first order logic (fol) first order predicate calculus
TRANSCRIPT
First order logic (FOL)
first order predicate calculus
D Goforth - COSC 4117, fall 2006 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
Formal Definitionof
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 2006 9
Propositional logic vs. FOL
Propositional
Propositions (t/f)Connectives sentences
FOLObjects, functionsRelations on objects (t/f)Connectives sentencesQuantifiers
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 statementMother(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