1,2) & ~setnch(2,1)  ~wumpus(2,2) • wumpus(1,3)  stench(2,1) & stench(2,3) & stench(1,4) Agent that reason logically

7. 1,4 2,4 3,4 4,4 A = Agent B = Breeze G = Glitter, Gold OK = Safe square P = Pit S = Stench V = Visited W = Wumpus 1,4 2,4 3,4 4,4 1,3 2,3 3,3 4,3 1,3 2,3 3,3 4,3 1,2 2,2 3,2 4,2 1,2 2,2 3,2 4,2 P ? OK 1,1 2,1 3,1 4,1 1,1 2,1 3,1 4,1 A V A B OK OK OK OK • Figure 6.3 The first step taken by the agent in the wumpus world. • The initial situation, after percept [None, None, None, None, None]. • After one move, with percept [None, Breeze, None, None, None]. Inference in Wumpus world(I) Agent that reason logically

8. 1,4 2,4 3,4 4,4 A = Agent B = Breeze G = Glitter, Gold OK = Safe square P = Pit S = Stench V = Visited W = Wumpus 1,4 2,4 3,4 4,4 P ? 1,3 2,3 3,3 4,3 1,3 2,3 3,3 4,3 W ! A W! P ? S G B S 1,2 2,2 3,2 4,2 1,2 2,2 3,2 4,2 A V V OK OK OK OK B B 1,1 2,1 3,1 4,1 1,1 2,1 3,1 4,1 V V V V P ! OK OK OK OK • Figure 6.4 Two later stages in the progress of the agent. • After the third move, with percept [Stench, None, None, None, None]. • After the fifth move, with percept [Stench, Breeze, Glitter, None, None]. Inference in Wumpus world(II) Agent that reason logically

9. Representation, Reasoning, and Logic • Syntax : the possible configurations that constitute sentences • Semantics : the facts in the world to which the sentences refer Agent that reason logically

10. The logical reasoning Figure 6.5 The connection between sentences and facts is provided by the semantics of the language. The property of one fact following from some other facts is mirrored by the property of one sentence being entailed by some other sentences. Logical inference generates new sentences that are entailed by existing sentences. Agent that reason logically

11. Inference I • Entailment :: generation of new sentences that are necessarily true, given that the old sentences are true • Soundness, truth-preserving :: An inference procedure that generates only entailed sentences  modus ponens <-> abduction • KB├i,  is derived from KB by I • Proof :: a sound inference procedure Agent that reason logically

12. Inference II • Completeness :: an inference procedure that can find a proof for any sentence that is entailed • Proof :: specifying the reasoning steps that are sound • Valid :: if and only if all possible interpretations in all possible worlds • Tautologies, analytic sentences :: valid sentences • Satisfiable :: if and only if there is some interpretation in some world for which it is true • Unsatisfiable :: a sentence that is not satisfiable Agent that reason logically

13. Logics • Boolean logic • Symbols represent whole propositions (facts) • Boolean connectives • First-order logic • objects, predicates • connectives, quantifiers Agent that reason logically

14. Wrong logical reasoning FIRST VILLAGER: We have found a witch. May we burn her? ALL: A witch! Burn her! BEDEVERE: Why do you think she is a witch? SECOND VILLAGER: She turned me into a newt. BEDEVERE: A newt? SECOND VILLAGER (after looking at himself for some time): I got better. ALL: Burn her anyway. BEDEVERE: Quiet! Quiet! There are ways of telling whether she is a witch. BEDEVERE: Tell me … What do you do with witches? ALL: Burn them. BEDEVERE: And what do you burn, apart from witches? FOURTH VILLAGER: … Wood? BEDEVERE: So why do witches burn? SECOND VILLAGER: (pianissimo) Because they’re made of wood? BEDEVERE: Good. ALL: I see. Yes, of course. BEDEVERE: So how can we tell if she is made of wood? FIRST VILLAGER: Make a bridge out of her. BEDEVERE: Ah … but can you not also make bridges out of stone? ALL: Yes, of course … um … er … BEDEVERE: Does wood sink in water? ALL: No, no, it floats. Throw her in the pond. BEDEVERE: Wait. Wait … tell me, what also floats on water? ALL: Bread? No, no no. Apples … gravy … very small rocks … BEDEVERE: No, no no. KING ARTHUR: A duck! (They all turn and look at ARTHUR. BEDEVERE looks up very impressed.) BEDEVERE: Exactly. So … logically … FIRST VILLAGER (beginning to pick up the thread): If she .. Weight the same as a duck … she’s made of wood. BEDEVERE: And therefore? ALL: A witch! Agent that reason logically

