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First Order Logic (FOL)

- Syntax
- Semantics
- Wumpus world example
- Ontology (ont to be logica word) kinds

of things one can talk about in the language

Why first-order logic?

- We saw that propositional logic is limited

because it only makes the ontological commitment

that the world consists of facts. - Difficult to represent even simple worlds like

the Wumpus world - e.g.,
- dont go forward if the Wumpus is in front of

you takes 64 rules.

First-order logic (FOL)

- Ontological commitments
- Objects wheels, doors, engines, cars, Dave, Joe
- Relations Inside(car1, Dave), Beside(Dave, Joe)
- Functions WeightOf(car1)
- Properties IsOpen(door1), IsOn(engine1)
- Taxonomic Properties Car(car1), Person(Dave),

Person(Joe) - Relations return True or False
- Functions return a single objects, otherwise they

wouldnt be functions. - WeightOf(x) returns a single weight
- BrotherOf(x) is not a function. You may have

several brothers. But, NaturalMotherOf(x) is a

function.

Semantics

- There is a correspondence between
- functions, which return values
- predicates (or relations), which are true or

false - Function fatherOf(Mary) Bill
- Predicate fatherOfP(Mary, Bill)
- The point is that every function can have a

corresponding predicate associated with it.

Examples

- One plus two equals three
- Objects
- Relations
- Properties
- Functions
- Squares neighboring the Wumpus are smelly
- Objects
- Relations
- Properties
- Functions

Examples

- One plus two equals three
- Objects Numbers, one, two, three, one plus two
- Relations equals(one, two)
- Properties Integer(one)
- Functions plus (one, two) returns one plus

two , the object obtained by applying

function plus to one and two - three is another name for this object)
- Squares neighboring the Wumpus are smelly
- Objects squares, Wumpus, square1,1, square1,2
- Relations neighboring(square1,1, square1,2)
- Properties smelly(square1,1)
- Functions rt-neighborOf(square1,1) returns

square1,2

FOL Syntax of basic elements

- Constant symbols 1, 5, A, B, USC, JPL, Alex,

Manos, - Predicate symbols gt, friend, student, colleague,

- Function symbols , sqrt, schoolOf,
- Variables x, y, z, next, first, last,
- Connectives ?, ?, ?, ?
- Quantifiers ?, ?
- Equality

FOL Atomic sentences

- AtomicSentence ? Predicate(Term, ) Term Term
- Term ? Function(Term, ) Constant Variable
- Examples
- SchoolOf(Manos)
- Colleague(TeacherOf(Alex,cs561), TeacherOf(Manos,

cs571)) - gt(( x y), x)

FOL Complex sentences

- Sentence ? AtomicSentence Sentence

Connective Sentence Quantifier Variable,

Sentence ? Sentence (Sentence) - Examples
- S1 ? S2, S1 ? S2, (S1 ? S2) ? S3, S1 ? S2, S1?

S3 - Colleague(Paolo, Maja) ? Colleague(Maja, Paolo)

Student(Alex, Paolo) ? Teacher(Paolo, Alex)

Semantics of atomic sentences

- Sentences in FOL are interpreted with respect to

a model - Model contains objects and relations among them
- Terms refer to objects (e.g., Door1, Alex,

MotherOf(Paolo)) - Constant symbols refer to objects
- Predicate symbols refer to relations
- Function symbols refer to functional Relations
- An atomic sentence predicate(term1, , termn) is

true iff the relation referred to by predicate

holds between the objects referred to by term1,

, termn

Example model

- Objects John, James, Mary, Alex, Dan, Joe, Anne,

Rich - Relation sets of tuples of objectsltJohn,

Jamesgt, ltMary, Alexgt, ltMary, Jamesgt, ltDan,

Joegt, ltAnne, Marygt, ltMary, Joegt, - E.g. Parent relation -- ltJohn, Jamesgt, ltMarry,

Alexgt, ltMary, Jamesgtthen Parent(John, James)

is true Parent(John, Mary) is false

Quantifiers

- Expressing sentences about collections of objects

without enumeration (naming individuals) - E.g., All Trojans are clever Someone in the

class is sleeping - Universal quantification (for all) ?x R(x)
- Existential quantification (there exists) ?x R(x)

Universal quantification (for all) ?

