AI_05_First Order Logic.pptx
- 1. LECTURE 5 First Order Logic Instructor : Yousef Aburawi Cs411 -Artificial Intelligence Misurata University Faculty of Information Technology Spring 2022/2023
- 2. First Order Logic Outline I. Syntax of FOL II. Quantifiers III. Model for FOL IV. Assertions & queries in FOL
- 3. I. Propositional Logic: Strength and Weakness Programming languages lack a general mechanism for deriving facts from other facts. They lack the expressiveness required to handle partial information. Propositional logic addresses the above issues. It also has compositionality – the meaning of a sentence is a function of the meanings of its parts. ¬ rain ∨ ¬ outside ∨ wet It lacks the expressive power to describe an environment with many objects. 𝐵1,1 ⇔ (𝑃1,2 ∨ 𝑃2,1) 𝐵1,2 ⇔ (𝑃1,1 ∨ 𝑃1,3 ∨ 𝑃2,2) ⋮ // Squares adjacent to pits are breezy. Propositional logic assumes the world contains facts only.
- 4. Combining Formal and Natural Languages Objects: people, houses, cars, trees, colors, days, … Relations: unary properties such as big, windy, … 𝑛-ary properties such as bigger than, parent of, on, owns, … Functions: square of, best friend, age, … First-order logic built around objects and relations capable of expressing facts about some or all objects
- 6. Alphabet of First-Order Logic Logical symbols connectives: ∧, ∨, ⇒, ⇔, ¬ quantifiers: ∀ (universal quantification), ∃ (existential quantification) parenthesis: (, ) and punctuation , variables: 𝑥, 𝑦, 𝑧, … ; 𝑥1, 𝑥2, … equality: = Non-logical symbols constants: Socrates, Turing, 1, earth, … predicate symbols: true, false Father(𝑥, 𝑦) // 𝑥 is father of 𝑦 Female(𝑥) // 𝑥 is female function symbols: gcd(𝑥, 𝑦) // greatest common divisor of 𝑥 and 𝑦 FatherOf(𝑥) // father of 𝑥
- 7. Terms and Atomic Sentences Terms variables: 𝑥, 𝑦, 𝑧, … ; 𝑥1, 𝑥2, … constants: Socrates, Turing, 1, earth, … functions: gcd(𝑥, 𝑦), FatherOf(𝑥), … 𝑓(𝑥1, 𝑥2, … , 𝑥𝑛) function symbol terms Atomic sentences predicates: true, false Mother(Aphrodite, Harmonia) Male(John) FatherOf(Apollo) = Zeus term equalities
- 8. Complex Sentences made of atomic sentences using logical connectives Father(𝑥, 𝑦) ⇒ Male(𝑥) Likes(Mary, John) ⇔ Likes(John, Mary) (Parent(𝑥, 𝑦) ∧ Parent(𝑦, 𝑧)) ⇒ GrandParent(𝑥, 𝑧) Female(𝑥) ∨ ¬Mother(𝑥, 𝑦) universal quantification ∀𝑥 Circle 𝑥 ⇒ Ellipse(𝑥) // Every circle is an ellipse. ¬∀𝑥 Likes 𝑥, sushi // Not everyone likes sushi. existential quantification ∃𝑥 Star 𝑥 ∧ ¬ (𝑥 = Sun) ∃𝑥 Whale 𝑥 ∧ (Age(𝑥) = 200) ∀𝑥 Integer 𝑥 ⇒ (Even 𝑥 ∨ Odd 𝑥 ) // Every integer is either even or odd. // There are stars other than the sun. // Some whales live to 200 years.
