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Logical Reasoning and Knowledge Representation

Learn about the principles of knowledge representation, logical reasoning, and inference in a variety of domains. Explore how sentences and facts are connected and how logical inference can generate new knowledge.

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Logical Reasoning and Knowledge Representation

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  1. 9주 강의 지식표현 Agent that reason logically

  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

  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)

  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

  5. Figure 6.2 A typical wumpus world Specifying the environment

  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)

  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)

  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)

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

  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.

  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

  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

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

  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!

  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

  16. Commitments Formal languages and their and ontological and epistemological commitments

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

  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.

  19. Semantics Truth table showing validity of a complex sentence

  20. Validity and Inference Truth tables for five logical connectives

  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

  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.

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

  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

  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

  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

  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

  28. 숙제 • 6.3, 6.6, 6.7, 6.9, 6.10, 6.12, 6.15, 6.16

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