Artificial Intelligence Lecture 11: Logical Agents and Propositional Logic Reasoning - Pro, Study notes of Computer Science

A portion of cs 416 artificial intelligence lecture notes covering logical agents, propositional logic reasoning, logical equivalences, inference rules, monotonicity of knowledge base, resolution, and completeness. It includes examples and an algorithm for resolution, as well as an introduction to horn clauses, forward chaining, and backward chaining.

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CS 416
Artificial Intelligence
Lecture 11
Logical Agents
Chapter 7
Midterm Exam
Midterm will be on Thursday, March 13th
It will cover material up until Feb 27th
Reasoning w/ propositional logic
Remember what we’ve developed so far
Logical sentences
And, or, not, implies (entailment), iff (equivalence)
Syntax vs. semantics
Truth tables
Satisfiability, proof by contradiction
Logical Equivalences
Know these equivalences
Reasoning w/ propositional logic
Inference Rules
Modus Ponens:
Whenever sentences of form α=> βand αare given
the sentence βcan be inferred
–R
1: Green => Martian
–R
2: Green
Inferred: Martian
Reasoning w/ propositional logic
•Inference Rules
And-Elimination
Any of conjuncts can be inferred
–R
1: Martian ^ Green
Inferred: Martian
Inferrred: Green
•Use truth tables if you want to confirm inference
rules
pf3
pf4
pf5

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CS 416

Artificial Intelligence

Lecture 11

Logical Agents

Chapter 7

Midterm Exam

• Midterm will be on Thursday, March 13 th

• It will cover material up until Feb 27 th

Reasoning w/ propositional logic

• Remember what we’ve developed so far

  • Logical sentences
  • And, or, not, implies (entailment), iff (equivalence)
  • Syntax vs. semantics
  • Truth tables
  • Satisfiability, proof by contradiction

Logical Equivalences

• Know these equivalences

Reasoning w/ propositional logic

• Inference Rules

  • Modus Ponens:
    • Whenever sentences of form α => β and α are given the sentence β can be inferred - R 1 : Green => Martian - R 2 : Green - Inferred: Martian

Reasoning w/ propositional logic

•Inference Rules

  • And-Elimination
    • Any of conjuncts can be inferred
      • R 1 : Martian ^ Green
      • Inferred: Martian
      • Inferrred: Green

•Use truth tables if you want to confirm inference

rules

Example of a proof

~P ~B

B P?

P? P?

P?

Example of a proof

~P ~B

B ~P

P? P?

~P

Constructing a proof

• Proving is like searching

  • Find sequence of logical inference rules that lead to

desired result

  • Note the explosion of propositions
    • Good proof methods ignore the countless irrelevant propositions

Monotonicity of knowledge base

•Knowledge base can only get larger

  • Adding new sentences to knowledge base can only make it get larger - If (KB entails α) - ((KB ^ β) entails α)
  • This is important when constructing proofs
    • A logical conclusion drawn at one point cannot be invalidated by a subsequent entailment

How many inferences?

• Previous example relied on application of inference

rules to generate new sentences

  • When have you drawn enough inferences to prove

something?

  • Too many make search process take longer
  • Too few halt logical progression and make proof process incomplete

Resolution

•Unit Resolution Inference Rule

  • If m and li are complementary literals

Example of resolution

•Proof that there is not a pit in P1,2: ~P 1,

  • KB ^ P1,2 leads to empty clause
  • Therefore ~P (^) 1,2 is true

Formal Algorithm

Horn Clauses

• Horn Clause

  • Disjunction of literals with at most one is positive
    • (~a V ~b V ~c V d)
    • (~a V b V c V ~d) Not a Horn Clause

Horn Clauses

• Can be written as a special implication

  • (~a V ~b V c) Ù (a ^ b) => c
    • (~a V ~b V c) == (~(a ^ b) V c) … de Morgan
    • (~(a ^ b) V c) == ((a ^ b) => c … implication elimination

Horn Clauses

• Permit straightforward inference determination

  • Forward chaining
  • Backward chaining

Horn Clauses

• Permit determination of entailment in linear time

(in order of knowledge base size)

Forward Chaining Forward Chaining

• Properties

  • Sound
  • Complete
    • All entailed atomic sentence will be derived

• Data Driven

  • Start with what we know
  • Derive new info until we discover what we want

Backward Chaining

• Start with what you want to know, a query (q)

• Look for implications that conclude q

• Goal-Directed Reasoning