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 Every A is B.
    I Particular affirmative Some A is B.
    E Universal negative No A is B.
    O Particular negative Some A is not 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 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 subtype-supertype 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.

    A lattice, a tree, and an acyclic graph

    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 Typesnonalcoholichot alcoholiccaffeinicsparkling
    HerbTea x x      
    Coffee x x   x  
    MineralWater x       x
    Wine     x    
    Beer     x   x
    Cola x     x x
    Champagne     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.

    Lattice of beverages

    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:

    Figure 9 shows the revised lattice with the new nodes and attributes.

    Revised lattice

    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 first-order 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:

    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:

    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.

    John is going to Boston by 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:

    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 two-wheeled 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:

    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.


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