Building, Sharing, and Merging Ontologies
John F. Sowa
The art of ranking things in genera and species is of no small importance and very much assists
our judgment as well as our memory. You know how much it matters in botany, not to mention
animals and other substances, or again moral and notional entities as some call them. Order
largely depends on it, and many good authors write in such a way that their whole account could
be divided and subdivided according to a procedure related to genera and species. This helps one
not merely to retain things, but also to find them. And those who have laid out all sorts of notions
under certain headings or categories have done something very useful.
Gottfried Wilhelm von Leibniz, New Essays on Human Understanding
Abstract. For centuries, philosophers have sought universal categories for classifying everything
that exists, lexicographers have sought universal terminologies for defining everything that can be
said, and librarians have sought universal headings for storing and retrieving everything that has
been written. During the 1970s, the ANSI SPARC committee proposed the three-schema
architecture for defining and integrating the database systems that manage the world economy.
Today, the semantic web has enlarged the task to the level of classifying, labeling, defining,
finding, integrating, and using everything on the World Wide Web, which is rapidly becoming the
universal repository for all the accumulated knowledge, information, data, and garbage of
humankind. This talk surveys the issues involved, the approaches that have been successfully
applied to small systems, and the ongoing efforts to extend them to distributed, interconnected,
rapidly growing, heterogeneous systems.
Contents:







1. What is Ontology?
2. Some Modern Systems
3. Trees, Lattices, and Other Hierarchies
4. Notations for Logic
5. Ontology Sharing and Merging
6. Glossary
References
This paper consists of excerpts from previously published articles by John F. Sowa, updated with
new material about ongoing projects on ontology and their implications for databases, knowledge
bases, and the semantic web. For more background on these and related topics, see the book
Knowledge Representation.
1. What is Ontology?
The subject of ontology is the study of the categories of things that exist or may exist in some
domain. The product of such a study, called an ontology, is a catalog of the types of things that are
assumed to exist in a domain of interest D from the perspective of a person who uses a language L
for the purpose of talking about D. The types in the ontology represent the predicates, word senses,
or concept and relation types of the language L when used to discuss topics in the domain D. An
uninterpreted logic is ontologically neutral: It imposes no constraints on the subject matter or the
way the subject is characterized. By itself, logic says nothing about anything, but the combination
of logic with an ontology provides a language that can express relationships about the entities in the
domain of interest.
Aristotle's Categories. The word ontology comes from the Greek ontos for being and logos for
word. It is a relatively new term in the long history of philosophy, introduced by the 19th century
German philosophers to distinguish the study of being as such from the study of various kinds of
beings in the natural sciences. The traditional term for the types of beings is Aristotle's word
category, which he used for classifying anything that can be said or predicated about anything. In
the first treatise in his collected works, Aristotle presented ten basic categories, which are shown at
the leaves of the tree in Figure 1. That tree is based on a diagram by the Viennese philosopher
Franz Brentano (1862).
Figure 1: Aristotle's categories
To connect the categories of Figure 1, Brentano added some terms taken from other works by
Aristotle, including the top node Being and the terms at the branching nodes: Accident, Property,
Inherence, Directedness, Containment, Movement, and Intermediacy.
Genus and Differentiae. The oldest known tree diagram was drawn in the 3rd century AD by the
Greek philosopher Porphyry in his commentary on Aristotle's categories. Figure 2 shows a version
of the Tree of Porphyry, as it was drawn by the 13th century logician Peter of Spain. It illustrates
the subcategories under Substance, which is called the supreme genus or the most general
supertype.
Figure 2: Tree of Porphyry
Despite its age, the Tree of Porphyry has many features that are considered quite modern.
Following is Porphyry's description:
Substance is the single highest genus of substances, for no other genus can be found that is prior to
substance. Human is a mere species, for after it come the individuals, the particular humans. The
genera that come after substance, but before the mere species human, those that are found between
substance and human, are species of the genera prior to them, but are genera of what comes after
them.
Aristotle used the term διαφορα (in Latin, differentia) for the properties that distinguish different
species of the same genus. Substance with the differentia material is Body and with the differentia
immaterial is Spirit. The technique of inheritance is the process of merging all the differentiae
along the path above any category: LivingThing is defined as animate material Substance, and
Human is rational sensitive animate material Substance. Aristotle's method of defining new
categories by genus and differentiae is fundamental to artificial intelligence, object-oriented
systems, the semantic web, and every dictionary from the earliest days to the present.
Syllogisms. Besides his categories for representing ontology, Aristotle developed formal logic as a
precise method for reasoning with them and about them. His major contribution was the invention
of syllogisms as formal patterns for representing rules of inference. The following table lists the
names of the four types of propositions used in syllogisms and the corresponding sentence patterns
that express them.
Type
Name
Pattern
A
Universal affirmative
I
Particular affirmative Some A is B.
E
Universal negative
No A is B.
O
Particular negative
Some A is not B.
Every A is B.
With letters such as A and B in the sentence patterns, Aristotle introduced the first known use of
variables in history. Each letter represents some category, which the Scholastics called praedicatum
in Latin and which became predicate in English. If necessary, the verb form is may be replaced by
are, has, or have in order to make grammatical English sentences. Although the patterns may look
like English, they are limited to a highly stylized or constrained syntax, which is sometimes called
controlled natural language. Such language can be read as if it were natural language, but the
people who write it must have some training before they can write it correctly. The advantage of
controlled language is that it can be automatically analyzed by computer and be translated to logic.
