First order logic (FOL)

1. First order logic (FOL) first order predicate calculus

2. Why another system? • procedural / declarative difference • algorithmic vs data representation BUT • propositional logic is inadequate • representation • weak, too specific, lacks expressive power • reasoning • inference is OK but brittle to real world conditions (errors, assumptions, unknowns) D Goforth - COSC 4117, fall 2006

3. First order logic vs propositional • basis of reasoning • propositional logic: statements • first order logic: OBJECT-ORIENTED • objects • relations • functions • statements about objects, relations and functions • possible values of statements • true, false, unknown D Goforth - COSC 4117, fall 2006

4. Other systems of logic • extensions of first order logic • temporal: facts are true/false/unknown for a period of time • probabilistic: facts are true or false but known with a certain probability • fuzzy logic: facts are partially true • meta-systems: higher order logics – reasoning about logic systems D Goforth - COSC 4117, fall 2006

5. FOL • Domain of objects • Functions of objects (other objects -Domain is closed) • Relations among objects • Properties of objects (unary relations) • Statements about objects, relations and functions D Goforth - COSC 4117, fall 2006

6. Objects in FOL • Constants – names of specific objects • E.g., Doreen, Gord, William, 32 • Functions – Father(Doreen), Age(Gord), Max(23,44) • variables – a, b, c, … for statements about unidentified objects or general statements D Goforth - COSC 4117, fall 2006

7. FOL - example • Domain {Art, Bill, Carol, Doreen} • Functions of objects: • Mother(Art) identifies an object • Relations: • Siblings (Bill, Carol) true or false • Properties of objects (unary relations) • IsStudent(Carol) true or false • Statements about domain: • Mother(Bill) = Mother(Carol) true or false D Goforth - COSC 4117, fall 2006

8. Formal Definition of FOL Statement about relation or property OR Equivalence of objects Reference to an object Statements about sets of objects Relation or property

9. Propositional logic vs. FOL Propositional Propositions (t/f) Connectives  sentences FOL Objects, functions Relations on objects (t/f) Connectives  sentences Quantifiers D Goforth - COSC 4117, fall 2006

10. symbols in FOL • objects (constants), functions, predicates • BIGGEST PROBLEM LEARNING FOL: DIFFERENCE BETWEEN FUNCTIONS AND PREDICATES • interpretations specify meaning of each symbol (intended interpretation) • models determine truth of sentences • e.g. if symbols Doreen and Mother(Art) refer to same object then statement Mother(Art) = Doreen is true D Goforth - COSC 4117, fall 2006

11. The quantifiers • allow statements about many objects • apply to sentence containing variable • universal : true for all substitutions for the variable • existential : true for at least one substitution for the variable D Goforth - COSC 4117, fall 2006

12. The quantifiers • examples: • x: Mother(Art) = x • x y: Mother(x)=Mother(y) => Sibling(x,y) • y x: Mother(y) = x • x y: Mother(y) = x (not! nest carefully) D Goforth - COSC 4117, fall 2006

13. Manipulating quantifiers • de Morgan’s laws • existential is generalized “OR” ~x: S(x) <=> x: ~S(x) • universal is generalized “AND” ~ x: S(x) <=> x: ~S(x) D Goforth - COSC 4117, fall 2006

14. Example domain - kinship • objects – people • functions • Mother(x), Father(x) • predicates • Female(x), Parent(x,y), Spouse(x,y) • definitions (compound sentences in KB) • x: Male(x) <=> ~ Female(x) [depends on domain!] • x y : y = Mother(x) <=> Female(y)^Parent(y,x) • x y : y = Father(x) <=> Male(y)^Parent(y,x) • define these: • child, grandparent, sibling, brother D Goforth - COSC 4117, fall 2006