Knowledge%20Representation

1
Knowledge Representation
In which we show how to use first-order logic to
represent the most important aspects of the real
world.
This is a treatise of chapter 12 of Russel
Norvig AIMA (Ed.3) The notation for FOL will be
in implicative normal form, which makes the
examples executable.
2
Categories and Objects
The organization of objects into categories is a
vital part of KR. Important relationships are
subclass relation (AKO - a kind of)
ltcategorygt AKO ltcategorygt. instance
relation ( ISA - is a) ltobjectgt ISA
ltcategorygt.
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The upper ontology()
Anything
AbstractObjects
GeneralizedEvents Sets Numbers
Representations Intervals
Places PhysicalObjects
Processes Categories Sentences
Measurements Moments Things
Stuff
Times Weights
Animals Agents Solid Liquid Gas


Humans
() This is AIMA s version. Other authors have
other partitions. See bus_semantics
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Bus semantics
study ako activity. subset
ako thing. version ako
abstract. accident ako activity.
activation ako activity. start of
it addition ako abstract. address
ako place. advice ako
abstract. age ako year.
(measure) agreement ako abstract.
air ako place. kan bussen gå i
lufta ? analysis ako abstract. animate
ako agent. application ako
activity. area ako measure.
.. (750 ako items)
thing top node unkn ako thing.
event ako thing. set
ako thing. cardinality ako number. member
ako thing. amount ako thing.
abstract ako thing. activity
ako thing. agent ako thing.
company ako agent. e.g.
TT contact ako thing. content
ako thing. group ako
thing. identity ako thing.
information ako thing. list
ako thing. mass ako thing.
gt stuff measure ako thing. meta
ako thing. lexical object
object ako thing. class.
part ako thing. place
ako thing. river ako thing.
(water but no area) story ako thing.
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Categories
Category is a kind of set and denotes a set of
objects. A category has a set of properties that
is common to all its members. Categories are
formally represented in logic as predicates, but
we will also regard categories as a special kind
of objects. We then actually introduce a
restricted form of second order logic, since the
the terms that occur may be predicates.
Example Elephants and Mammals are
categories. The set denoted by Elephants is a
subset of the set denoted by Mammals. The set
of properties common to Elephants is a superset
of the set of properties common to Mammals.
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Taxonomy
Subcategory relations organize categories into a
taxonomy or taxonomic hierarchy. Other names
are type hierarchy or class hierarchy . We
state that a category is a subcategory of another
category by using the notation for subsets
Basketball ? Ball We will also use the
notation ako(basketball,ball).
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Category representations
There are two choices of representing categories
in first order logic predicates and objects.
That is, we can use the predicate Basketball(b)
or we can reify the category as an object
basketball. We could then write
member(x,basketball) or x ?
basketball We will also use the notation
isa(x,basketball). Basketball is a subset or
subcategory of Ball, which is abbreviated
Basketball ? Ball We will also use the
notation ako(basketball,ball).
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Inheritance
Categories serve to organize and simplify the
knowledge base through inheritance. If we say
that all instances of Food is edible (edible is
in the property set of Food), and if we assert
that Fruit is a subcategory of Food, and Apple
is a subcategory of Fruit, then we know that
every apple is edible. We say that the individual
apples inherit the property of edibility, in this
case from their membership in the Food category.
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Reifying properties
An individual object may have a property. For
example, a specific ball, BB9 can be round. In
ordinary FOL, we write
Round(BB9). As for categories, we can regard
Round as higher order object, and say
BB9 has the property Round We will also use
the notation hasprop(BB9,round).
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Reifying Property Values
Some properties are determined by an attribute
and a value. For example, the diameter of my
basketball BB9 has diameter 9.5 Diameter(BB9
)9.5 We can also use the notation
has(bb9,diameter,9.5). An alternative
representation for properties , when regarded as
Boolean attributes is has(BB9,round,true)
. In the same manner, we can express that a red
ball has colour red. has(BB9,colour,red).
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Logical expressions on categories
An object is a member of a category
isa(bb9,basketball). A category is a subclass of
another category ako(basketball,ball). All
members of a category have som properties
isa(X,basketball) gt hasprop(X,round). Members
of a category can be recognized by some
properties, for example hasprop(X,orange)
and hasprop(X,round) and has(X,diameter,9.5)
and isa(X,ball) gt isa(X,basketball) A category
as a whole has some properties
isa(teacher,profession). Here, it is a fallacy
to conclude that isa(tore,teacher) and
isa(teacher,profession)gtisa(tore,profession).
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Category Decompositions
We can say that both Male and Female is a
subclass of Animal, but we have not said that a
male cannot be a female. That is expressed by
Disjoint(Male,Female) If we know that all
animals are either male or female, (they exhaust
the possibilities) Exhaustive(Male,Female,
Animals). A disjoint exhaustive decomposition is
known as a partition Partition(Male,Female,
Animals).
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Physical Compositions
