6.2 KiB
6.2 KiB
Slide 1: Learning Objectives
- Understand when the order of quantifiers is important.
- Understand how ∀ and ∃ are connected.
- Use and remember scoping rules.
- Identify bound and free variables in formulae.
- Establish the truth of formulae in predicate logic.
- Understand why predicate logic is described as undecidable.
- Understand the difference between zero-order, first-order, and higher-order predicate logics.
Slide 2: Objectives & Recap
- Quantifiers and their alternative view.
- Distributive laws of quantifiers.
- De Morgan's laws with quantifiers.
- Scope of quantifiers.
- Bound and free variables.
Slide 3: The Universal Quantifier (∀)
- Pronounced as "for all".
- Example: ∀x (human(x) ⇒ mortal(x)) = All humans are mortal.
- In terms of conjunction: ∀x p(x) ≡ p(x₁) ∧ p(x₂) ∧... ∧ p(xₙ)
Slide 4: The Existential Quantifier (∃)
- Pronounced as "there exists" or "for some".
- Example: ∃x (human(x) ∧ happy(x)) = Some humans are happy.
- In terms of disjunction: ∃x p(x) ≡ p(x₁) ∨ p(x₂) ∨... ∨ p(xₙ)
Slide 5: Distributive Laws of Quantifiers
- ∀x (p(x) ∧ q(x)) ≡ (∀x p(x)) ∧ (∀x q(x))
- ∃x (p(x) ∨ q(x)) ≡ (∃x p(x)) ∨ (∃x q(x))
Slide 6: The Order of Quantification
- Example 1 (Everyone loves someone): ∀x ∃y loves(x, y)
- Example 2 (There is someone loved by everyone): ∃y ∀x loves(x, y)
Slide 7: De Morgan's Laws and Quantifiers
- ¬∃x p(x) ≡ ∀x ¬p(x)
- ¬∀x p(x) ≡ ∃x ¬p(x)
Slide 8: Scope of Quantifiers
- In absence of brackets, scope extends to the end of the formula.
- Brackets can enforce different scoping patterns.
Slide 9: Bound and Free Variables
- Bound variable: occurrence introduced by a quantifier within its scope.
- Example:
Formula Variable status child(x) Only one occurrence of x, free. ∀x child(x) ∧ clever(x) Both occurrences of x are bound by the same quantifier. ((∀x child(x)) ∧ clever(x)) The first occurrence of x is bound, the second is free.
- Example:
- Free variable: occurrence not within any quantifier's scope.
Slide 10: Meaning of Bound Variables
- The meaning of a bound variable does not depend on its name.
- Example:
∀x (child(x) ∧ clever(x)) ⇒ ∃y loves(y, x) ∀B (child(B) ∧ clever(B)) ⇒ ∃C loves(C, B)
- Example:
Slide 11: Meaning of Free Variables
- Free variables denote unknowns or unspecified objects.
- Example:
∀x (child(x) ∧ clever(x)) ⇒ x is loved. ∀x (child(x) ∧ clever(x)) ⇒ z is loved.
- Example:
Slide 12: Exercise
- Identify bound and free variables in the formula:
∃x taller(y, x) ∃x ∃y taller(x, y) ∧ taller(x, z) - Solution:
- In the first formula: x is bound, y is free.
- In the second formula: y is bound by ∃y, z is free. Both occurrences of x are bound and refer to the same variable.
Slide 13: The Equality Symbol (=)
- ⊢ Richard has at least two brothers:
∃x ∃y (brother(x, richard) ∧ brother(y, richard) ∧ ¬(x = y)) - Definition of sibling using parent:
∀x ∀y sibling(x, y) ≡ (¬(x = y) ∧ ∃m ∃f ¬(m = f) ∧ parent(m, x) ∧ parent(f, x) ∧ parent(m, y) ∧ parent(f, y))
Slide 14: Establishing the Truth Values of Formulae
- Example (slide 21 and 22):
- Individuals: Ahmed, Khan, Patel, Scott.
- Properties: male, tall, short.
- Formula: ∀x (male(x) ⇒ tall(x) ∨ short(x))
- Truth table shows the formula is false (Patel is male but not tall or short).
Slide 15: Exercise
- Given individuals, properties, and true propositions as in slide 23.
- Evaluate the truth of: ∀x ¬male(x) ⇒ short(x)
- Solution (slide 24):
x male(x) ¬male(x) short(x) ¬male(x) ⇒ short(x) Ahmed T F F F T Khan F T T T T Patel T F F F T Scott T F T F F - The formula is false (Patel is male but not short).
Slide 16: Example Involving ∃
- Given individuals, properties, and true propositions as in slide 25.
- Formula: ∃x (male(x) ∧ ¬tall(x) ⇒ short(x))
- Truth table (slide 26):
x male(x) ¬tall(x) short(x) ¬tall(x) ⇒ short(x) male(x) ∧ ¬tall(x) ⇒ short(x) Ahmed T F F F T T Khan F F T F F F Patel T F F F F F Scott T T T F T F - The formula is true (Scott is male, not tall but short).
Slide 17: Exercise
- Given individuals, properties, and true propositions as in slide 27.
- Evaluate the truth of: ∃x (male(x) ∧ (tall(x) ∨ ¬short(x)))
- Solution (slide 28):
x male(x) tall(x) short(x) ¬short(x) tall(x) ∨ ¬short(x) male(x) ∧ (tall(x) ∨ ¬short(x)) Ahmed T T F T T T T Khan F F T F F F F Patel T T F F T T F Scott T F T F F F F - The formula is true (Ahmed and Patel are males, tall or not short).
Slide 29: Predicate Logic is Undecidable
- Universal quantification introduces computational impossibility when testing truth values with an infinite number of possible values.
Slide 30: First-order Predicate Logic
- Quantifiers refer only to objects (constants), not predicate or function names.
- Propositional logic is zero-order logic.
Slide 31: Summary
- Order of quantifiers matters when both ∀ and ∃ are present.
- Quantifiers are connected through negation, obey De Morgan's laws.
- Scope of quantifiers extends to the end of the formula without brackets.
- Bound variables are introduced by a quantifier within its scope; free variables are not within any quantifier's scope.
Slide 32: Reading, References and Acknowledgements
- Reading from Artificial Intelligence textbook by Russell and Norvig.
- References: Introductory Logic and Sets for Computer Scientists by Nissanke.