To make the rules easier to remember, the medieval Scholastics developed a system of mnemonics
for naming and classifying them. They started by assigning the vowels A, I, E, and O to the four
basic types of propositions. The letters A and I come from the first two vowels of the Latin word
affirmo (I affirm), and the letters E and O come from the word nego (I deny). These letters are the
vowels used in the names of the valid types of syllogisms. The following table shows examples of
the four types of syllogisms named Barbara, Celarent, Darii, and Ferio. The three vowels in each
name specify the types of propositions that are used as the two premises and the conclusion.
Barbara
A: Every animal is material.
A: Every human is an animal.
A:  Every human is material.
Celarent
E: No spirit is a body.
A: Every human is a body.
E:  No spirit is a human.
Darii
A: Every beast is irrational.
I: Some animal is a beast.
I:  Some animal is irrational.
Ferio
E: No plant is rational.
I: Some body is a plant.
O:  Some body is not rational.
Barbara, Celarent, Darii, and Ferio are the four types of syllogisms that make up Aristotle's first
figure. Another fifteen types are derived from them by rules of conversion, which change the order
of the terms or the types of statements. Barbara and Darii are the basis for the modern rule of
inheritance in type hierarchies. Celarent and Ferio are used to detect and reason about constraints
and constraint violations in a type hierarchy. Those four rules are also the foundation for a subset of
first-order logic called description logic, two versions of which are DAML and OIL.
2. Some Modern Systems
Philosophers often build their ontologies from the top down with grand conceptions about
everything in heaven and earth. Programmers, however, tend to work from the bottom up. For their
database and AI systems, they often start with limited ontologies or microworlds, which have a
small number of concepts that are tailored for a single application. The blocks world with its
ontology of blocks and pyramids has been popular for prototypes in robotics, planning, machine
vision, and machine learning.
For the Chat-80 question-answering system, David Warren and Fernando Pereira designed an
ontology for a microworld of geographical concepts. The hierarchy in Figure 3 shows the Chat-80
categories, which were used for several related purposes: for reasoning, they support inheritance of
properties from supertypes to subtypes; for queries, they map to the fields and domains in a
database; and for language analysis, they determine the constraints on permissible combinations of
nouns, verbs, and adjectives. Yet Figure 3 is specialized for a single application: rivers and roads
are considered subtypes of lines; and bridges, towns, and airstrips are treated as single points.
Figure 3: Geographical categories in the Chat-80 system
For Chat-80, the restrictions illustrated in Figure 3 simplified both the analyzer that interpreted
English questions and the inference engine that computed the answers. But the simplifying
assumptions that were convenient for Chat-80 would obscure or eliminate details that might be
essential for other applications.
Although many database and knowledge-based systems are considerably larger than Chat-80, the
overwhelming majority of them have built-in limitations that prevent them from being merged and
shared with other projects. Banks, for example, have a large number of similar concepts, such as
CheckingAccount, SavingsAccount, Loan, and Mortgage. Yet when two banks merge, there are so
many inconsistencies in the detailed specifications of those concepts that the resulting database is
always the disjoint union of the original databases. Any actual merging is usually accomplished by
canceling all the accounts of one type from one bank, transferring the funds, and recreating totally
new accounts in the format of the other bank.
Conceptual Schema. The need for standardized ways of encoding knowledge has been recognized
since the 1970s. The American National Standards Institute (ANSI) proposed that all pertinent
knowledge about an application domain should be collected in a single conceptual schema
(Tsichritzis & Klug 1978). Figure 4 illustrates an integrated system with a unified conceptual
schema at the center. Each circle is specialized for its own purposes, but they all draw on the
common application knowledge represented in the conceptual schema. The user interface calls the
database for query and editing facilities, and it calls the application programs to perform actions
and provide services. Then the database supports the application programs with facilities for data
sharing and persistent storage. The conceptual schema binds all three circles together by providing
the common definitions of the application entities and the relationships between them.
Figure 4: Conceptual schema as the heart of an integrated system
For more than twenty years, the conceptual schema has been important for integrated application
design, development, and use. Unfortunately, there were no full implementations. Yet partial
implementations of some aspects of the conceptual schema have formed the foundation of several
important developments: the fourth generation languages (4GLs); the object-oriented
programming systems (OOPS); and the tools for computer-aided software engineering (CASE).
Each of these approaches enhances productivity by using and reusing common data declarations for
multiple aspects of system design and development. Each of them has been called a solution to all
the world's problems; and each of them has been successful in solving some of the world's
problems. But none of them has achieved the ultimate goal of integrating everything around a
unified schema. One programmer characterized the lack of integration in a poignant complaint:
Any one of those tools by itself is a tremendous aid to productivity.
But any two of them together will kill you.
The latest attempt to integrate all the world's knowledge is the semantic web. So far, its major
contribution has been to propose XML as the common syntax for everything. That is useful, but the
problems of syntax are almost trivial in comparison to the problems of developing a common or at
least a compatible semantics for everything.
Large Ontologies. At the opposite extreme from a microworld limited to a small domain, the Cyc
system (Lenat & Guha 1990; Lenat 1995) was designed to accommodate all of human knowledge.
Its very name was taken from the stressed syllable of the word encyclopedia. Figure 5 shows two
dozen of the most general categories at the top of the Cyc hierarchy. Beneath those top levels, Cyc
contains about 100,000 concept types used in the rules and facts encoded in its knowledge base.
Figure 5: Top-level categories used in Cyc
The following three projects have developed the largest ontologies that are currently available:



Cyc. The Cyc project is the largest sustained effort to develop a broad-coverage ontology
with detailed axioms and definitions for each concept. Over 100 person-years of effort have
been spent on hand-crafting a hierarchy of 100,000 concept types with over a million
associated axioms. A free version of the Cyc knowledge base, called OpenCyc, will be
available in July, 2001.