- ? ltvariablesgt ltsentencegt
- Every one in the cs561 class is smart ? x

In(cs561, x) ? Smart(x) - ? P corresponds to the conjunction of

instantiations of PIn(cs561, Manos) ?

Smart(Manos) ? In(cs561, Dan) ? Smart(Dan) ?

In(cs561, Bush) ? Smart(Bush)

Universal quantification (for all) ?

- ? is a natural connective to use with ?
- Common mistake to use ? in conjunction with ?

e.g ? x In(cs561, x) ? Smart(x)means every

one is in cs561 and everyone is smart

Existential quantification (there exists) ?

- ? ltvariablesgt ltsentencegt
- Someone in the cs561 class is smart ? x

In(cs561, x) ? Smart(x) - ? P corresponds to the disjunction of

instantiations of P(In(cs561, Manos) ?

Smart(Manos)) ? (In(cs561, Dan) ? Smart(Dan)) ?

(In(cs561, Bush) ? Smart(Bush))

Existential quantification (there exists) ?

- ? is a natural connective to use with ?
- Common mistake to use ? in conjunction with ?

e.g ? x In(cs561, x) ? Smart(x)is true if

there is anyone that is not in cs561! - (remember, false ? true is valid).

Properties of quantifiers

Not all by one person but each one at least by one

Proof?

Negation Laws for Quantified Statements

- ? x P(x) ltgt ? x P(x)
- ? x P(x) ltgt ? x P(x)

Informal Proof

- In general we want to prove
- ? x P(x) ltgt ? x P(x)
- ? x P(x) ((? x P(x))) ((P(x1) P(x2)

P(xn)) ) (P(x1) v P(x2) v v P(xn)) ) - ? x P(x) P(x1) v P(x2) v v P(xn)
- ? x P(x) (P(x1) v P(x2) v v P(xn))

Example sentences

- Brothers are siblings .
- Sibling is transitive.
- Ones mother is ones siblings mother.
- A first cousin is a child of a parents

sibling.

Example sentences

- Brothers are siblings ? x, y Brother(x, y) ?

Sibling(x, y) - Sibling is transitive? x, y, z Sibling(x, y)

? Sibling(y, z) ? Sibling(x, z) - Ones mother is ones siblings mother? m, c

Mother(m, c) ? Sibling(c, d) ? Mother(m, d) - A first cousin is a child of a parents

sibling? c, d FirstCousin(c, d) ? ? p, ps

Parent(p, d) ? Sibling(p, ps) ? Parent(ps, c)

Example sentences

- Ones mother is ones siblings mother? m, c,d

Mother(m, c) ? Sibling(c, d) ? Mother(m, d) - ? c,d ?m Mother(m, c) ? Sibling(c, d) ? Mother(m,

d)

Translating English to FOL

- Every gardener likes the sun.
- ? x gardener(x) gt likes(x,Sun)
- You can fool some of the people all of the time.
- ? x ? t (person(x) time(t)) gt can-fool(x,t)

Translating English to FOL

- You can fool all of the people some of the time.
- ? x ? t (person(x) time(t)) gt
- can-fool(x,t)
- All purple mushrooms are poisonous.
- ? x (mushroom(x) purple(x)) gt poisonous(x)

Translating English to FOL

- No purple mushroom is poisonous.
- (? x) purple(x) mushroom(x) poisonous(x)
- or, equivalently,
- (? x) (mushroom(x) purple(x)) gt poisonous(x)

Translating English to FOL

- There are exactly two purple mushrooms.
- (? x)(? y) mushroom(x) purple(x) mushroom(y)

purple(y) (xy) (? z) (mushroom(z)

purple(z)) gt ((xz) v (yz)) - Deb is not tall.
- tall(Deb)

Translating English to FOL

- X is above Y if X is on directly on top of Y or

else there is a pile of one or more other objects

directly on top of one another starting with X

and ending with Y. - (? x)(? y) above(x,y) ltgt (on(x,y) v (? z)

(on(x,z) above(z,y)))

Equality

- A sibling is another child of ones parents
- ? x,y Sibling(x,y) ? (xy) ? ? p Parent(p, x)

? Parent(p, y)

Higher-order logic?

