Download First-Order Logic Tutorial

CS 305 – Introduction to
Artificial Intelligence (AI)
A First-Order Logic Tutorial
1
Purpose

This tutorial was created to help explain
first-order logic
2
The Basics

Constant symbols: B, X, Jane, Ralph


A specific object in the world
Predicate symbols: Parent(John, Jane),
Brother(Jack, Jill)

Refers to a relation in a model


Ex: John is Jane’s Parent
Ex: Jack is Jill’s brother
3
The Basics (continued)

Function Symbols: FatherOf(John),
PetOf(Jill)

An object is directly related to exactly one
other object


Ex: The object John is directly related to the
object John by the relationship FatherOf. That
object may be called Ralph.
Ex: The object Jill is directly related to another
object called Sparky by the relation PetOf.
4
The Basics (continued)

Terms: A, Jane, Ralph, FatherOf(John),
PetOf(Jill)


Atomic Sentences: Parent(John, Jane),
Brother(Jack, Jill)


Logical representation that refers to an particular
object
Predicate symbol followed parenthesized list of
terms
Complex Sentences:

Constructed by using logical connectives

Connectives include: ,,,,
5
Examples of Sentences


Understanding how to construct basic
sentences is key to using FOL
Ex: YoungerThan(Joe, Jim)


Parent(Adam, Zoe)  Parent(Jane, Adam)


This sentence says “Joe is younger than Jim”
This sentence says “Adam is Zoe’s parent and
Jane is Adam’s Parent”
Ex: Cat(Morris)  Likes(Morris, catnip)

This sentence says “If Morris is a cat then he likes
catnip”
6
Create your own sentences

Given the following English sentences,
create FOL sentences




Fido is Jack’s pet.
Jill owns a car.
Tweety is a yellow bird and is Bob’s pet.
Tom is a cat and Jerry is a bird.
7
Create your own sentences

English: Fido is Jack’s pet.


English: Jill owns a car.


FOL: Owns(Jill,car)
English: Tweety loves bird seed and he is
Bob’s pet.


FOL: Pet(Fido, Jack)
FOL: Loves(Tweety,bird seed)  Pet(Tweety,Bob)
English: Tom is a cat and Jerry is a bird.

FOL: Cat(Tom)  Bird(Jerry)
8
Representing Collections

How can you represent of an entire
collection of objects using FOL?

Refer to “All people…” or “There is a
person…”


Ex: How can one state “All dogs like bones”
using FOL?
Ex: How can one state “Someone likes liver”
using FOL?
9
Quantifiers

Universal Quantifier: 


“For all….”
Ex: “All dogs like bones”



FOL: xDog(x)  Likes(x,bones)
This reads “For any object x, if x is a dog then x likes
bones”
Ex: “Anything with a beak and feathers is a bird”


FOL: xHasBeak(x)  HasFeathers(x)  Bird(x)
This reads “For any object x, if x has a beak and x has
feathers then x is a bird”
10
Quantifiers

Existential Quantifier: 


“There exists….”
Ex: “There is someone who likes liver”



FOL: xLikes(x, liver)
This reads “There is an object x that likes liver”
Ex: “Mary owns a coat”

xOwns(Mary,x)  Coat(x)
11
Using Inference FOL (Proofs)

Recognize which rules you can use.


The primary rules you can use are the Inference
Rules on pages 211-212 (these rules apply to FOL
as well as propositional logic)
In order to apply these rules, you must also have
rules to handle (get rid of) quantifiers.

The rules to accomplish this are discussed on pages 272273
12
Simple Proof

Suppose we have the following
situation:


Anything that barks is a dog. Fido barks.
We wish to prove that Fido is a dog.


This fact is obvious to us but how can a
computer prove this?
See the next slide
13
Simple Proof (continued)

Step 1: State the facts you know in FOL

We know anything that barks is a dog.
State this fact in FOL:

a. xBarks(x)  Dog(x)


This says “If x barks then x is a dog”. We are given
this fact.
We know that Fido barks. State this fact in
FOL:

B. Barks(Fido)
14
Simple Proof (continued)

Step 2: Remove all quantifiers


Apply the Universal Instantiation inference
rule to remove the universal quantifier ()
in sentence a
The result is:

c. Barks(Fido)  Dog(Fido)
15
Simple Proof (continued)

So far we have the following sentences:




A. xBarks(x)  Dog(x)
B. Barks(Fido)
C. Barks(Fido)  Dog(Fido)
Step 3: See what inference rules can be
applied.



Think about what we want to do: Eliminate the
implication in C, leaving the sentence “Dog(Fido)”
How can we so this? Use Modens Ponens
D. Dog(Fido)
16
Simple Proof (continued)

Our entire proof looks like:
A. xBarks(x)  Dog(x)
State a given fact
B. Barks(Fido)
State a given fact
C. Barks(Fido)  Dog(Fido)
Apply Universal Instantiation on C
D. Dog(Fido)
Apply Modus Ponens to B and C
17