One object can be a part of another
object. Example, declaring direct parts
part(bucharest,romania).
part(romania,eastern_europe).
part(europe,earth). We can make a transitive
extension partof part(Y,Z) and partof(X,Y)
gt partof(X,Z). and reflexive ()
partof(X,X). Therefore we can conclude that
partof(bucharest,earth) () depending on
definition
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Bunch
It is also useful to define composite objects
with definite parts but no particular structure.
For example, we might say The apples in the bag
weigh two pounds It is adviced that we dont
regard these apples as the set of (all) apples,
but instead define them as a bunch of apples. For
example, if the apples are Apple1,Apple2 and
Apple3, then BunchOf(Apple1,Apple2,Apple3)
Denotes the composite object with three apples
as parts, not elements.
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More about bunches
The category of deodorant actually denotes the
set of all deodorants in the world. A subset of
this, e.g. mydeodorants is a subset of these,
i.e. ako(mydeodorants,deodorant). However,
we are often interested in a set as being a
composite object, with properties connected to
the collected properties of the elements,
like the cardinalty of the set, weight, price
etc. To make this distinction, we can use the
concept "bunch". A bunch is both similar to a
category and to an unstructured composite
object. mydeodorants isa bunch of
deodorant (objects) isa(mydeodorants,deodo
rant). We shall introduce the notation as a
combination bunch(mydeodorants,deodorant).
We also allow various expressions for the
bunch objects bunch(apple1,apple2,apple2,
apple). bunch(apples,tomatoes,fruit).
partof(twopoundoftomatoes,sometomatoesandbeef).
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Substances and Objects
The real world can be seen as consisting of
primitive objects (particles) and composite
objects. A common characteristic is that they can
be counted (individuated) However, there are
objects that cannot be individuated like
Butter, Water, Nitrogen, Wine, Grass, etc. They
are called stuff, and are denoted in English
without articles or quantifiers (not a
water). Typically, they can be divided without
loosing their intrinsic properties. When you take
a part of a substance, you still have the same
substance. isa(X,butter) and partof(Y,X)gtisa(Y,
butter). We can say that butter melts at 30
degrees centigrade isa(X,butter)
gthas(X,meltingpoint,30).
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Measures, Abstracts ,Mentals
In the common world, objects have height, mass,
cost and so on. The values we assign for these
properties are called measures. We imagine that
the universe includes abstract measure objects
such as length. Measure objects are given as a
number of units, e.g. meters. Logically, we can
combine this with unit functions Length(L1)
Inches(1.5) Centimeters(3.81) Another way is
to use predicates Length(L1,1.5,inches)
Abstract concepts like autonomy, quality are
difficult to represent without seeking artificial
measurements. (e.g. IQ). Mental concepts are
beliefs, thoughts, feelings etc.
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Reasoning systems for categories
Semantic networks and Description Logics are two
closely related systems for reasoning with
categories. Both can be described using
logic. Semantic networks provide graphical aids
of visualizing the knowledge base, together with
efficient algoritms for inferring properties of
an object on the basis of its category
membership. Description logics provide a formal
language for constructing and combining category
definitions, and efficient algorithms for
deciding subsets and superset relationships
between categories.
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Semantic Networks Example
Mammal
ako
Legs
Person
2
HaveMother
ako
ako
Female
Male
isa
isa
brother
Legs
1
John
Mary
sister
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Link types in semantic nets
There are 3 types of entities in a semantic
nets categories, objects and values (other
than these) Then there could be 9 different types
of relations between these. They are drawn with
certain conventions. Note that objects can act as
values also. category ? category
ako(C1,C2) every C1 is a.k.o.
C2 category ? category haveatt(C1,R,C2)
every C1has a R a.k.o C2 category ? value
have(C1,R,V) every C1 has attribute
value R V object ? category
isa(O,C) O is a C object ? object
has(O1,R,O2) O1 has relation R to
O2 object ? value has (O,R,V)
O has attribute value RV object ? object
partof (O1,O2) O1 is a part of O2
In addition, we have all kinds of relations
between values. value
V1 gt 2V2 5
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Further comments on link types
We know that persons have female persons as
mothers, but we cannot draw a HasMother link from
Persons to FemalePersons because HasMother is a
relation between a person and his or her mother,
and categories do not have mothers. For this
reason, we use a special notation the double
boxed link. In logic, we have given it the name
haveatt,e.g. haveatt(person,mother,femalep
erson). Compare this to
haveatt(lion,mother,lioness).
We also want to express that persons normally
have two legs.As before, we must be careful not
to assert that categories have legs instead we
use a single-boxed link. In logic, we have given
it the name have,e.g.
have(person,legs,2).
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Content of semantic net
Logic representation ako(person,mammal). ako(fe
male,person). ako(male,person). haveatt(person,mo
ther,female). have(person,legs,2). has(john,legs
,1). isa(mary,female). isa(john,male). has(john
,sister,mary). has(mary,brother,john).
A paraphrase of the knowledge All persons are
mammals All females are persons