15. Ontological and epistemological commitments • Ontological commitments :: to do with the nature of reality • Propositional logic(true/false), Predicate logic, Temporal logic • Epistemological commitments :: to do with the possible states of knowledge an agent can have using various types of logic • degree of belief • fuzzy logic Agent that reason logically

16. Commitments Formal languages and their and ontological and epistemological commitments Agent that reason logically

17. Propositional Logic • logical constant : true/false • propositional symbols : P, Q • parentheses : (P & Q) • logical connectives : &(conjuction), v(disjunction), ->(implication), <->(equivalence), ~(negation) Agent that reason logically

18. Grammar Sentence  AtomicSentence | ComplexSentence AtomicSentence True |False | P | Q | R | … ComplexSentence  ( Sentence ) | Sentence Connective Sentence | Sentence Connective   |  |  |  Figure 6.8 A BNF (Backus-Naur Form) grammar of sentences in propositional logic. Agent that reason logically

19. Semantics Truth table showing validity of a complex sentence Agent that reason logically

20. Validity and Inference Truth tables for five logical connectives Agent that reason logically

21. Models • Any world in which a sentence is true under a particular interpretation • Entailment :: a sentence  is entailed by a knowledge base KB if the models of the KB are all models of  • The set of models of P & Q is the intersection of the models of P and the models of Q Agent that reason logically

22. Inference Rules for propositional logic  => ,   1  2  …  n • Modus Ponens or Implication-Elimination: (From an implication and the premise of the implication, you can infer the conclusion.) • And-Elimination: (From a conjunction, you can infer any of the conjuncts.) • And-Introduction: (From a list of sentences, you can infer their conjunction.) • Or-Introduction: (From a sentence, you can infer its disjunction with anything else at all.) • Double-Negation Elimination: (From a doubly negated sentence, you can infer a positive sentence.) • Unit Resolution: (From a disjunction, if one of the disjuncts is false, then you can infer the other one is true.) i 1, 2, …, n 1  2  …  n i 1  2  …  n     ,    • Resolution: (This is the most difficult. Because  cannot be both true and false, one of the other disjucts must be true in one of the premises. Or equivalently, implication is transitive.)   ,       => ,  =>  or equivalently      =>  Figure 6.13 Seven inference for propositional logic. The unit resolution rule is a special case of the resolution rule, which in turn is a special case of the full resolution rule for first-order logic discussed in Chapter 9. Agent that reason logically

23. Complexity of propositional inference • NP-complete • Monotonicity • If KB1╞  then (KB1 ∪ KB2) ╞  • Horn clause logic • polynomial time complexity • P1∧P2∧….∧Pn ⇒ Q Agent that reason logically

24. Wumpus world • Initial state ~S1,1 ~B1,1 ~S2,1 B2,1 S1,2 ~B1,2 • Rule R1: ~S1,1 -> ~W1,1 & ~W1,2 & ~W2,1 R2: ~S2,1 -> ~W1,1 & ~W2,1 & ~W2,2 & ~W3,1 R3: ~S1,2 -> ~W1,1 & ~W1,2 & ~W2,2 & ~W1,3 R4: S1,2 -> W1,3 V W1,2 V W2,2 V W1,2 Agent that reason logically