- 9. Syntax of First-Order Logic
- 10. II. Scope of a Quantifier Quantifiers ∀ and ∃ have the lowest precedence. ∀𝑥 𝑃 𝑥 ⇒ 𝑄 𝑥 ≡ ∀𝑥 (𝑃 𝑥 ⇒ 𝑄 𝑥 ) scope of ∀ ∀𝑥 𝑃 𝑥 ⇒ 𝑄 𝑥 ∨ ∃𝑦 𝑅(𝑥, 𝑦) ∨ 𝑆 𝑦 ∧ 𝑇 𝑥, 𝑦 ≡ ∀𝑥 (𝑃 𝑥 ⇒ (𝑄 𝑥 ∨ ∃𝑦 (𝑅 𝑥, 𝑦 ∨ (𝑆 𝑦 ∧ 𝑇 𝑥, 𝑦 )))) scope of ∃ scope of ∀ Each of ∀ and ∃ quantifies the remaining scope of the innermost pair of parentheses containing it. ⋯ ⋯ ⋯ ⋯ ∀𝑥 ⋯ ⋯ ⋯ ⋯ scope of ∀
- 11. Free and Bound Variables A variable occurrence is free in a formula if it is not quantified. A variable occurrence is bound in a formula if it is quantified. ∀𝑥 Father(𝑥, 𝑦) ⇒ Male(𝑥) 𝑥 is bound while 𝑦 is free. ¬∀𝑥∃𝑦∃𝑧∀𝑠∀𝑡 𝑃(𝑥, 𝑦, 𝑧, 𝑠, 𝑡) 𝑥, 𝑦, 𝑧, 𝑠, 𝑡 are all bound ∀𝑥 ∀𝑦 (𝑃(𝑥) ⇒ 𝑄(𝑥, 𝑓(𝑦), 𝑧)) 𝑥, 𝑦 are bound while 𝑧 is free. 𝑃 𝑥 ⇒ ∃𝑥 𝑄(𝑥) Free and bound variables can have the same name. free bound 𝑃 𝑥 ⇒ (∃𝑥 𝑄 𝑥 ) ∧ 𝑅(𝑥) same free variable different bound variable
- 12. Nested Quantifiers ∀𝑥∃𝑦 Student 𝑥 ∧ Course(y) ∧ Enrolled(𝑥, 𝑦) ∀𝑥∃𝑦 Brother 𝑥, 𝑦 ⇒ Sibiling(𝑥, 𝑦) Order matters for quantifiers of different types: ∀𝑥∃𝑦 Loves 𝑥, 𝑦 // Everybody (𝑥) loves somebody (𝑦) ∃𝑥∀𝑦 Loves 𝑦, 𝑥 // There is someone (𝑥) whom everyone (𝑦) loves. ∃𝑥∃𝑦 Loves 𝑥, 𝑦 ≡ ∃𝑦∃𝑥 Loves 𝑥, 𝑦 ∀𝑥∀𝑦 (Brother 𝑥, 𝑦 ⇒ Sibiling(𝑥, 𝑦)) ≡ ∀𝑦∀𝑥 (Brother 𝑥, 𝑦 ⇒ Sibiling(𝑥, 𝑦)) but not for those of the same type and appearing next to each other:
- 13. Connections Between ∀ and ∃ Through ¬ ∀𝑥 ¬Likes 𝑥, Parsnips ≡ ¬ ∃𝑥 Likes 𝑥, Parsnips ∀𝑥 Likes 𝑥, Icecream ≡ ¬ ∃𝑥 ¬ Likes 𝑥, Icecream De Morgan’s rules still apply: ¬∀𝑥 𝑃(𝑥) ≡ ∃𝑥 ¬𝑃(𝑥) ¬∃𝑥 𝑃 𝑥 ≡ ∀𝑥 ¬𝑃(𝑥) ¬∀𝑥∃𝑦∃𝑧∀𝑠∀𝑡 𝑃(𝑥, 𝑦, 𝑧, 𝑠, 𝑡) ≡ ∃𝑥¬∃𝑦∃𝑧∀𝑠∀𝑡 𝑃(𝑥, 𝑦, 𝑧, 𝑠, 𝑡) ≡ ∃𝑥∀𝑦¬∃𝑧∀𝑠∀𝑡 𝑃(𝑥, 𝑦, 𝑧, 𝑠, 𝑡) ≡ ∃𝑥∀𝑦∀𝑧¬∀𝑠∀𝑡 𝑃(𝑥, 𝑦, 𝑧, 𝑠, 𝑡) ≡ ∃𝑥∀𝑦∀𝑧∃𝑠¬∀𝑡 𝑃(𝑥, 𝑦, 𝑧, 𝑠, 𝑡) ≡ ∃𝑥∀𝑦∀𝑧∃𝑠∃𝑡 ¬𝑃(𝑥, 𝑦, 𝑧, 𝑠, 𝑡) Move negation inward, flipping the quantifiers:
- 14. III. Model for First-Order Logic Sentences are true with respect to a model 𝑀. The model 𝑀 contains objects (called domain elements) and interpretations of symbols. constant symbols → objects in domain 𝐷 predicate symbols → relations function symbols → functional relations Each predicate 𝑃(𝑥1, … , 𝑥𝑘) is mapped to a relation over 𝐷. Each function 𝑓(𝑥1, … , 𝑥𝑘) is mapped to a function to 𝐷.