EDR. The Electronic Dictionary Research project in Japan has developed a dictionary with
over 400,000 concepts, with their mappings to both English and Japanese words. Although
the EDR project has many more concepts than Cyc, it does not provide as much detail for
each one. http://www.iijnet.or.jp/edr/
WordNet. George Miller and his colleagues (Miller 1995; Fellbaum 1998) developed
WordNet as a hierarchy of 166,000 word form and sense pairs. WordNet doesn't have as
much detail as Cyc or as broad coverage as EDR, but it is the most widely used ontology for
natural language processing, largely because it has long been easily accessible over the
Internet. http://www.cogsci.princeton.edu/~wn/
Cyc has the most detailed axioms and definitions; it is an example of an axiomatized or formal
ontology. EDR and WordNet are usually considered terminological ontologies. The difference
between a terminological ontology and a formal ontology is one of degree: as more axioms are
added to a terminological ontology, it may evolve into a formal or axiomatized ontology. For
definitions of these and other terms used to describe the methods for building, sharing, and merging
ontologies, see the glossary in Section 6 of this paper.
3. Trees, Lattices, and Other Hierarchies
Porphyry began the practice of drawing trees to represent hierarchies of categories, but more
general acyclic graphs are needed to represent an arbitrary partial ordering, such as the subtypesupertype relation between categories. In this paper, Figures 1, 2, and 3 are trees, in which every
node except the top has a single parent node. Figure 5 for the Cyc ontology is an acyclic graph, in
which some nodes have more than one parent. Such graphs support multiple inheritance, since a
node can inherit properties from any or all of its parents. Figure 6 shows three kinds of graphs for
representing partial orderings: a tree, a lattice, and an arbitrary acyclic graph. To simplify the
drawings, a common convention is to omit the arrows that show the direction of the ordering and to
assume that the lower node represents a subtype of the higher node.
Figure 6: A lattice, a tree, and an acyclic graph
The term hierarchy is often used indiscriminately for any partial ordering. Some authors use the
term hierarchy to mean a tree, and tangled hierarchy to mean an acyclic graph that is not a tree. In
general, every tree is an acyclic graph, and every lattice is also an acyclic graph; but most lattices
are not trees, and most trees are not lattices. In fact, the only graphs that are both trees and lattices
are the simple chains (which are linearly ordered). Formally, a lattice is a mathematical structure
consisting of a set L, a partial ordering such as the subtype-supertype relation, and two operators
that represent the supremum or least common upper bound and the infimum or greatest common
lower bound. For more detail about lattices and related structures, see the tutorial on math and
logic.
Figure 7 shows a hierarchy of top-level categories defined by Sowa (2000), based on the
distinctions observed by a number of philosophers, especially Charles Sanders Peirce and Alfred
North Whitehead. The categories are derived by combinations of three ways of partitioning or
subdividing the top category T: Physical or Abstract (P, A); Independent, Relative, or Mediating (I,
R, M); Continuant or Occurrent (C, O). Each of the other categories is a synonym for the
combination of categories from which it was derived: Object, for example, could be represented by
the abbreviation PIC for Physical Independent Continuant; and Purpose would be AMO for
Abstract Mediating Occurrent. At the bottom of Figure 7, the absurd type , which represents the
contradictory conjunction of all categories. It completes the hierarchy by serving as a subtype of
every other type.
Figure 7: Hierarchy generated by the top three distinctions
To avoid making the diagram too cluttered, the hierarchy in Figure 7 omits some of the possible
combinations. The full lattice would be generated by taking all possible combinations of the three
basic distinctions, but in many lattices, some of the possible combinations are not meaningful. The
following table of beverages, which is taken from a paper by Michael Erdmann (1998), illustrates a
typical situation in which many combinations do not occur. Some combinations are impossible,
such as a beverage that is simultaneously alcoholic and nonalcoholic. Others are merely unlikely,
such as hot and sparkling.
Attributes
Concept Types nonalcoholic hot alcoholic caffeinic sparkling
HerbTea
x
x
Coffee
x
x
MineralWater
x
Wine
x
x
x
x
Beer
Cola
Champagne
x
x
x
x
x
x
Table of beverage types and attributes
To generate the minimal lattice for classifying the beverages in the above table, Erdmann applied
the method of formal concept analysis (FCA), developed by Bernhard Ganter and Rudolf Wille
(1999) and implemented in an automated tool called Toscana. Figure 8 shows the resulting lattice;
attributes begin with lower-case letters, and concept types begin with upper-case letters.
Figure 8: Lattice constructed by the method of formal concept analysis
In Figure 8, beer and champagne are both classified at the same node, since they have exactly the
same attributes. To distinguish them more clearly, wine and champage could be assigned the
attribute madeFromGrapes, and beer the attribute madeFromGrain. Then the Toscana system
would automatically generate a new lattice with three added nodes:



Wine would be alcoholic&madeFromGrapes.
Beer would be sparkling&alcoholic&madeFromGrain.
Champagne would be sparkling&alcoholic&madeFromGrapes.
Figure 9 shows the revised lattice with the new nodes and attributes.
Figure 9: Revised lattice with new attributes
Note that the attribute nonalcoholic is redundant, since it is the complement of the attribute
alcoholic. If that attribute had been omitted from the table, the FCA method would still have
constructed the same lattice. The only difference is that the node corresponding to the attribute
nonalcoholic would not have a label. In a lattice for a familiar domain, such as beverages, most of
the nodes correspond to common words or phrases. In Figure 9, the only node that does not
correspond to a common word or phrase in English is sparkling&alcoholic.
