First order logic in knowledge representation
First-order logic allows for more expressive power than propositional logic by representing objects, relations, and functions in the world. It includes constants like names, predicates that relate objects, functions, variables, logical connectives, equality, and quantifiers. Relations can represent properties of single objects or facts about multiple objects. Models represent interpretations of first-order logic statements graphically. Terms refer to objects as constants or functions. Atomic sentences make statements about objects using predicates. Complex sentences combine atomic sentences with connectives. Universal quantification asserts something is true for all objects, while existential quantification asserts something is true for at least one object.
First order logic in knowledge representation
- 1. FIRST ORDER LOGIC IN KNOWLEDGE REPRESENTATION Ishara Athukorala EP 1310 11/03/16 1
- 2. Why First Order Logic Propositional logic has limited expressive power. • Propositional logic assumes that the world contains facts. • First-order logic assumes the world contains, • Objects: people, houses, numbers, colors, baseball games, wars, … • Relations: red, round, prime, brother of, bigger than, part of, comes between, … • Functions: father of, best friend, one more than, … 11/03/16 2
- 3. Syntax of First Order Logic • Constants: john,apples • Predicates: likes(john, apples) • Functions: likes(john, fruit_of(apple_tree)) • Variables: likes(X, apples) • Connectives:¬, ⇒, ∧, ∨, ⇔ • Equality:= • Quantifiers: True for all objects (Universal) Exists at least one object (Existential) 11/03/16 3
- 4. Relations • Some relations are properties: they state some fact about a single object: Round(ball), Prime(7). • n-ary relations state facts about two or more objects: Married(John,Mary), LargerThan(3,2). • Some relations are functions: their value is another object: Plus(2,3), Father(Dan). 11/03/16 4
- 5. Models for FOL: Graphical Example 11/03/16 5
- 6. Terms • A logical expression that refers to an object. • Two types of terms, 1. Constant symbols 2. Function symbols 11/03/16 6
- 7. Atomic Sentences • Atomic sentences state facts using terms and predicate symbols P(x,y) interpreted as “x is P of y”. Examples: • Brother_of(Mary,Pete) is false • Brother_of(Pete,Brother(Pete)) is True. Binary relation Function 11/03/16 7
- 8. Complex Sentences • Complex sentences with connectives ( ( ), ) ( ( ))Brother LeftLeg Richard John Democrat Bush¬ ∨ objects connectives Binary relation function properties 11/03/16 8
- 11. Some Examples… • All purple mushrooms are poisonous. ∀x (mushroom(x) ∧ purple(x)) → poisonous(x) • Every gardener likes the sun. ∀x gardener(x) → likes(x,Sun) • Clinton is not tall. ¬tall(Clinton) 11/03/16 11
- 12. Some Examples… • “On Mondays and Wednesdays I go to John’s house for dinner” ∀X.((is_mon(X) V is_wed(X)) -> eat_meal(me,houseOf(John),X)) • “Every rose has a thorn” ∀X(rose(X) -> EY.(has(X,Y) ∧ thorn(Y))) 11/03/16 12
- 13. References • Benson Mates, Elementary Logic, OUP, New York 1972 (Library of Congress Catalog Card no.74-166004) • Elliot Mendelson, Introduction to Mathematical Logic, Van Nostran Reinholds Company, New York 1964 11/03/16 13