Ontologies

1
Ontologies

• The formal definition of ontology is
• (1) a branch of metaphysics concerned with the
  nature and relations of being and
• (2) a particular theory about the nature of being
  or the kinds of existents
• The term comes from philosophy
• In AI, we define an ontology as
• A representation vocabulary, often specialized to
  some domain (or subject matter)
• An alternative AI view is that the ontology
  represents either common sense knowledge or
  knowledge to support natural language
• This latter view is more restrictive in terms of
  usage and more general in terms of the kinds of
  knowledge that is to be represented

2
Why Ontologies?

• Recall the variety of knowledge representations
  used in AI during the 1960s and 1970s
• Imagine that expert system 1 performs automotive
  engine diagnosis and expert system 2 designs new
  automotive engines that have superior gas mileage
• Can the two systems communicate with each other?
  Can one draw upon knowledge from the other?
• probably not, but why not?
• Given the number of expert systems, and the sheer
  amount of knowledge modeled, it seems ridiculous
  to not be able to share it across problem solvers
• The ontology is supposed to provide a mechanism
  for knowledge sharing
• knowledge-based systems
• search engines
• Internet agents
• natural language reasoners
• CYC (common sense reasoner) and the applications
  that it supports

3
Ontology as Content Theory

• AI researchers have often viewed AI from a
  mechanism theory point of view
• To get AI to work, we have to implement systems
  that do things so that AI should be the study of
  how people solve problems
• On the other hand, some view the central problem
  of AI in terms of representation
• What knowledge do we need to solve the given
  problem?
• The ontology promotes this latter view, AI as a
  study of content theory
• Content theory means that we study and theorize
  over the nature of the content needed for a
  system, that is, the nature of knowledge
• This is not to say that the knowledge
  representation problem is solved
• We must still make a commitment in our ontology
  in terms of the form of representation, but we
  may select unwisely
• In the past, the primary representations for
  ontologies have been
• predicate calculus
• frame systems
• semantic networks

4
Types of Knowledge

• One of the problems that early expert systems had
  was that there was no way to distinguish between
  what a system knew from what a system used in
  terms of knowledge
• Consider the auto-mechanic diagnostic system,
  while we can ask it questions like what is the
  cause of these symptoms? we cannot ask it do
  you know what a spark plug does? or tell me
  everything you know about car engines
• There are different ways to define what a system
  knows
• David Marr defined 3 levels of knowledge
• what problems can be solved by the system
  (computational level)
• the strategy that is used (algorithmic level)
• how the knowledge is represented/implemented
  (implementation level)
• Allen Newell instead defined 2 levels
• what the system knows (knowledge level)
• how the system knows it (symbol level)
• With distinctions such as these, we can
  differentiate between what is in an ontology and
  how the ontology is going to work

5
What Should an Ontology Store?

• Since an ontology is an attempt to store
  knowledge neutral of its use, we need to define
  the types of knowledge that an ontology will
  store
• The ontology needs to know about things in the
  domain
• classes and the relationships between classes
  (class/subclass for instance, but also other
  relationships such as classes that are siblings
  of the same parent class), and also instances of
  classes
• features/attributes, the values that can be
  stored in those features, the range and types for
  those values as well as restrictions that might
  be placed on those values
• properties found within the domain (e.g., laws of
  the domain, relationships between classes and
  features, inference rules)
• Domain specific elements may also be applied
• What we dont want to do is place only
  problem-specific knowledge in the ontology
• For instance, if we are creating an ontology of
  medicine for diagnostic purposes, the medical
  knowledge should be in a neutral form so that it
  can also be used for reasoning over drug
  interactions, drug development, and medical
  experiments

6
Upper Ontology

• An ontology will start with the most basic
  concepts at the root of the tree
• Entity or Thing is commonly used as a root node
• There is disagreement over the root nodes
  children
• CYC individual object, intangible, represented
• GUM configuration, element, sequence
• Wordnet living, nonliving
• Sowa concrete, process, object abstract
• Given such divergent approaches for upper
  ontology, how can we possibly agree upon an
  entire domain ontology?
• There are some concepts that do seem universal
• objects exist, objects have attributes which have
  values which can change over time, values have
  properties
• objects have relationships with other objects,
  contain parts, and participate in or perform
  processes
• there are states that objects can be in, and
  states can change over time
• there are events that occur during time instants
  where events cause other events and/or cause
  objects to change states

