First order logic (FOL) first order predicate calculus

D Goforth - COSC 4117, fall 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 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 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 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 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 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

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

D Goforth - COSC 4117, fall 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 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 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 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 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 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