All males are persons
Persons have a mother who is female Persons have
normally 2 legs John has 1 leg Mary isa female
John is a male John has a sister Mary Mary has
a brother John
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Inheritance and inference in semantic nets
The rules of inheritance can now be automated
using our logic representation of semantic nets.
isa(X,Y) and ako(Y,Z) gt isa(X,Z). have(Y,R,V)
and isa(X,Y) gt has(X,R,V). haveatt(C,R,M) and
isa(X,C) and has(X,R,V) gt isa(V,M). With these
definitions, we can prove that Mary has two legs,
even if this information is not explicitly
represented.
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Example of inheritance
PROOF has(mary,legs,2) because
have(person,legs,2) and isa(mary,person) hav
e(person,legs,2) is true isa(mary,person)
because isa(mary,female) and
ako(female,person) isa(mary,female) is
true ako(female,person) is true
isa(X,Y)and ako(Y,Z)gtisa(X,Z) have(X,Y,Z)and
isa(A1,X)gthas(A1,Y,Z) haveatt(X,Y,C) and
isa(A1,X) and has(A1,Y,V) gt
isa(V,C). tgtako(person,mammal) tgtako(female,per
son) tgtako(male,person) tgthaveatt(person,mother,
female) tgthave(person,legs,2) tgtisa(mary,female)
tgtisa(john,male) tgthas(john,legs,1) tgthas(john
,sister,mary) tgthas(mary,brother,john)
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Multiple inheritance in a hierarchy
An attribute of a class can be given an attribute
value. This is a default value that can be
redefined in any of its subclasses. In the
example, it is stated that persons normally have
2 legs, while John, who is a person, has 1 leg.
Actually, there are conflicting evidence, but the
ambiguity is solved by stating that the innermost
definition, most specific definition takes place,
i.e. Johns definition. This will be made precise
later.
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Multiple Inheritance
We have described a type of semantic nets that
can be characterized as a hierarchy Each
category is a subclass of exactly one
superclass. However, things are not that
simple. Take for instance a girl, who is both a
child and a female. Such networks are called a
heterarchy. Inheritance in heterarchies is a bit
more complicated. The following semantic network
is an extension of the previous example.
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Example of heterarchy
What is the height of Mary,Alice,Robert and John?
Mammal
175
ako
Height
Legs
Person
2
HaveMother
ako
ako
170
ako
Height
Height
180
100
Height
Female
Male
Child
omitted
isa
Boy
Girl
Legs
John
1
Mary
isa
isa
Alice
Robert
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Heterarchy in Logic
isa(mary,female). isa(john,male). isa(alice,girl)
. isa(robert,boy). has(john,legs,1). have(person
,legs,2). have(person,height,175). have(female,hei
ght,170). have(child,height,100).
have(male,height,180). omitted for sake
of demonstration
ako(person,mammal). ako(female,person). ako(male,p
erson). ako(child,person). ako(girl,female). ako(
girl,child). ako(boy,male). ako(boy,child).
haveatt(person,legs,number). haveatt(person,height
,centimeter).
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Inheritance results
1. Mary has legs 2 default
inheritance from person 2. John has legs 1
explicitly stated 3. Mary has height 170
default inheritance from female 4. John has
height 175 default inheritance from
person 5. Robert has height 100 default
inheritance from child , priority over

inheritance from person 6. Alice has height
100 and height170. contradicting values, no
priority
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Description Logics

• Description logics are notations that are
  designed to make it easier to describe
  definitions and properties of categories.
• The most important inference task is
• subsumption checking if one category is a
  subset of another
• classification checking if an object belongs
  to a category
• consistency checking if a description is
  satisfiable

Small example Bachelor
And(Unmarried,Adult,Male) The equivalent
first-order logic woul be Bachelor(x) ?
Unmarried(x) ? and Adult(x) ? Male(x)
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Description Logic example
Example Describe the set of men with at
least three sons who are all unempoyed and
married to doctors and at most two daughters who
are all professors in physics ot math departments
And(Man,AtLeast(3,Son),AtMost(2,Daughter),
All(Son,And(Unemployed,Married,All(Spouse,Doctor))
), All(Daughter,And(Professor,Fills(Departmen
t,Phusics,Math)))).
As the example shows, Description Logic is a kind
of variable-free logic that can be translated to
FOL (exercise)