Lattices are especially important for representing ontologies and for revising, refining, and sharing
ontologies. They are just as useful at the lower levels of the ontology as they are at the topmost
levels. Each addition of a new distinction or differentia results in a new lattice, which is called a
refinement of the previous lattice. The first lattices were introduced by Leibniz, who generated all
possible combinations of the basic distinctions. A refinement generated by FCA contains only the
minimal number of nodes needed to accommodate the new attribute and its subtypes. Leibniz's
method would introduce superfluous nodes, such as hot & caffeinic & sparkling &
madeFromGrapes. The FCA lattices, however, contain only the known concept types and likely
generalizations, such as sparkling & alcoholic. For this example, Leibniz's method would generate
a lattice of 64 nodes, but the FCA method generates only 14 nodes. A Leibniz-style of lattice is the
ultimate refinement for a given set of attributes, and it may be useful when all possible
combinations must be considered. But the more compact FCA lattices avoid the nonexistent
combinations.
4. Notations for Logic
To express anything that has been or will be represented requires a universal language  one that
can represent anything and everything that can be said. Fortunately, universal languages do exist.
There are two kinds:
1. Natural languages. Everything in the realm of human experience that can be expressed at all
can be expressed in a natural language, such as English, French, Japanese, or Swahili.
Natural languages are general enough to explain and comment upon any artificial language,
mathematical notation, or programming language ever conceived or conceivable. They are
even general enough to serve as a metalanguage that can explain themselves or other natural
and artificial languages.
2. Logic. Everything that can be stated clearly and precisely in any natural language can be
expressed in logic. There may be aspects of love, poetry, and jokes that are too elusive to
state clearly. But anything that can be implemented on a digital computer in any
programming language can be specified in logic.
Although anything that can be stated clearly and precisely can be expressed in logic, attaining that
level of precision is not always easy. Yet logic is all there is: every programming language,
specification language, and requirements definition language can be defined in logic; and nothing
less can meet the requirements for a complete definitional system.
The problem of relating different systems of logic is complex, but it has been studied in great
depth. In one sense, there has been a de facto standard for logic for over a century. In 1879, Gottlob
Frege developed a tree notation for logic, which he called the Begriffsschrift. In 1883, Charles
Sanders Peirce independently developed an algebraic notation for predicate calculus, which with a
change of symbols by Giuseppe Peano, is the most widely used notation for logic today.
Remarkably, these two radically different notations have identical expressive power: anything
stated in one of them can be translated to the other without loss or distortion.
Even more remarkably, the classical first-order logic (FOL) that Frege and Peirce developed a
century ago has proved to be a fixed point among all the variations that logicians and
mathematicians have invented over the years. FOL has enough expressive power to define all of
mathematics, every digital computer that has ever been built, and the semantics of every version of
logic including itself. Fuzzy logic, modal logic, neural networks, and even higher-order logic can
be defined in FOL. Every textbook of mathematics or computer science attests to that fact. They all
use a natural language as the metalanguage, but in a form that can be translated to two-valued firstorder logic with just the quantifiers  and  and the basic Boolean operators. Besides expressive
power, first-order logic has the best-defined, least problematical model theory and proof theory,
and it can be defined in terms of a bare minimum of primitives: just one quantifier (either  or )
and one or two Boolean operators. Even subsets, such as Horn-clause logic or Aristotelian
syllogisms, are more complicated, in the sense that more detailed definitions are needed to specify
what cannot be said in those subsets than to specify everything that can be said in full FOL.
The power of FOL and the prestige of its adherents have not deterred philosophers, logicians,
linguists, and computer scientists from developing other logics. For various purposes, modal logics,
higher-order logics, and other extended logics have many desirable properties:

More natural translations. In FOL, an English sentence like It may rain would be translated
to an awkward paraphrase: Of all the states of affairs in the set of causal successors of the
present, there exists at least one in which it is raining. In modal logic, the lengthy preamble
about states of affairs would be replaced by a single symbol .


Fewer axioms. The axiom of mathematical induction, with its quantifier (P:Predicate),
requires more expressive power than pure first-order logic. In FOL, it can only be eliminated
by brute force  by replacing the variable P with a separate first-order axiom for every
predicate that P might represent. Since there are infinitely many possible predicates, that
strategy replaces one second-order axiom with an infinity of first-order axioms. Many
logicians find an infinity of axioms to be more distasteful than the complexities of higherorder logic.
More efficient computation. A complex logic can sometimes simplify knowledge
representation by transferring much of the detail from the axioms to the rules of inference.
This transfer may improve the speed of computation, since the rules of inference are more
likely to be compiled, but the axioms are more likely to be interpreted.
To computer scientists, these arguments sound like the familiar trade-offs between compile time
and execution time: the rules of inference of the more complex logics can be compiled, while the
larger number of axioms required for the simpler logics are executed less efficiently by an
interpreter. In fact, this analogy leads to one way of resolving the disputes: first-order logic can be
used as a metalanguage for defining the other kinds of logic. In effect, FOL becomes the unifying
metalanguage that defines, relates, and supports an open-ended variety of extended logics. Then for
various applications, the implementers can make a decision to compile the definitions or execute
them interpretively.
Since the semantics of FOL was firmly established by Alfred Tarski's model theory in 1935, the
only thing that has to be standardized is notation. But notation is a matter of taste that raises the
most heated arguments and disagreements. To minimize the arguments, the NCITS L8 committee
on Metadata has been developing two different notations with a common underlying semantics.
ndard, and any concrete notation that conforms to the abstract syntax can be used as an equivalent.
