Semantic Data Chapter 2 : Introduction to first order...
Transcript of Semantic Data Chapter 2 : Introduction to first order...
Semantic Data
Chapter 2 : Introduction to first order logic
Jean-Louis Binot
12/02/2020Semantic Data1
Course content outline
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Sources and recommended readings
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Why study logic?
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Types of logics
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Agenda
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Model-theoretic semantics2
First order logic1
Decidability4
Satisfiability, validity, entailment3
Precursors of first order logic
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A very short reminder of proposition logic
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◼ ⊤ ⊥
◼ Π
◼ ∧ ∨ → ↔
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∈ Π
⊤ ⊥
∧ ∨ → ↔
∧ ∨ → ↔
∨ → ∧ ↔ ∨
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A very short reminder of proposition logic ./.
∧ ∨ → ↔
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Some standard equivalences in propositional logic
❑ ∧ ≡ ∧ ∧
❑ ∨ ≡ ∨ ∨
❑ ∧ ∧ ≡ ∧ ∧ ∧
❑ ∨ ∨ ≡ ∨ ∨ ∨
❑ ≡
❑ → ≡ →
❑ → ≡ ∨
❑ ↔ ≡ → ∧ →
❑ ∧ ≡ ∨
❑ ∨ ≡ ∧
❑ ∧ ∨ ≡ ∧ ∨ ∧ ∧ ∨
❑ ∨ ∧ ≡ ∨ ∧ ∨ ∨ ∧
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Limits of proposition logic
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∀ →
∃ ∧
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Vocabulary of first order logic
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◼ → ↔
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∀
∃
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Syntax of first order logic
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∨
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∀ → ∃ ∧
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First order logic: formal syntax
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◼ ⊤) ⊥)
◼ ¬ ∧ ∨ → ↔
◼ ∀ ∃
: ∀ ∃, ¬ ∧ ∨, →, ↔.
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Examples of terms and formulas
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∧
∃ ∧
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Intuitive understanding of quantifiers and common mistakes
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◼ ∀
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→
∧
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∃ ∧
∃ →
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Nested quantifiers
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∀ ∀ →
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∀ ∃
∃ ∀
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De Morgan’s rules for quantifiers
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❑ ∀ ∃
∨ ↔ ∧ ∀ ↔ ∃
∧ ↔ ∨ ∀ ↔ ∃
∧ ↔ ∨ ∀ ↔ ∃
∨ ↔ ∧ ∃ ↔ ∀
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Free and bound variables, quantifier scope
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∀ ∃
◼ ∀ ∃
◼ ∀ ∃
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∀ ∀
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Caution note
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∀ ∨ ∃
∀ ∨ ∃
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Why is it called first order logic ?
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∃
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Agenda
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Model-theoretic semantics2
First order logic1
Decidability4
Satisfiability, validity, entailment3
Model theoretic semantics
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I I
❑ I I
I ⊨23
I
IF
Ingredients of model-theoretic semantics
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Model theoretic semantics of propositional logic (reminder)
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❑ I
◼ I
❑ I I
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II
◼ ⊤I
⊥I
◼I I
◼ ∧ I I I
◼ ∨ I I I
◼ → I I I
◼ ↔ I I I
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Semantics of FOL: intuition
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Interpretation domain : example
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❑ I
◼I I I
I I
◼I
◼I
I1
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II
I2
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First order logic: semantics
I ΔI I α
❑ ΔI
❑I
◼I ∈ ΔI
◼I ΔI →
◼I ΔI → ΔI
❑ α α ∈ ΔI
❑ tI FI I
ΔI ΔI
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First order logic semantics : terms
I
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◼I
◼ α I α
◼I I I I I
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First order logic semantics : formulas
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◼I I I I I
❑ → ↔
◼
❑
I I ∈ ΔI
◼ ∀ I ∈ ΔI I
◼ ∃ I ∈ ΔI I
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Agenda
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Model-theoretic semantics2
First order logic1
Decidability4
Satisfiability, validity, entailment3
Fundamental semantic concepts
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Satisfiability and validity
❑ I I ⊨ I
◼ I
◼ I I I ⊭
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❑ ⊨
❑ ⊭
◼ ⊨ ⊭
∨
∨
∧
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Satisfiability for set of formulas
❑ I
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I
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I ∈ ⊭
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∧ ∧
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Entailment and logical equivalence
❑ ⊨
◼
❑ ⊨
◼ ∅ ⊨ ⊨
❑ ≡
I I I
⊨
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Some useful results
❑ ⊨ ⊨ → →
❑ ⊨ ⊭ ∧ ∧
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Logical consequence and knowledge-based inferences
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⊨
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Updated framework of reference
Data integration component
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Monotonicity, closed-world and open-world assumptions
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⊨ ∧ ⊨
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◼ ⊨ ∧ ⊨
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Agenda
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Model-theoretic semantics2
First order logic1
Decidability4
Satisfiability, validity, entailment3
Questions regarding the use of FOL for reasoning
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1. Knowledge representation in FOL
∀ ∀ ∧ ∧ →
∀ → ∨
∀ ∀ ↔
∀ ∧ → ∨ ∨ ∨
∀ ∧ → ↔ ∃
∀ ∧ → ↔ ∃
∀ ∧ ⇒ ⇔
∀ →
∧
∧
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1. Knowledge representation in FOL
∀ ∀ ∧ ∧ →
∀ → ∨
∀ ∀ ↔
∀ ∧ → ∨ ∨ ∨
∀ ∧ → ↔ ∃
∀ ∧ → ↔ ∃
∀ ∧ ⇒ ⇔
∀ →
∧
∧
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✓
✓
✗
▪
▪
2. Logical reasoning
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⊨
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A truth table enumeration algorithm for PL entailment
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Decision problems
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Decidability of propositional logic
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◼ ⊨ ⊭ ∧ ∧
◼ ⊨ ⊭
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◼ ≡ ⊨ ⊨
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Undecidability of first order logic
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Practical approaches for reasoning with first order logic
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∧ →
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Summary
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References
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THANK YOU
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