CS.462 Artificial Intelligence SOMCHAI THANGSATHITYANGKUL Lecture 05 : Knowledge Base & First Order Logic
2 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
3 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)
4 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 is on a line, you know KB If inference rules are sound, then KB Modu s ponen s And- introduct ion And- eliminat ion Modu s tolens
5 Natural deduction example StepFormulaDerivation 1 P Q Given 2 PRPR 3 (Q R) S Given Prove S
6 Natural deduction example KB: 1. JerryGivingLecture (TodayIsTuesday TodayIsThursday) 2. JerryGivingLecture Prove: TodayIsTuesday
7 StepFormulaDerivation 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.
8 Propositional Resolution
9 Propositional Resolution Example
10 Resolution tree KB : (A C D) (A D E) (A C) Prove : (D E) Negated conclusion : (D E) Convert KB in the CNF, So we have KB: 1.( A C D) 2.(A D E) 3.( A C) 4. D 5. E
11 Resolution tree
12 Try this (P → Q) → Q, (P → P) → R, (R → S) → ¬(S → Q) Prove R
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