To determine conformance, two concrete notations are also being standardized at the same time:


Knowledge Interchange Format (KIF). This is a linear notation for logic with an easily
parsed syntax and a restricted character set that is intended for interchange between
heterogeneous computer systems.
Conceptual Graphs (CGs). This is a graphic notation for logic based on the existential graphs
of C. S. Peirce augmented with features from linguistics and the semantic networks of AI. It
has been designed for a smoother mapping to and from natural languages and as a
presentation language for displaying logic in a more humanly readable form.
Both KIF and CGs have identical expressive power, and anything stated in either one can be
automatically translated to the other. For the standards efforts, any other language that can be
translated to or from KIF or CGs while preserving the basic semantics has an equivalent status.
To illustrate the KIF and CG notations, Figure 10 shows a conceptual graph that represents the
sentence "John is going to Boston by bus." The CG has four concept nodes: [Go], [Person: John],
[City: Boston], and [Bus]. It has three conceptual relation nodes: (Agnt) relates [Go] to the agent
John, (Dest) relates [Go] to the destination Boston, and (Inst) relates [Go] to the instrument bus.
Figure 10: CG for "John is going to Boston by bus."
In addition to the graphic display form shown in Figure 10, there is also a formally defined
conceptual graph interchange form (CGIF), which serves as a linear representation that can be
conveniently stored and exchanged between different implementations:
[Go: *x] [Person: 'John' *y] [City: 'Boston' *z] [Bus: *w]
(Agnt ?x ?y) (Dest ?x ?z) (Inst ?x ?z)
The CGIF notation also has a very direct mapping to KIF:
(exists ((?x Go) (?y Person) (?z City) (?w Bus))
(and (Name ?y 'John) (Name ?z 'Boston)
(Agnt ?x ?y) (Dest ?x ?z) (Inst ?x ?w)))
For a list of the relations that connect the concepts corresponding to verbs to the concepts of their
participants, see the web page on thematic roles.
The CG in Figure 10 corresponds to a logical form that has only two operators: conjunction and the
existential quantifier. To illustrate negation and the universal quantifier, the following table shows
the four proposition types used in syllogisms and their representation in CGIF and KIF.
Pattern
CGIF
KIF
Every A is B.
[A: @every *x] [B: ?x] (forall ((?x A)) (B ?x))
Some A is B.
[A: *x] [B: ?x]
(exist ((?x A)) (B ?x))
No A is B.
~[ [A: *x] [B: ?x] ]
(not (exist ((?x A)) (B ?x)))
Some A is not B. [A: *x] ¬[ [B: ?x] ]
(exist ((?x:A)) (not (B ?x))
The four statement types illustrated in the above table represent the kinds of statements used in
syllogisms, which are a small subset of full first-order logic. A larger subset, called Horn-clause
logic, is used in the if-then rules of expert systems. Following is an example of such a rule, as
express in the language Attempto Controlled English (ACE):
If a borrower asks for a copy of a book
and the copy is available
and LibDB calculates the book amount of the borrower
and the book amount is smaller than the book limit
and a staff member checks out the copy to the borrower
then the copy is checked out to the borrower.
The ACE language can be read as if it were English, but it is a formal language that can be
automatically translated to logic. Following is the translation to CGIF:
(Named [Entity: *f] [String: "LibDB"])
[If: (Of [Copy: *b] [Book])
(Of [BookAmount: *g] [Borrower: *a])
[BookLimit: *i] [StaffMember: *k]
[Event: (AskFor ?a ?b)]
[State: (Available ?b)]
[Event: (Calculate ?f ?g)]
[State: (SmallerThan ?g ?i)]
[Event: (CheckOutTo ?k ?b ?a)]
[Then: [State: (CheckedOutTo ?b ?a)]]]
And following is the translation to KIF:
(exist ((?f entity))
(and (Named ?f 'LibDB)
(forall ((?a borrower) (?b copy) (?c book) (?g bookAmount)
(?i bookLimit) (?k staffMember) (?d ?h ?l ?e ?j))
(if (and (of ?b ?c) (of ?g ?a)
(event ?d (askFor ?a ?b))
(state ?e (available ?b))
(event ?h (calculate ?f ?g))
(state ?j (smallerThan ?g ?i))
(event ?l (checkOutTo ?k ?b ?a)) )
(exist (?m) (state ?m (checkedOutTo ?b ?a))) ))))
Besides the combinations used in syllogisms and Horn-clause logic, conceptual graphs and KIF
support all the possible combinations permitted in first-order logic with equality. They can also be
used as metalevel languages, which can be used to represent a much richer version of logic,
including modal and intentional logics. For more examples, see the translation of English sentences
to CGs, KIF, and predicate calculus. For the theoretical foundation of these extensions, see the
book Knowledge Representation by John Sowa.
5. Ontology Sharing and Merging
Knowledge representation is the application of logic and ontology to the task of constructing
computable models for some application domain. Each of the three basic fields  logic, ontology,
and computation  presents a different class of problems for knowledge sharing:

Logic. Different implementations support different subsets and variations of logic. Sharing
information between them can usually be done automatically if the information can be
expressed in the common subset. Other kinds of transfers may be possible, but some of the
information may be lost or modified.


Ontology. Different systems may use different names for the same kinds of entities; even
worse, they may use the same names for different kinds. Sometimes, two entities with
different definitions are intended to be the same, but the task of proving that they are indeed
the same may be difficult or impossible.
Computation. Even when the names and definitions are identical, computational or
implementational side effects may cause the same knowledge to behave differently in
different systems. In some implementations, the order of entering rules and data may have an
effect on the possible inferences and the results of computations. Sometimes, the side effects
may cause a simple inference on one system to get hung up in an endless loop on another
system.