7
Beyond an Upper Ontology

• We will typically want to represent a specific
  domain of interest which will be the entities in
  the domain and how the entities relate
• Entities can be at a
• class level (e.g., living thing, emotional state,
  event)
• specific classes (e.g., human being, love, rock
  concert)
• And instances (e.g., Frank Zappa, my love of
  Frank Zappa music, the Frank Zappa concert that I
  saw in 1988)
• Attributes (features) of each class (as discussed
  two slides back)
• The vocabulary of the domain
• Relations for how entities relate to each other
• is-a class/subclass relationship
• instance class/instance relationship (or should
  we have both class/subclass and class/instance
  relationships be denoted with is-a?)
• has-a component/part relationship
  (alternatively, we could use part-of or contains)
• is-in spatial relationship (or possibly
  temporal relationship)

8
Example Ontology Sports

• Question who decides what should be represented
  and how?
• here, the ontology is tangled (multiple parents)
• we may disagree about whether there should be
  more than events here (players are not events,
  sports are not events)

9
Example Ontology CS Dept.
Organization Department School
University Program ResearchGroup
Institute Publication Article
TechnicalReport JournalArticle
ConferencePaper UnofficialPublication
Book Software Manual
Specification Work Course
Research Schedule
Person Worker Faculty
Professor AssistantProfessor
AssociateProfessor VisitingProfessor Lecture
r PostDoc Assistant
ResearchAssistant
TeachingAssistant AdministrativeStaff
Director Chair Professor Dean
Professor ClericalStaff SystemsStaff
Student UndergraduateStudent
GraduateStudent
10
Example Continued Relationships
Relation Argument 1
Argument 2
publicationAuthor Publ
ication Person publicationDate Publication .DAT
E publicationResearch Publication Research softwa
reVersion Software .STRING softwareDocumentatio
n Software Publication teacherOf Faculty Cou
rse teachingAssistantOf TeachingAssistant Course
takesCourse Student Course age Person .NUMB
ER emailAddress Person .STRING head Organizat
ion Person undergraduateDegreeFrom Person Unive
rsity doctoralDegreeFrom Person University advis
or Student Professor alumnus Organization P
erson affiliateOf Organization
Person researchInterest Person Research
researchProject ResearchGroup Research listedCou
rse Schedule Course tenured
Professor .TRUTH
11
Example Continued Sample Inferences

• Suborganizations are transitive
• If subOrganization(x,y) and subOrganization(y,z),
  then subOrganization(x,z).
• Affiliates are invertable
• If affiliatedOrganization(x,y), then
  affiliatedOrganization(y,x).
• Membership transfers through suborganizations
• If member(x,m) and subOrganization(x,y), then
  member(y,m).
• These are domain-independent properties
• On the previous slide, we had used already
  defined classes to help us define the
  relationships Person, .DATE, .STRING, .NUMBER,
  University, etc
• Domain independent inferences should be available
  for these domain-independent properties but we
  have to make sure we apply them correctly
• For instance, teacher-student relationships are
  not transitive, and a student being enrolled in a
  course may or may not represent a membership
  relation

12
Representing Methods

• An ontology does not have to be limited to domain
  knowledge
• we may also include how to perform processes in
  the domain
• we often refer to such processes as methods
• Below we see several ontologies which draw upon
  various problem solving methods (PSMs)
• some of the methods available are Propose
  Revise (PR) and Cover and Differentiate (CD)

13
Representational Format

• In order to create a machine-usable ontology
• we need algorithms to manipulate and infer over
  the data
• algorithms require a specific form of data
  structure
• what structures (representations) should we use?
• Cyc uses a first-order predicate calculus form
• (Teacher_of CSC_625 Fox)
• (Musician Frank_Zappa)
• (Band_member Genesis Phil_Collins)
• OWL uses an XML approach with tags defined by the
  ontology author(s)
• Should inferences be captured in rule-based form
  or other?
• A Cyc inference might look like this
• (implies (hasDegreeIn ?PERSON ?DEGREE)
  (hasAttribute ?PERSON CollegeGraduate))
• These issues have not been resolved
• nor do they seem likely to be resolved because
  people are using different representations in
  their various ontological research (Ontolingua,
  Cycl, OWL, KQML, KIF, Protégé-II, etc)

