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Transcript
CS.462 Artificial Intelligence SOMCHAI THANGSATHITYANGKUL Lecture 05 : Knowledge Base & First Order Logic Knowledge base • A knowledge base KB is a set of sentences. Example KB: JerryGivingLecture (TodayIsTuesday TodayIsThursday) JerryGivingLecture • It is equivalent to a single long sentence: the conjunction of all sentences (JerryGivingLecture (TodayIsTuesday TodayIsThursday)) JerryGivingLecture 2 Entailment • Entailment is the relation of a sentence logically follows from other sentences. |= |= if and only if, in every interpretation in which is true, is also true • Deduction theorem: |= if and only if is valid (always true) 3 Natural Deduction • Proof is a sequence of sentences First ones are premises (KB) Then, you can write down on line j the result of applying an inference rule to previous lines When f is on a line, you know KB f If inference rules are sound, then KB f Modu s ponen s Modu s tolens AndAndintroduct eliminat 4 ion ion Natural deduction example Prove S Step Formula Derivation 1 PQ Given 2 PR Given 3 (Q R) S Given 5 Natural deduction example • KB: 1. JerryGivingLecture (TodayIsTuesday TodayIsThursday) 2. JerryGivingLecture Prove: TodayIsTuesday 6 Step Formula Derivation 1 JerryGivingLecture (TodayIsTuesday TodayIsThursday) Given 2 JerryGivingLecture Given 3 JerryGivingLecture (TodayIsTuesday TodayIsThursday) Biconditional elimination to 1. 4 (TodayIsTuesday TodayIsThursday) JerryGivingLecture Biconditional elimination to 1. 5 JerryGivingLecture (TodayIsTuesday TodayIsThursday) Contrapositive to 4. 7 Propositional Resolution 8 Propositional Resolution Example 9 Resolution tree • • • • KB : (A CD) (ADE) (A C) Prove : (DE) Negated conclusion : (DE) Convert KB in the CNF, So we have KB: 1. 2. 3. 4. 5. (A C D) (A D E) (A C) D E 10 Resolution tree 11 Try this (P → Q) → Q , (P → P) → R , (R → S) → ¬(S → Q) Prove R 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36