25. Finding the wumpus • Inference process • Modus ponens : ~S1,1 and R1  ~W1,1 & ~W1,2 & ~W2,1 • And-Elimination ~W1,1 ~W1,2 ~W2,1 • Modus ponens and And-Elimination: ~W2,2 ~W2,1 ~W3,1 • Modus ponens S1,2 and R4  W1,3 V W1,2 V W2,2 V W1,1 Agent that reason logically

26. Inference process(cont.) • unit resolution ~W1,1 and W1,3 V W1,2 V W2,2 V W1,1  W1,3 V W1,2 V W2,2 • unit resolution ~W2,2 and W1,3 V W1,2 V W2,2  W1,3 V W1,2 • unit resolution ~W1,2 and W1,3 V W1,2  W1,3 Agent that reason logically

27. Translating knowledge into action • A1,1 & EastA & W2,1 -> ~Forward EastA :: facing east • Propositional logic is not powerful enough to solve the wumpus problem easily Agent that reason logically

28. 숙제 • 6.3, 6.6, 6.7, 6.9, 6.10, 6.12, 6.15, 6.16 Agent that reason logically

29. First-order Logic

30. Limitation of propositional logic • A very limited ontology •  to need to the representation power •  first-order logic Agent that reason logically

31. First-order logic • A stronger set of ontological commitments • A world in FOL consists of objects, properties, relations, functions • Objects people, houses, number, colors, Bill Clinton • Relations  brother of, bigger than, owns, love • Properties  red, round, bogus, prime • Functions father of, best friend, third inning of Agent that reason logically

32. Examples • “One plus two equals three” • objects :: one, two, three, one plus two • Relation :: equal • Function :: plus • “Squares neighboring the wumpus are smelly • Objects :: wumpus, square • Property :: smelly • Relation :: neighboring Agent that reason logically

33. First order logics • Objects와 relations • 시간, 사건, 카테고리 등은 고려하지 않음 • 영역에 따라 자유로운 표현이 가능함  ‘king’은 사람의 property도 될 수 있고, 사람과 국가를 연결하는 relation이 될 수도 있다 • 일차술어논리는 잘 알려져 있고, 잘 연구된 수학적 모형임 Agent that reason logically

34. Syntax and Semantics Sentence  AtomicSentence | Sentence Connective Sentence | Auantifier Variable,…Sentence | Sentence | (Sentence) AtomicSentence  Predicate(Term,…) | Term=Term TermFunction (Term,…) | Constant | Variable Connective   |  |  |  Quantifier   |  Constant  A | X1 | John | … Variable  a | x | s | … Predicate  Before | HanColor | Raining | … Function  Mother | LeftLegOf | … Figure 7.1 The syntax of first-order logic (with equality) in BNF (Backus-Naur Form). Agent that reason logically

35. 예 • Constant symbols :: A, B, John, • Predicate symbols :: Round, Brother • Function symbols :: Cosine, FatherOf • Terms :: King John, Richard’s left leg • Atomic sentences :: Brother(Richard,John), Married(FatherOf(Richard), MotherOf(John)) • Complex sentences :: Older(John,30)=>~younger(John,30) Agent that reason logically

36. Quantifiers • World = {a, b, c} • Universal quantifier (∀) ∀x Cat(x) => Mammal(x)  Cat(a) => Mammal(a) & Cat(a) => Mammal(a) & Cat(a) => Mammal(a) • Existential quantifier (∃) ∃x Sister(x, Sopt) & Cat(x) Agent that reason logically

37. Nested quantifiers • ∀x,y Parent(x,y) => Child(y,x) • ∀x,y Brother(x,y) => Sibling(y,x) • ∀x∃y Loves(x,y) • ∃y∀x Loves(x,y) Agent that reason logically