14
Ontological Commitments

• As stated earlier, the various ontology formats
  make different commitments to the upper ontology
• This means that they are already somewhat
  incompatible
• The question is what primitives need to be
  predefined for an ontology?
• Just about everyone agrees that ontologies should
  support class/subclass relationship and therefore
  there should be some mechanism for inheritance
• What other primitives should our ontology have?
• OWL for instance makes a commitment to the
  following primitives
• URIs, class, properties (domain, range, type),
  equivalences (i.e., setting two classes to be
  equivalent)
• Cyc has a greater set of primitives that include
• spatial primitives, temporal primitives, language
  primitives (e.g., grammatical roles)

15
Are Primitives Universal?

• Early work in knowledge representation in AI
  already demonstrated that researchers want to use
  different primitives
• Semantic networks contained these primitives
• isa (class/subclass), instance, has-part, object
  (meaning a physical object), agent, beneficiary,
  state, and properties, also mathematical
  relations ()
• Roger Schank defined primitives that were all
  verb-related for his Scripts/Conceptual
  Dependencies
• ATRANS (transfer of abstract relationship),
  PTRANS (transfer of physical location), PROPEL,
  MOVE, GRASP, INGEST, EXPEL, MTRANS (transfer of
  mental information), MBUILD (build new
  information) SPEAK, ATTEND
• Upper ontology implies that there is a set of
  primitive primitives
• That is, primitives that should underlie all
  knowledge and uses of knowledge

16
The Envisioned Uses of Ontologies

• Sharing common understanding of the structure of
  information
• consider web sites that contain medical
  information and e-commerce services computer
  agents could extract and aggregate information
  from different sites to answer user queries or to
  use the data as input to other applications
• Enabling reuse of domain knowledge
• when one organization of researchers or
  developers creates an ontology, others can reuse
  it for their own domains and purposes
• Making explicit domain assumptions
• hard-coding assumptions about the world in
  programming-language code makes assumptions hard
  to find, hard to understand, hard to change
• Separating domain knowledge from operational
  knowledge
• Analyzing domain knowledge

17
Using an Ontology

• How will we use our ontologies?
• To enhance search engines beyond keyword searches
• I want to search for peer-reviewed publications
  and so I search for articles, but someone has
  defined their peer-reviewed papers as journal
  articles and their articles include technical
  reports
• To annotate multimedia data files
• we cant currently search for the content of
  image or sound files
• To annotate design components
• imagine an expert system that needs to replace
  component 1 with component 2 based on the
  component functions and sizes
• Intelligent agents
• so that two agents can find a common vocabulary
  to communicate together
• To support ubiquitous computing endeavors

18
Examples How Ontologies are Used

• Question answering
• If coded in the ontology, simply look up and
  respond
• class/subclass information (primarily through
  inheritance)
• attribute value response/database extraction
• If inferences are available then select and apply
  appropriate inference
• for example, given that A isa B and B is part of
  a C, we might want to know if A is part of a C

19
Continued

• Processing
• We might want to transform or manipulate some set
  of knowledge
• this will again require some special purpose
  inference or process, for example from a
  biological ontology
• if either X biosynthesis or X catabolism
  exists then the parent X metabolism must also
  exist
• Translation
• The knowledge of one ontology might be used in
  conjunction with another ontology and therefore
  terms in one ontology might need translating into
  the proper format/language/terms of the other
• special purposes methods might be available to
  translate terms
• otherwise, common terms must be identified
  between the ontologies so that related terms can
  then be recognized even if the related terms go
  by different names

20
Cyc Common Sense Reasoning

• An attempt to construct a common sense
  knowledge-base
• Effort of about 20 years of coding, a
  person-century according to the earlier article
  (more than that to this point)
• 5,000,000 common sense facts and rules
• General-purpose knowledge-base to be used with
  other applications
• other applications are to sit on top of CYC so
  that, when necessary, these system can delve
  deeper into common sense/general-purpose
  knowledge when the special-purpose knowledge is
  not adequate
• CycL is the language used to encode Cycs KB
• Originally, it was a frame-like system (loosely
  organized objects), but modified to use predicate
  calculus
• Rules represented in free-form ways depending on
  their specific intentions with uncertainty
  represented only through relative rankings
• P is more likely than Q
• Categories of facts and rules arranged
  hierarchically using a directed graph (instead of
  a tree)