Although these three aspects of knowledge representation pose different kinds of problems, they
are interdependent. For applications in library science, humans usually process the knowledge.
Therefore, the major attention has been directed toward standardizing the terminology used to
classify and find the information. For artificial intelligence, where the emphasis is on computer
processing, the major attention has been directed to deep, precise axiomatizations suitable for
extended computation and deduction. But as these fields develop, the requirements are beginning to
overlap. More of the librarians' task is being automated, and the AI techniques are being applied to
large bodies of information that have to be sorted, searched, and classified before extended
deductions are possible.
To address such problems, standards bodies, professional societies, and industry associations have
developed standards to facilitate sharing. Yet the standards themselves are part of the problem.
Every field of science, engineering, business, and the arts has its own specialized standards,
terminology, and conventions. Yet the various fields cannot be isolated: medical instruments, for
example, must be compatible with the widely divergent standards developed in the medical,
pharmaceutical, chemical, electrical, and mechanical engineering fields. And medical computer
systems must be compatible with all of the above plus the standards for billing, inventory,
accounting, patient records, scheduling, email, networks, databases, and government regulations.
The first requirement is to develop standards for relating standards.
The problems of aligning the terms from different ontologies are essentially the same as the
problems of aligning words from the vocabularies of different natural languages. As an example,
Figure 11 shows the concept type Know, which represents the most general sense of the English
word know, and two of its subtypes. On the left are the German concept type Wissen and the
French concept type Savoir, which correspond to the English sense of knowing-that. On the right
are the German Kennen and the French Connâitre, which correspond to the English sense of
knowing-some-entity.
Figure 11: Refinement of Know and its French and German equivalents
Figure 12 shows a more complex pattern for the senses of the English words river and stream and
the French words fleuve and rivière. In English, size is the feature that distinguishes river from
stream; in French, a fleuve is a river that flows into the sea, and a rivière is either a river or a stream
that flows into another river. In translating French into English, the word fleuve maps to the French
concept type Fleuve, which is a subtype of the English type River. Therefore, river is the closest
one-word approximation to fleuve; if more detail is necessary, it could also be translated by the
phrase river that runs into the sea. In the reverse direction, river maps to River, which has two
subtypes: one is Fleuve, which maps to fleuve; and the other is BigRivière, whose closest
approximation in French is the word rivière or the phrase grande rivière.
Figure 12: Hierarchy for River, Stream, and their French equivalents
Even when two languages have words that are roughly equivalent in their literal meanings, they
may be quite different in salience. In the type hierarchy, Dog is closer to Vertebrate than to Animal.
But since Animal has a much higher salience, people are much more likely to refer to a dog as an
animal than as a vertebrate. To illustrate the way salience affects word choice, Figure 13 shows part
of the hierarchy that includes the English Vehicle and the Chinese Che, which is represented by the
character at the top of the hierarchy. That character was derived from a sketch of a simple twowheeled cart: the vertical line through the middle represents the axle, the horizontal lines at the top
and bottom represent the two wheels, and the box in the middle represents the body. Over the
centuries, that simple concept has been generalized to represent all wheeled conveyances for
transporting people or goods.
Figure 13: Hierarchy for English Vehicle and Chinese Che
The English types Car, Taxi, Bus, Truck=Lorry, and Bicycle are subtypes of Vehicle. The Chinese
types do not exactly match the English ones: Che is a supertype of Vehicle that includes Train
(HuoChe), which is not usually considered a vehicle in English. The type QiChe includes Taxi
(ChuZuQiChe) and Bus (GongGongQiChe) as well as Car, which has no specific word that
distinguishes it from a taxi or bus.
In English, the specific words car, bus, or taxi are commonly used in speech, and the generic
vehicle would normally be used only in a technical context, such as traffic laws. In Chinese,
however, the word che is the most common term for any kind of a vehicle. When the specific type
is clear from the context, a Chinese speaker would simply say Please call me a che, I'm waiting for
the 5 o'clock che, or I parked my che around the corner. The fact that che is both a stand-alone
word and a component of all its subtypes enhances its salience; and the fact that chuzuqiche and
gonggongqiche are four-syllable words decreases their salience. Therefore, it would sound
unnatural to use the word chuzuqiche, literally the exact equivalent of taxi, to translate the sentence
Please call me a taxi. In translations from Chinese, the type Che would have to be specialized to an
appropriate subtype in order to avoid sentences like I parked my wheeled conveyance around the
corner.
Misalignments between ontologies arise from a variety of cultural, geographical, linguistic,
technical, and even random differences. Geography probably contributes to the French distinction,
since the major rivers in France flow into the Atlantic or the Mediterranean. In the United States,
however, there are major rivers like the Ohio and the Misouri, which flow into the Mississippi. The
Chinese preference for one-syllable morphemes that can either stand alone or form part of a
compound leads to the high salience for che, while the English tendency to drop syllables leads to
highly salient short words like bus and taxi from omnibus and taxicab.
The issues illustrated in Figures 11, 12, and 13 represent inconveniences, but they do not create
inconsistencies in the merged ontology. They can be resolved by refining one or both ontologies by
adding more concept types that represent the union of all the distinctions in both ontologies that
were merged. Figure 14 shows a "bowtie" inconsistency that sometimes arises in the process of
aligning two ontologies.
