Agent that reason logically 지식표현 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 2 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 3 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 4 Specifying the environment Figure 6.2 A typical wumpus world Agent that reason logically 5 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 6 Inference in Wumpus world(I) 1,4 2,4 3,4 4,4 1,3 2,3 3,3 4,3 1,2 2,2 3,2 4,2 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 1,2 2,2 3,2 4,2 3,1 4,1 P? OK 1,1 A OK 2,1 3,1 4,1 1,1 OK V OK 2,1 A B OK Figure 6.3 The first step taken by the agent in the wumpus world. (a) The initial situation, after percept [None, None, None, None, None]. (b) After one move, with percept [None, Breeze, None, None, None]. Agent that reason logically 7 Inference in Wumpus world(II) 1,4 2,4 1,3 3,4 4,4 2,3 3,3 4,3 2,2 3,2 4,2 3,1 4,1 W! 1,2 A OK 1,1 V OK OK 2,1 B V OK 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 W ! 2,3 A 3,3P ? S G B 1,2 S 2,2 3,2 V V OK OK 1,1 2,1 B 3,1 V V P! OK OK 4,3 4,2 4,1 Figure 6.4 Two later stages in the progress of the agent. (a) After the third move, with percept [Stench, None, None, None, None]. (b) After the fifth move, with percept [Stench, Breeze, Glitter, None, None]. Agent that reason logically 8 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 9 The logical reasoning Aentences World Facts Follows Semantics Semantics Representation Entails Sentence Fact 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 10 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 11 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 12 Logics Boolean logic Symbols represent whole propositions (facts) Boolean connectives First-order logic objects, predicates connectives, quantifiers Agent that reason logically 13 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? Agent that reason logically ALL: A witch! 14 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 15 Commitments Formal languages and their and ontological and epistemological commitments Language Ontological Commitment (What exists in the world) Epistemological Commitment (What an agent believes about facts) Propositional logic First-order logic Temporal logic Probability theory Fuzzy logic facts facts, objects, relations times facts degree of truth true/false/unknown true/false/unknown true/false/unknown degree of belief 0…1 degree of belief 0…1 Agent that reason logically 16 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 17 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 18 Semantics Truth table showing validity of a complex sentence P H PH (P H) ┐H ((P H) ┐H)P False False True True False True False True False True True True False False True False True True True True Agent that reason logically 19 Validity and Inference Truth tables for five logical connectives P Q ┐P PQ PQ PQ PQ False False True True False True False True True True False False False False False True False True True True True True False True Agent that reason logically True False False True 20 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 21 Inference Rules for propositional logic 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 with anything else at all.) 1 2 … n Double-Negation Elimination: (From a doubly negated Unit Resolution: (From a disjunction, if one of the disjuncts is false, then you can infer the other one is true.) i Or-Introduction: (From a sentence, you can infer its disjunction sentence, you can infer a positive sentence.) i 1, 2, …, n 1 2 … n conjunction.) 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. 22 Agent that reason logically 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 23 Wumpus world Initial state ~S1,1 ~S2,1 S1,2 ~B1,1 B2,1 ~B1,2 Rule R1: R2: R3: R4: ~S1,1 ~S2,1 ~S1,2 S1,2 -> ~W1,1 & ~W1,2 & ~W2,1 -> ~W1,1 & ~W2,1 & ~W2,2 & ~W3,1 -> ~W1,1 & ~W1,2 & ~W2,2 & ~W1,3 -> W1,3 V W1,2 V W2,2 V W1,2 Agent that reason logically 24 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 25 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 26 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 27 숙제 6.3, 6.6, 6.7, 6.9, 6.10, 6.12, 6.15, 6.16 Agent that reason logically 28 First-order Logic Limitation of propositional logic A very limited ontology to need to the representation power first-order logic Agent that reason logically 30 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 31 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 32 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 33 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 34 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 35 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 36 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 37 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 38 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 39 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 40 -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 41 ∃! (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 42 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 43 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 44 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 45 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 46 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 47 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 48

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