21
Cyc Ontology

• Categories of categories
• Categories of individuals (tangible and
  intangible objects)
• Categories of scripts (typical actions and
  events)
• physiological actions, problem solving actions,
  communications, rites of passage, work, hobby,
  natural phenomena
• Entities are categorized as
• Relationships (predicates)
• Attributes
• Lexical items (words, parts of speech/tense,
  etc)
• Proper nouns (people, places, )
• Microtheories (explained on the next slide)
• Miscellany
• Special-purpose inferences and heuristics (for
  efficiency)
• Temporal and spacial reasoning using qualitative
  or naïve physics
• Domain specific axioms (e.g., medical diagnostic
  rules)
• Inferences for specific syntactic structures
• General-purpose axioms to be applied when
  special-purpose axioms are not available or do
  not work

22
Naïve Physics

• A well researched problem is how do humans reason
  naively
• if I toss something in the air, what happens to
  it? You dont apply physics, you use common
  sense
• Naïve Physics, also called qualitative reasoning,
  is something that we might want to model in an
  ontology
• when I pour liquid from one cup into another half
  filled cup, what is the result?
• when I let go of an object, what happens?
• In these cases we use naïve reasoning ideas
  instead of physics
• We can devise some simple rules
• pouring from a non-empty cup into a non-empty cup
  results in the second cup containing more and
  possibly overfilling
• unless supported, items will fall
• There are qualitative equations to model these
  situations that we might want to use in place of
  quantitative (physics) equations

23
Qualitative Reasoning Examples

• We will use qualitative state changes to describe
  the current state of an entity
• Physics reasoning might include such states as
  increase, decrease, steady, empty, between and
  full and use simple inference rules like
• Empty Empty Empty Empty Full Full
• Empty Between Between Full Between
  Overflow
• Between Between Between, Full
• Temporal reasoning might include states like
  before, during, after, overlap, coincide, etc
  with rules like
• If X is before Y and Y is before Z then X is
  before Z
• If X coincides with Y then X begins at the same
  time Y begins and X ends at the same time Y ends
• If X is during Y then Y begins before X and Y
  ends after X
• Spatial reasoning might have states like left-of,
  right-of, on-top-of, underneath, obstructed-by,
  inside-of, etc and rules like
• If A is on top of B and A is heavy, then B cannot
  be moved
• If A is inside of B and B is moved to position
  P1, then A is moved to P1

24
Microtheories

• Microtheories partition CYCs knowledge base into
  different domains/concepts/problems and belief
  states
• microtheories are also called contexts
• Examples include medical diagnosis,
  manufacturing, weather during the winter, what to
  look for when buying a car, etc
• information can be lifted from one context to
  another
• Each microtheory has its own
• categories (hierarchy of related items)
• predicates (some are shared between
  microtheories, however one might use the same
  predicate as another with different numbers of
  parameters!)
• axioms/inference methods
• assumptions
• Reasoning within a microtheory might be thought
  of as a separate belief state although in reality
  a microtheory works much like a partitioned name
  space
• Microtheories may overlap or be independent

25
Using Contexts

• Assertions are true within a given context but
  not universally
• the rules behind dining in restaurants differ
  from those of dining at home
• The context is specified in a statement
• A statement may be true in one context and false
  in another
• for instance, an assumption that apples cost 30
  cents may be false in the context of the
  depression-era US
• We might have to lift elements from one context
  into another
• this provides a mechanism for reasoning about
  items in different contexts
• when lifting items from one context to another,
  assumptions, vocabulary, axioms and other
  elements that differ must be resolved in the new
  context
• For example
• a mother with a child will be expected to behave
  a certain way
• but she would be expected to behave like anyone
  else in a grocery store
• under exceptional situations, we lift behavior
  from the mother/child context to override the
  behavior in the grocery store