Figure 14: A bowtie inconsistency between two ontologies
On the left of Figure 14, Circle is represented as a subtype of Ellipse, since a circle can be
considered a special case of an ellipse in which both axes are equal. On the right is a representation
that is sometimes used in object-oriented programming languages: Ellipse is considered a subclass
of Circle, since it has more complex methods. If both ontologies were merged, the resulting
hierarchy would have an inconsistency. To resolve such inconsistencies, some definitions must be
changed, or some of the types must be relabeled. In most graphics systems, the mathematical
definition of Circle as a subtype of Ellipse is preferred because it supports more general
transformations.
6. Glossary
This glossary summarizes the terminology of methods and techniques for defining, sharing, and
merging ontologies. These definitions, which were written by John F. Sowa, are based on
discussions in the ontology working group of the NCITS T2 Committee on Information
Interchange and Interpretation.
alignment.
A mapping of concepts and relations between two ontologies A and B that preserves the
partial ordering by subtypes in both A and B. If an alignment maps a concept or relation x in
ontology A to a concept or relation y in ontology B, then x and y are said to be equivalent.
The mapping may be partial: there could be many concepts in A or B that have no
equivalents in the other ontology. Before two ontologies A and B can be aligned, it may be
necessary to introduce new subtypes or supertypes of concepts or relations in either A or B in
order to provide suitable targets for alignment. No other changes to the axioms, definitions,
proofs, or computations in either A or B are made during the process of alignment.
Alignment does not depend on the choice of names in either ontology. For example, an
alignment of a Japanese ontology to an English ontology might map the Japanese concept Go
to the English concept Five. Meanwhile, the English concept for the verb go would not have
any association with the Japanese concept Go.
differentiae.
The properties, features, or attributes that distinguish a type from other types that have a
common supertype. The term comes from Aristotle's method of defining new types by
stating the genus or supertype and stating the differentiae that distinguish the new type from
its supertype. Aristotle's method of definition has become the de facto standard for natural
language dictionaries, and it is also widely used for AI knowledge bases and object-oriented
programming languages. For a discussion and comparison of various methods of definition,
see the notes on definitions by Norman Swartz.
formal ontology.
A terminological ontology whose categories are distinguished by axioms and definitions
stated in logic or in some computer-oriented language that could be automatically translated
to logic. There is no restriction on the complexity of the logic that may be used to state the
axioms and definitions. The distinction between terminological and formal ontologies is one
of degree rather than kind. Formal ontologies tend to be smaller than terminological
ontologies, but their axioms and definitions can support more complex inferences and
computations. The two major contributors to the development of formal ontology are the
philosophers Charles Sanders Peirce and Edmund Husserl. Examples of formal ontologies
include theories in science and mathematics, the collections of rules and frames in an expert
system, and specification of a database schema in SQL.
hierarchy.
A partial ordering of entities according to some relation. A type hierarchy is a partial
ordering of concept types by the type-subtype relation. In lexicography, the type-subtype
relation is sometimes called the hypernym-hyponym relation. A meronomy is a partial
ordering of concept types by the part-whole relation. Classification systems sometimes use a
broader-narrower hierarchy, which mixes the type and part hierarchies: a type A is
considered narrower than B if A is subtype of B or any instance of A is a part of some
instance of B. For example, Cat and Tail are both narrower than Animal, since Cat is a
subtype of Animal and a tail is a part of an animal. A broader-narrower hierarchy may be
useful for information retrieval, but the two kinds of relations should be distinguished in a
knowledge base because they have different implications.
identity conditions.
The conditions that determine whether two different appearances of an object represent the
same individual. Formally, if c is a subtype of Continuant, the identity conditions for c can
be represented by a predicate Idc. Two instances x and y of type c, which may appear at
different times and places, are considered to be the same individual if Idc(x,y) is true. As an
example, a predicate IdHuman, which determines the identity conditions for the type
HumanBeing, might be defined by facial appearance, fingerprints, DNA, or some
combination of all those features. At the atomic level, the laws of quantum mechanics make
it difficult or impossible to define precise identity conditions for entities like electrons and
photons. If a reliable identity predicate Idt cannot be defined for some type t, then t would be
considered a subtype of Occurrent rather than Continuant.
integration.
The process of finding commonalities between two different ontologies A and B and
deriving a new ontology C that facilitates interoperability between computer systems that are
based on the A and B ontologies. The new ontology C may replace A or B, or it may be used
only as an intermediary between a system based on A and a system based on B. Depending
on the amount of change necessary to derive C from A and B, different levels of integration
can be distinguished: alignment, partial compatibility, and unification. Alignment is the
weakest form of integration: it requires minimal change, but it can only support limited kinds
of interoperability. It is useful for classification and information retrieval, but it does not
support deep inferences and computations. Partial compatibility requires more changes in
order to support more extensive interoperability, even though there may be some concepts or
relations in one system or the other that could create obstacles to full interoperability.
Unification or total compatibility may require extensive changes or major reorganizations of
A and B, but it can result in the most complete interoperability: everything that can be done
with one can be done in an exactly equivalent way with the other.
knowledge base.
An informal term for a collection of information that includes an ontology as one component.
Besides an ontology, a knowledge base may contain information specified in a declarative
language such as logic or expert-system rules, but it may also include unstructured or
unformalized information expressed in natural language or procedural code.
lexicon.
A knowledge base about some subset of words in the vocabulary of a natural language. One
component of a lexicon is a terminological ontology whose concept types represent the word
senses in the lexicon. The lexicon may also contain additional information about the syntax,
spelling, pronunciation, and usage of the words. Besides conventional dictionaries, lexicons
include large collections of words and word senses, such as WordNet from Princeton
University and EDR from the Japan Electronic Dictionary Research Institute, Ltd. Other
examples include classification schemes, such as the Library of Congress subject headings or
the Medical Subject Headers (MeSH).
mixed ontology.
