CS.462Artificial Intelligence

SOMCHAI THANGSATHITYANGKUL

Lecture 05 : Knowledge Base & First Order Logic

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Knowledge base• A knowledge base KB is a set of sentences. Ex

ample KB: JerryGivingLecture (TodayIsTuesday TodayIsThursday) JerryGivingLecture• It is equivalent to a single long sentence: the conjunction of all sentences ( JerryGivingLecture (TodayIsTuesday TodayIsThursday)) JerryGivingLecture

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Entailment

• Entailment is the relation of a sentenc e logically

follows from other sentences. |= |= if and only if, in every interpreta

tion in which is true, is also true• Deduction theorem: |= if and only

if is valid (always true)

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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

Modus

ponens

And-introduct

ion

And-eliminat

ion

Modus

tolens

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Natural deduction example

Step Formula Derivation

1 P Q Given

2 PR Given

3 (Q R) S Given

Prove S

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Natural deduction example• KB:

1. JerryGivingLecture (TodayIsTuesd ay TodayIsThursday)

2. JerryGivingLectureProve: TodayIsTuesday

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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.

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Propositional Resolution

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Propositional Resolution Example

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Resolution tree• KB : (A CD) (ADE) (A C)• Prove : (DE)• Negated conclusion : (DE)• Convert KB in the CNF, So we have

KB:1. (A C D)2. (A D E)3. (A C)4. D5. E

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Resolution tree

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Try this (P → Q) → Q , (P → P) → R , (R → S) → ¬(S → Q) Prove R

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