26
What Does Cyc Know?
27
Some of Cycs Entries
Thing is the "universal collection" the
collection which, by definition, contains
everything there is. Every individual object,
every other collection. Everything that is
represented in the Knowledge Base ("KB") and
everything that could be represented in the KB.
Cyc is designed to support representing any
imaginable concept in a form that is immediately
compatible with all other representations in the
KB, and directly usable by computer
software. Spatial Things have a spatial extent or
location relative to some other Spatial Thing or
in some embedding space. Note that to say that an
entity is a member of this collection is to
remain agnostic about two issues. First, a
Spatial Thing may be partially tangible (e.g.
Texas-State) or wholly intangible (e.g. the
Arctic Circle or a line mentioned in a geometric
theorem). Second, although we do insist on
location relative to another spatial thing or in
some embedding space, a Spatial Thing might or
might not be located in the actual physical
universe. It is far from clear that all Spatial
Things are so located an ideal platonic circle
or a trajectory through the phase space of some
physical system (e.g.) might not be.
28
An organization is a group whose group-members
are intelligent agents. In each organization,
certain relationships and obligations exist
between the members of the organization, or
between the organization and its members.
Organizations include both informal and legally
constituted organizations. Each organization can
undertake projects, enter into agreements, own
property, and do other tasks characteristic of
agents consequently, an organization is a kind
of agent. Notable organizations include legal
government organization, commercial
organization, and geopolitical entity. Social
Behavior, in Cyc, is made up of social
occurrences. A social occurrence is an action in
which two or more agents take part. In many
cases, social occurrences involve communication
among the participating agents. Some social
occurrences have very elaborate role structures
(e.g. a typical lawsuit), while others have
fairly simple role structures (e.g. greeting a
colleague at work).
29
Part of Cycs Ontology
30
Example Cyc Code
Predicates (objectHasColor  Rover 
TanColor)(memberStatusOfOrganization
Norway  Nato FoundingMember) Defining a
predicate (isa residesInDwelling
BinaryPredicate) (arg1Isa residesInDwelling
Animal) (arg2Isa residesInDwelling
ShelterConstruction) Functions (BorderBetwe
enFn Sweden Norway) returns T or
F (GroupFn Agent) returns the group storing
Agent Using a quantifier (forAll ?X
(implies (owns Fred ?X)
(objectFoundInLocation ?X FredsHouse)))
(implies (isa ?A Animal) (thereExists
?M (and (mother ?A ?M) (isa ?M
FemaleAnimal))))
31
Some Examples of Cyc Inferences

• You have to be awake to eat
• You can usually see peoples noses but not their
  hearts
• You cannot remember events that have not yet
  happened yet
• If you cut a lump of peanut butter in half, each
  half is also a lump of peanut butter, but if you
  cut a table in half, neither half is a table
• If you are carrying a container that's open on
  one side, you should carry it with the open end
  up
• Vampires dont exist (but one microtheory states
  that Dracula is a vampire)
• The U.S.A. is a big country
• When people die, they stay dead

32
Cyc Today and Tomorrow

• Cyc exists in two formats, Cyc as we have
  discussed, and OpenCyc
• OpenCyc is a public domain portion of Cyc that
  has been debugged
• It consists of about 300,000 items rather than
  the 5 million of Cyc
• Cyc is currently used as a platform for which
  larger applications are constructed where
  applications can fall back on the common sense
  and ontological knowledge of Cyc
• This is being used in a variety of situations
  including
• military software and national security (see for
  instance http//www.cyc.com/doc/white_papers/TKB-I
  A2005.pdf)
• network security (see http//www.cyc.com/doc/white
  _papers/IAAI-05-CycSecure.pdf)
• in support of the semantic web and natural
  language understanding (for instance, see
  http//www.cyc.com/doc/white_papers/FLAIRS06-Appli
  cationOfCycToWordSenseDisambiguation.pdf)
• Cyc contains a learning environment that allows
  users to help it add knowledge and refine
  knowledge
• go to http//game.cyc.com/
• Beyond this, Cyc is attempting to learn new
  information by examining web pages and capturing
  the knowledge in its ontology

33
Another Ontology WordNet

• WordNet, like Cyc, is a massive listing of terms
  that encode general worldly knowledge
• In the case of WordNet, all knowledge pertains to
  English words
• definitions
• synonyms
• hypernyms and hyponyms
• The knowledge is in a natural language form
  rather than a machine understandable form
• But like Cyc, WordNet is intended to be used by
  other software (agents) however what they can use
  WordNet for is limited
• E.g., seeing how related two words are by
  following and counting synonym links