An ontology in which some subtypes are distinguished by axioms and definitions, but other
subtypes are distinguished by prototypes. The top levels of a mixed ontology would normally
be distinguished by formal definitions, but some of the lower branches might be
distinguished by prototypes.
partial compatibility.
An alignment of two ontologies A and B that supports equivalent inferences and
computations on all equivalent concepts and relations. If A and B are partially compatible,
then any inference or computation that can be expressed in one ontology using only the
aligned concepts and relations can be translated to an equivalent inference or computation in
the other ontology.
primitive.
A category of an ontology that cannot be defined in terms of other categories in the same
ontology. An example of a primitive is the concept type Point in Euclid's geometry. The
meaning of a primitive is not determined by a closed-form definition, but by axioms that
specify how it is related to other primitives. A category that is primitive in one ontology
might not be primitive in a refinement of that ontology.
prototype-based ontology.
A terminological ontology whose categories are distinguished by typical instances or
prototypes rather than by axioms and definitions in logic. For every category c in a
prototype-based ontology, there must be a prototype p and a measure of semantic distance
d(x,y,c), which computes the dissimilarity between two entities x and y when they are
considered instances of c. Then an entity x can classified by the following recursive
procedure:




Suppose that x has already been classified as an instance of some category c, which
has subcategories s1,...,sn.
For each subcategory si with prototype pi, measure the semantic distance d(x, pi , c).
If d(x, pi , c) has a unique minimum value for some subcategory si, then classify x as
an instance of si, and call the procedure recursively to determine whether x can be
further classified by some subcategory of si.
If c has no subcategories or if d(x, pi , c) has no unique minimum for any si, then the
classification procedure stops with x as an instance of c, since no finer classification is
possible with the given selection of prototypes.
As an example, a black cat and an orange cat would be considered very similar as instances
of the category Animal, since their common catlike properties would be the most significant
for distinguishing them from other kinds of animals. But in the category Cat, they would
share their catlike properties with all the other kinds of cats, and the difference in color
would be more significant. In the category BlackEntity, color would be the most relevant
property, and the black cat would be closer to a crow or a lump of coal than to the orange cat.
Since prototype-based ontologies depend on examples, it is often convenient to derive the
semantic distance measure by a method that learns from examples, such as statistics, cluster
analysis, or neural networks.
Quine's criterion.
A test for determining the implicit ontology that underlies any language, natural or artificial.
The philosopher Willard van Orman Quine proposed a criterion that has become famous: "To
be is to be the value of a quantified variable." That criterion makes no assumptions about
what actually exists in the world. Its purpose is to determine the implicit assumptions made
by the people who use some language to talk about the world. As stated, Quine's criterion
applies directly to languages like predicate calculus that have explicit variables and
quantifiers. But Quine extended the criterion to languages of any form, including natural
languages, in which the quantifiers and variables are not stated as explicitly as they are in
predicate calculus. For English, Quine's criterion means that the implicit ontological
categories are the concept types expressed by the basic content words in the language: nouns,
verbs, adjectives, and adverbs.
refinement.
An alignment of every category of an ontology A to some category of another ontology B,
which is called a refinement of A. Every category in A must correspond to an equivalent
category in B, but some primitives of A might be equivalent to nonprimitives in B.
Refinement defines a partial ordering of ontologies: if B is a refinement of A, and C is a
refinement of B, then C is a refinement of A; if two ontologies are refinements of each other,
then they must be isomorphic.
semantic factoring.
The process of analyzing some or all of the categories of an ontology into a collection of
primitives. Combinations of those primitives generate a hierarchy, called a lattice, which
includes the original categories plus additional ones that make it more symmetric. The
techniques of semantic factoring can be applied to any level of an ontology from the highest,
most general concept types to the lowest, most specialized types. The methods can be
automated, as in formal concept analysis, which is a systematic technique for deriving a
lattice of concept types from low-level data about individual instances.
semiotic.
The study of signs in general, their use in language and reasoning, and their relationships to
the world, to the agents who use them, and to each other. It was developed independently by
the logician Charles Sanders Peirce, who called it semeiotic, and by the linguist Ferdinand de
Saussure, who called it sémiologie; other variants are the terms semiotics and semiology.
Peirce developed semiotic into a rich, highly nuanced foundation for formal ontology,
starting with three metalevel categories, which he called Firstness, Secondness, and
Thirdness. Specialized examples of these categories include Aristotle's triad of Inherence,
Directedness, and Containment in Figure 1 and the triad of Independent, Relative, and
Mediating in Figure 7. One of Peirce's most famous examples is the triad of Icon, Index, and
Symbol.
terminological ontology.
An ontology whose categories need not be fully specified by axioms and definitions. An
example of a terminological ontology is WordNet, whose categories are partially specified
by relations such as subtype-supertype or part-whole, which determine the relative positions
of the concepts with respect to one another but do not completely define them. Most fields of
science, engineering, business, and law have evolved systems of terminology or
nomenclature for naming, classifying, and standardizing their concepts. Axiomatizing all the
concepts in any such field is a Herculean task, but subsets of the terminology can be used as
starting points for formalization. Unfortunately, the axioms developed from different starting
points are often incompatible with one another.
unification.
A one-to-one alignment of all concepts and relations in two ontologies that allows any
inference or computation expressed in one to be mapped to an equivalent inference or
computation in the other. The usual way of unifying two ontologies is to refine each of them
to more detailed ontologies whose categories are one-to-one equivalent.
References
All bibliographical references have been moved to the combined bibliography for this web site.
Send comments to John F. Sowa.
Last Modified: 01/18/2009 17:34:46