Lecture 8: First-Order Logic

Lecture 8: First-Order Logic¶

AIMA Chapter 8 — 1 hour¶

Why First-Order Logic?¶

• Propositional logic: Cannot express “all,” “some,” “relations”

• FOL: Objects, relations, functions, quantifiers

• More expressive than propositional logic

Learning Objectives¶

• Understand syntax and semantics of first-order logic

• Use quantifiers (∀, ∃) and terms

• Represent knowledge in FOL

• Apply knowledge engineering process

FOL: Basic Elements¶

• Objects: People, numbers, etc.

• Relations: Brother(John, Mary)

• Functions: Mother(John) → person

• Predicates: True/false of objects

FOL: Syntax¶

• Constants: John, 2

• Variables: x, y

• Terms: Constants, variables, or f(t₁,...,tₙ)

• Atomic sentences: Predicate(term₁,...,termₙ)

• Complex: ¬, ∧, ∨, ⇒, ⇔, ∀, ∃

Quantifiers¶

• ∀x P(x): “For all x, P(x)”

• ∃x P(x): “There exists x such that P(x)”

• ∀x P(x): Equivalent to ¬∃x ¬P(x)

• De Morgan: ¬∀x P(x) ≡ ∃x ¬P(x)

FOL: Semantics¶

• Model: Domain, interpretation of symbols

• Interpretation: Maps constants to objects, predicates to relations

• Satisfaction: m ⊨ φ defined recursively

FOL: Kinship Domain¶

• Brother(x,y): x is brother of y

• Parent(x,y): x is parent of y

• Sibling(x,y): ∃p Parent(p,x) ∧ Parent(p,y)

• Grandparent: Parent(x,y) ∧ Parent(y,z)

Knowledge Engineering¶

1. Identify: Task, questions

2. Assemble: Relevant knowledge

3. Vocabulary: Decide on symbols

4. Encode: General knowledge, specific instance

5. Pose queries: Query inference procedure

6. Debug: Fix errors

Electronic Circuits Example¶

• Gates: AND, OR, NOT

• Terminals: Connected, signal

• OneOf: Each gate is exactly one type

• Consistency: Connected terminals have same signal

Summary¶

• FOL: Objects, relations, functions, quantifiers

• ∀, ∃: Universal and existential

• Knowledge engineering: Process for building KBs

• Next: Inference in FOL

References¶

• Russell & Norvig, AIMA 4e, Ch. 8

• Chapter PDF: chapters/chapter-08.pdf

Questions?¶

Next lecture: Inference in First-Order Logic (Chapter 9)