38. De Morgan’s Rule ∀x ~P  ~∃x P ~P&~Q  ~(P v Q) ~∀x P  ∃x ~P ~(P&Q)  ~P v ~Q ∀x P  ~∃x ~P P&Q  ~(~P v ~ Q) ∃x P  ~∀x ~P P v Q  ~(~P&~Q) Agent that reason logically

39. Equality • Identity relation • Father(John) = Henry • ∃x,y Sister(Spot,x) & Sister(Spot,y) & ~(x=y) ≠ ∃x,y Sister(Spot,x) & Sister(Spot,y) Agent that reason logically

40. Higher-order logic • ∀x,y (x=y)  (∀p p(x)  p(y)) • ∀f,g (f=g)  (∀x f(x) g(x)) ∀ Agent that reason logically

41. -expression • x,y x2 – y2 • -expression can be applied to arguments to yield a logical term in the same way that a function can be • (x,y x2 – y2)(25,24) = 252-242 = 49 • x,y Gender(x) ≠Gender(y) & Address(x) = Address(y) Agent that reason logically

42. ∃! (The uniqueness quantifier) • ∃!x King(x) • ∃x King(x) & ∀y King(y) => x=y world를 고려하여 보여주면 => object가 1, 2, 3개일 때 {a} w0  king={}, w1  king={a} w1만 model {a,b} w0  king={}, w1  king={a}, w2 {b}, w3  {a,b}  w1, w2만 model Agent that reason logically

43. Representation of sentences by FOPL • One’s mother is one’s female parent ∀m,c Mother(c)=m  Female(m) & Parent(m) • One’s husband is one’s male spouse ∀w,h Husband(h,w)  Male(h) & Spouse(h,w) • Male and female are disjoint categories ∀x Male(x)  ~Female(x) • A grandparent is a parent of one’s parent ∀g,c Grandparent(g,c)  ∃p parent(g,p) & parent(p,g) Agent that reason logically

44. Representation of sentences by FOPL • A sibling is another child of one’s parents ∀x,y Sibling(x,y)  x≠y & ∃p Parent(p,x) & Parent(p,y) • Symmetric relations ∀x,y Sibling(x,y)  Sibling(y,x) Agent that reason logically

45. The domain of sets (I) • The only sets are the empty set and those made by adjoining something to a set : ∀s Set(s)  (s=EmptySet) v (∃x,s2 Set(s2) & s=Adjoin(x,s2)) • The empty set has no elements adjoined into it. ~∃x,s Adjoin(x,s)=EmptySet • Adjoining an element already in the set has no effect ∀x,s Member(x,s)  s=Adjoin(x,s) • The only members of a set are the elements that were adjoined into it ∀x,s Member(x,s)  ∃y,s2 (s=Adjoin(y,s2) & (x=y v Member(x,s))) Agent that reason logically

46. The domain of sets (II) • A set is a subset of another if and only if all of the first set’s are members of the second set : ∀s1,s2 Subset(s1,s2)  (∀x Member(x,s1) => member(x,s2)) • Two sets are equal if and only if each is a subset of the other: ∀s1,s2 (s1=s2)  (Subset(s1,s2) & Subset(s2,s1)) Agent that reason logically

47. The domain of sets (III) • An object is a member of the intersection of two sets if and only if it is a member of each of sets : ∀x,s1,s2 Member(x,Intersection(s1,s2))  Member(x,s1) & Member(x,s2) • An object is a member of the union of two sets if and only if it is a member of either set : ∀x,s1,s2 Member(x,Union(s1,s2))  Member(x,s1) v Member(x,s2) Agent that reason logically

48. Asking questions and getting answers • Tell(KB, (∀m,c Mother(c)=m  Female(m) & Parent(m,c))) • …… • Tell(KB, (Female(Maxi) & Parent(Maxi,Spot) & Parent(Spot,Boots))) • Ask(KB,Grandparent(Maxi,Boots) • Ask(KB, ∃x Child(x, Spot)) • Ask(KB, ∃x Mother(x)=Maxi) • Substitution, unification, {x/Boots} Agent that reason logically