Knowledge Representation

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Knowledge Representation
Lila Rao Graham
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Definitions

• Knowledge Base
• A set of representations of facts about the
  world.
• Each representation is a sentence.
• Knowledge Representation
• Expressing knowledge in a form that can be
  manipulated by a computer.
• Inference Mechanism
• Generates new sentences that are necessarily true
  given that the old sentences are true.

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• Two aspects of the Knowledge Representation
  language
• 1. Formal system of defining the world
• Syntax
• What constitutes a sentence
• Semantics
• Meaning. Connects sentences to facts.
• KB entails a (KB a) means when all
• sentences in KB are true, a is true.
• 2. A proof theory
• Rules for determining all entailments.
• (KB -- a)

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KR Languages

• Logic
• Propositional Logic
• First Order Logic
• Production rules
• Structured Objects
• Semantic Nets
• Frames

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Propositional Logic Syntax

• A BNF (Backnus-Naur Form) grammar of sentences in
  propositional logic.
• Sentence ? Atomic Sentence ? Complex Sentence
• AtomicSentence ? True ? False
• ? P ? Q ? R ?
• ComplexSentence ? (Sentence)
• ?Sentence Connective Sentence
• ? ?Sentence
• Connective ? ? ? ? ? ? ? ?

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Propositional Logic Semantics
Truth table for the five logical connectives
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Propositional Logic Rules of Inference

• Modus Ponens/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.
• ?1 ? ?2 ? ... ? ?n
• ?I
• And-Introduction
• From a list you can infer their conjunction.
• ?1, ?2 , ... , ?n
• ?1 ? ?2 ? ... ? ?n

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Propositional Logic Rules of Inference

• Or-Introduction
• From a sentence, you can infer its disjunction
  with anything else at all.
• ?i
• ?1 ? ?2 ? ... ? ?n
• 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.
• ? ? ?, ? ?
• ?
• Resolution
• This is the most difficult. Because ? cannot be
  both true and false, one of the other disjuncts
  must be true in one of the premises. Or
  equivalently, implication is transitive.
• ? ? ?, ? ? ? ? ? ? ? ? ?, ? ? ?
• ? ? ? ?? ? ?

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Propositional Logic Inference

• Resolution refutation theorem prover
• Inference mechanism for propositional logic
• Refutation
• Recall that S ? is defined as
• Whenever all sentences in S are true, than ? is
  true as well.
• This is equivalent to saying that
• It impossible for S to be true and ? to be
  false.
• Or
• It is impossible for S and ?? to be true at the
  same time.
• In a refutation theorem prover, in order to prove
  ? from S, add ?? to S and try to derive a
  contradiction.

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• Algorithm to prove S ??
• 1. Rewrite S in clausal form.
• 2. Rewrite ?? in clausal form.
• 3. Using the various inference rules, derive the
  empty clause from the results of 1. and 2.

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Clausal Form or CNF

• Conjunctive Normal Form (CNF).
• Representing a sentence as a conjunction of
  disjunctions.
• To do this
• 1. Eliminate Implications
• Remember a ? b ? ?a ? b
• 2. Push negation inwards
• ?(a ? b) ? ?a ? ? b
• ?(a ? b) ? ?a ? ? b
• (De Morgans Laws)
• 3. Eliminate double negations
• 4. Push disjunctions into conjunctions
• a ? (b ? c) ? (a ? b) ? (a ? c)

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Example Converting to CNF

• Converting the following sentence to CNF
• (a ? ? b) ? (c ? d)
• Steps
• 1. Remove Implication
• ?(a ? ? b) ? (c ? d)
• 2. Push Negations Inwards
• ?a ? ? ? b ? (c ? d)
• 3. Eliminate Double Negations
• ?a ? b ? (c ? d)
• 4. Push Disjunctions into Conjunctions
• (?a ? b ? c) ? (?a ? b ? d)
• CNF
• Represented in KB as the 2 clauses
• ?a ? b ? c and ?a ? b ? d

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Example Resolution

• Prove that r follows from
• (p ? q) ? (r ? s) - (1)
• p ? ? s - (2)
• p ? q - (3)
• Solution
• 1. Clause (1) in CNF
• ? (p ? q) ? (r ? s)
• ?? ? p ? ? q ? r ? s - (1)
• Clause (2)
• ? p ? ? s - (2)
• Clause (3)
• p - (3)
• q - (4)

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Example (cont)

• 2. Clause of r is ? r - (5)
• 3. Using inference rules
• ?p ? ? q ? s - (6)
• from unit resolution rule of (1) and (5)
• ? q ? s - (7)
• from unit resolution of (3) and (6)
• s - (8)
• from (4) and (7)
• ? p - (9)
• from (2) and (8)
• - (10)
• from (3) and (9)
• Therefore r follows from the original clauses