- 15. Model Example Model for the family relationships of the Greek gods (incomplete). domain 𝐷 Zeus Ares Hera Aphrodite Harmonia Hermes Athena Artemis Dionysus Hades Apollo Poseidon Demeter Hephaestus Persephone ⋮ Father(Zeus, Hermes) Mother(Hera, Ares) Mother(Aphrodite, Harmonia) Father(Zeus, Athena) predicates Weapon(Zeus) // ≡ Thunderbolt Carry(Hermes) // ≡ Flute Weapon(Apollo) // ≡ BowAndArrows Carry(Aphrodite) // ≡ Apple ⋮ functions
- 16. Truth in First-Order Logic A predicate 𝑃 𝑡1, … , 𝑡𝑘 is true if the objects referred to by the terms 𝑡1, … , 𝑡𝑘 are in the relation referred to by the predicate. 𝑡1 = 𝑡2 is true if the two terms 𝑡1 and 𝑡2 refer to the same object. The semantics of sentences formed with logical connectives are identical to those in propositional logic. Quantifiers allow us to express properties of a collection of objects instead of enumerating them by name. ∀ (universal): “for all” ∃ (existential): “there exists”
- 17. Truths with Quantifications ∀𝑥 𝑃 𝑥 is true in a model 𝑀 iff 𝑃(𝑥) is true with 𝑥 assuming every object in the model ∀𝑥 Father(𝑥, 𝑦) ⇒ Male(𝑥) true (in every model) ∀𝑥 Ellipse(𝑥) ⇒ Circle 𝑥 true (in every model) ∃𝑥 𝑃 𝑥 is true in a model 𝑀 iff 𝑃(𝑥) is true with 𝑥 assuming some object in the model. ∃𝑥 Mother(𝑥, Ares) ∧ Mother(𝑥, Harmonia) false (in the model of Greek mythology) ∃𝑥 ¬ Likes 𝑥, sushi true (in a model that includes all the people in the world)
- 18. IV. Using FOL A domain is some part of the world about which we wish to express some knowledge. Knowledge engineering represents information about the world in a form that can be utilized by a computer to solve complex tasks such as: medical diagnosis dialog in a natural language etc. Knowledge representation (logical rules, semantic nets, etc.) Automated reasoning (inference engines, theorem provers, etc.)
- 19. Assertions and Queries in FOL Add sentences, called assertions, to a KB using TELL. TELL(KB, ∀𝑥∃𝑦 Brother 𝑥, 𝑦 ⇒ Sibiling(𝑥, 𝑦)) TELL(KB, Father(Zeus, Athena)) TELL(KB, Likes(John, Icecream)) Ask the KB questions using ASK. ASK(KB, Likes(John, Icecream)) Query: question asked Any query is entailed by the KB should be answered affirmatively.
- 20. Substitution Suppose another KB has the following predicates: Bird(Swan), Bird(Crane), Bird(Parrot), ASK(KB, ∃𝑥 Bird(𝑥)) To know what values of 𝑥 make the sentence true {𝑥/ Swan}, {𝑥/ Crane}, and {𝑥/ Parrot} substitution or binding list returns true Quantified query ASKVARS(KB, Bird(𝑥)) The query returns
- 21. The Kinship Domain Kinship relations are represented by binary predicates. ∀𝑚, 𝑐 Mother 𝑐 = 𝑚 ⇔ Female(𝑚) ∧ Parent(𝑚, 𝑐) ∀𝑤, ℎ Husband ℎ, 𝑤 ⇔ Male(ℎ) ∧ Spouse(ℎ, 𝑤) // One’s mother is the person’s female parent. // One’s husband is the person’s male spouse. ∀𝑝, 𝑐 Parent 𝑝, 𝑐 ⇔ Child(𝑐, 𝑝) // Parent and child are inverse relations. ∀𝑔, 𝑐 GrandParent 𝑔, 𝑐 ⇔ ∃𝑝 (Parent(𝑔, 𝑝) ∧ Parent(𝑝, 𝑐)) // A grand parent is a parent of one’s parent ∀𝑥, 𝑠 Sibling 𝑥, 𝑠 ⇔ 𝑥 ≠ 𝑠 ∧ ∃𝑝 (Parent(𝑝, 𝑥) ∧ Parent(𝑝, 𝑠)) // A sibling is another child of one’s parent Axioms
- 22. Axioms and Theorems Theorems are logical sentences entailed by axioms. ∀𝑥, 𝑦 Sibling 𝑥, 𝑦 ⇔ Sibling(𝑦, 𝑥) // entailed by // ∀𝑥, 𝑠 Sibling 𝑥, 𝑠 ⇔ 𝑥 ≠ 𝑠 ∧ ∃𝑝 (Parent(𝑝, 𝑥) ∧ Parent(𝑝, 𝑠)) ASK(KB, ∀𝑥, 𝑦 Sibling 𝑥, 𝑦 ⇔ Sibling(𝑦, 𝑥)) should return true. Axioms in a domain are logical sentences that are taken to be true without being derived.
- 23. Readings Chapters 8 of Textbox. 1-23
- 24. The End