34
And Another MikroKosmos

• Processes Spanish text dealing with mergers and
  acquisitions of companies
• The ontology covers
• domain knowledge
• worldly knowledge that might be applicable to the
  domain
• language-specific knowledge and NLU-related
  knowledge

35
OWL Ontology

• One of the leading competitors to Cyc
• OWL is an attempt at an ontology outside of the
  commonsense domain
• So there are fewer upper ontology commitments
• meaning fewer primitives than in CYC
• OWL is based on XML and RDF (Resource Description
  Framework) syntax
• OWL comes in three versions, OWL Full, OWL DL and
  OWL Lite depending on your needs
• We will only consider OWL Full
• The basic idea behind an OWL ontology is that you
  define class/subclass relationships, instances of
  classes, and class/instance properties
• The code that uses the ontology will implement
  how to reason over the classes

36
OWL Class Definitions

• You can define a class using one of six methods
• A new class definition with a specific URI
• An exhaustive enumeration of specific instances
  that make up the class population
• Restrictions on a previously defined class (thus,
  you narrow a prior class by restricting the
  available elements)
• restrictions can be based on values or
  cardinality
• Taking previously defined classes and union or
  intersect them together
• Taking a previously defined class and complement
  it (everything not in that class is in this
  class)
• for the latter five cases, the class can be an
  anonymous class (without name or URI)
• OWL has two predefined classes to start with,
  THING and NOTHING

37
Some Example Code
rdfparseType"Collection"









A class that is the intersection Of the two
classes that contain Tosca, Salome and
Turandon, Tosca
"
rdfabout"NorthAmerica"/ rdfabout"SouthAmerica"/ rdfabout"Australia"/ rdfabout"Antarctica"/ ss
The class of everything that is not meat
rdfabout"Meat"/ ass
The class of continents defined by enumeration
38
Defining Classes Using Restrictions

• allValuesFrom like for all in logic, includes
  all elements of a class that match the given
  value
• rdfresource"hasParent" /
  rdfresource"Human" /
• this restriction forms a class of individuals
  that has a parent whose value is human
• someValuesFrom to be in this class, the
  instance has to have the associated value as one
  of its attributes
• hasValue same as someValuesFrom but the value
  has to be semantically equivalent
• this might be used to define a specific instance
  as in
• maxCardinality, minCardinality to define valid
  ranges of numeric values
• rdfresource"hasParent" /
  rdfdatatype"xsdnonNegativeInteger"2
  ardinality
• this defines the class of entities that has no
  more than 2 parents where parent is a
  non-negative integer data type
• cardinality is used to define a specific
  expected value
• that is, both minimum and maximum

39
More Class Definitions

• Aside from defining a class, you can define
  classes to have certain relationships with other
  classes
• subClassOf
• equivalentClass to equate two classes as being
  equivalent (although not necessary equal)
• disjointWith mutual exclusiveness
• Properties (attributes) can have their own
  specifiers
• ObjectProperty
• DatatypeProperty
• InverseFunctionalProperty
• subPropertyOf
• equivalentProperty
• inverseOf
• TransitiveProperty
• subRegionOf
• SymmetricProperty
• And instances can have their own specifiers
• sameAs
• differentFrom
• AllDifferent

40
More OWL Examples


tiveProperty










ty rdfID"hasMother" rdfresource"hasParent"/
rdfresource"owlFunctionalProperty" /

/
41
Lets Develop an Example

• We will develop a food and wine ontology
• based on the web site http//protege.stanford.edu/
  publications/ontology_development/ontology101-noy-
  mcguinness.html
• First, determine the domain and scope
• types of wine, main types of food, which wines go
  with which foods
• Second, see if there is an ontology available so
  that we can reuse (some of) the knowledge
• also see if the knowledge is available in an easy
  to access or use form, for instance, go to
  www.wines.com to get a list of wines and their
  properties
• Enumerate the important terms of the domain
• for wines and food, we might start with a list
  like wine, grape, winery, location, (wine) color,
  body, flavor, sugar content, food, white meat,
  red meat
• Now define the classes and develop a class
  hierarchy from the terms
• top-down versus bottom-up approach, or some
  combination
• Define class properties
• note that some of the terms earlier are actually
  properties such as flavor, sugar content and
  location, color might be a class or a property
• define the facets of the properties (legal
  values, range of values, how values can be
  derived/computing or found)
• Create class instances

42
Example Continued