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1 AI=Knowledge Representation & Reasoning zSyntax zSemantics zInference Procedure yAlgorithm ySound? yComplete? yComplexity
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2 Some KR Languages zPropositional Logic zPredicate Calculus zFrame Systems zRules with Certainty Factors zBayesian Belief Networks zInfluence Diagrams zSemantic Networks zConcept Description Languages zNonmonotonic Logic
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3 Propositional Logic zSyntax yAtomic sentences: P, Q, … yConnectives: , , , zSemantics y Truth Tables zInference yModus Ponens yResolution ySoundness and completeness zComplexity issues.
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4 Semantics zSyntax: a description of the legal arrangements of symbols (Def “sentences”) zSemantics: what the arrangement of symbols means in the world Sentences Facts Sentences Representation World Semantics Inference
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5 Propsitional Logic: Syntax zAtoms zLiterals zSentences yAny literal is a sentence yIf S1 and S2 are sentences, then xThen (S1 S2) is a sentence xThen (S1 S2) is a sentence xThen (S1 S2) is a sentence xThen S1 is a sentence
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6 Propositional Logic: SEMANTICS zAn interpretation is an assignment to each variable either True or False. zAssignments to compound sentences are defined by the standard truth tables: zA propositional knowledge base says which sentences must be true in the world. P T T F F Q P T T F F Q P Q P Q P T FF F F TT T T F Q P T F T F
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7 Example Knowledge Base z(Smoke fire) Alarm zAlarm
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8 More Definitions zvalid = tautology = always true zsatisfiable = sometimes true zunsatisfiable = never true 1) smoke fire 2) smoke smoke 3) smoke fire fire 4)(smoke fire) ( smoke fire)
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9 Making Inferences zA knowledge base gives us partial information about the world: it constrains the world to a set of possible truth assignments. zBy inference, we decide what else holds in all of the truth assignments allowed by the knowledge base. zInference question: does KB = S ?
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10 Proof Procedures zTo decide whether KB = S, we can try to look for a proof of S from KB. zA proof procedure is some algorithm that we apply to a KB to produce its logical consequences. zA proof uses: ythe knowledge base, yaxiom schemas yinference rules.
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11 Soundness and Completeness zKB |- S: S is provable from KB. zA proof procedure is sound if: yIf KB |- S, then KB |= S. yThat is, the procedure produces only correct consequences. zA proof procedure is complete if: yIf KB |= S, then KB |- S. yThat is, the procedure produces all the consequences. zIdeally, the procedure should be sound and complete. (Ideals are nice in theory).
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12 Modus Ponens zFrom A and A B, infer B. zModus ponens with a few axiom schemas is sound and complete: y A (B A) yA (B C) ((A B) (A C)) y( A B) (B A) yMore in the book.
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13 Normal Forms zCNF = Conjunctive Normal Form zConjunction of disjuncts (each disjunct = “clause”) (P Q) R (P Q) R (P Q) R P Q R ( P Q) R ( P R) ( Q R)
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14 Resolution A B C, C D E A B D E zRefutation Complete yGiven an unsatisfiable KB in CNF, yResolution will eventually deduce the empty clause zProof by Contradiction yTo show = Q yShow { Q} is unsatisfiable!
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15 Resolution Example prove P (A B C) (B) ( B D) ( C A D) ( D P Q) ( Q)
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16 Computational Complexity zDetermining satisfiability is NP-complete. zEven when all clauses have at most 3 literals. zHence, also validity and entailment testing are NP-complete zIf all clauses have at most 2 literals, it is polynomial. zBut if the KB is in DNF, satisfiability is polynomial. yWhat does this tell us about transforming a CNF into a DNF knowledge base?
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17 Horn Clauses zIf every sentence in KB is of the form: Then Modus Ponens is –Polynomial time, and –Complete! A B C ... F Z equivalently A B C ... F Z Clause means a big disjunction At most one positive literal
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