- First-order logic allows us to quantify over

objects ( the first-order entities that exist in

the world). - Higher-order logic also allows quantification

over relations and functions. - e.g., two objects are equal iff all properties

applied to them are equivalent - ? x,y (xy) ? (? p, p(x) ? p(y))
- Higher-order logics are more expressive than

first-order however, so far we have little

understanding on how to effectively reason with

sentences in higher-order logic.

Rememberpropositionallogic

Reminder

- Ground term A term that does not contain a

variable. - A constant symbol
- A function applies to some ground term
- x/a substitution/binding list

Proofs

Proofs

- The three new inference rules for FOL (compared

to propositional logic) are - Universal Elimination (UE) (also called

Universal Instantiation) - for any sentence ?, variable x and ground term

?, - ?x ?
- ?x/?
- Existential Elimination (EE) (also called

Existential Instantiation) - for any sentence ?, variable x and constant

symbol k not in KB, - ?x ?
- ?x/k
- Existential Introduction (EI) (also called

Existential Generalization) - for any sentence ?, variable x not in ? and

ground term g in ?, - ?
- ?x ?g/x

Proofs

- The three new inference rules for FOL (compared

to propositional logic) are - Universal Elimination (UE) (also called

Universal Instantiation) - for any sentence ?, variable x and ground term

?, - ?x ? e.g., from ?x Likes(x, Candy) and

x/Joe - ?x/? we can infer Likes(Joe, Candy)
- Existential Elimination (EE) (also called

Existential Instantiation) - for any sentence ?, variable x and constant

symbol k not in KB, - ?x ? e.g., from ?x Kill(x, Victim) we can

infer - ?x/k Kill(Murderer, Victim), if Murderer

new symbol - Existential Introduction (EI) (also called

Existential Generalization) - for any sentence ?, variable x not in ? and

ground term g in ?, - ? e.g., from Likes(Joe, Candy) we can

infer - ?x ?g/x ?x Likes(x, Candy)

A Simple Proof

- Buffalo(Bob) Bob is a buffalo
- Pig(Pat) Pat is a pig
- ?x,y Buffalo(x) Pig(y) -gt Faster(x,y)

Buffaloes outrun pigs. - Prove Faster(Bob,Pat) Bob outruns Pat.
- Buffalo(Bob) Pig(Pat) -gt Faster(Bob,Pat)

3,UE xBob,yPat - Buffalo(Bob) Pig(Pat) 1,2 Conjunction
- Faster(Bob,Pat) 5,6 Modus Ponens QED!!

Another Proof (longer, but just as simple)

- C(x) means x is in this class.
- B(x) means x has read the book.
- P(x) means x has passed the class.
- ?x C(x) B(x) (Someone in the class has not

read the book) - ?x C(x) -gt P(x) (Everyone in the class passes)
- Prove ?x P(x) B(x) (Someone passed and didnt

read the book) - C(a) B(a) 1, EE (a is the skolem constant

must be new) - C(a) 4, And-Elimination (or simplification)
- C(a) -gt P(a) 2, Universal Instantiation
- P(a) 5,6 Modus Ponens
- B(a) 4, And-Elimination (or simplification)
- P(a) B(a) 7,8 Conjunction
- ?x P(x) B(x) 9, Existential generalization

QED!

More Inference Techniques

- Remember ?x P(x) ltgt ?x P(x) ?x P(x) ltgt ?x

P(x) - Say you are doing a proof 1. 2. 3. Prove ?x

P(x) 4. ?x P(x) Proof by contradiction5. ?x

P(x) 4, Equivalence6. P(a) 5, EE (a is

skolem)7. - DO YOUR EES FIRST! THEN DO YOU YOUR UES.

Logical agents for the Wumpus world

Remember generic knowledge-based agent

- TELL KB what was perceivedUses a KRL to insert

new sentences, representations of facts, into KB - ASK KB what to do.Uses logical reasoning to

examine actions and select best.

Using the FOL Knowledge Base

Set of solutions

Wumpus world, FOL Knowledge Base

Deducing hidden properties

Situation calculus

Describing actions

May result in too many frame axioms

Describing actions (contd)

Planning

Generating action sequences

empty plan

Recursively continue until it gets to empty plan

Summary