Foundational Aspects of Ontologies

Pascal Hitzler, Carsten Lutz, Gerd Stumme

9/2005

FachberichteINFORMATIKISSN 1860-4471

Universitat Koblenz-LandauInstitut fur Informatik, Universitatsstr. 1, D-56070 Koblenz

E-mail: researchreports@uni-koblenz.de ,

WWW: http://www.uni-koblenz.de/FB4/

Preface

Representing and reasoning with ontologies is the core technology for the Seman-tic Web, and is growing in importance in many other areas of Computer Sciencewhere structured and hierarchically organized knowledge is of importance. Whilethe need for ontological knowledge representation formalisms for practical appli-cations is abundant, it is apparent that only conceptually and mathematicallysound frameworks can provide the means for a significant technological advancein this area.

Indeed, formal and foundational aspects of ontologies are being studied inmany application domains in order to serve practical needs. It lies in the nature ofsuch fundamental research that a critical mass of different formal perspectivescan generate a cross-fertilization of ideas and applications. We believe that asignificant advance in understanding and establishing sound formal foundationsfor applied ontology research can this way be made. This workshop was setup to bring together researchers working on foundational aspects of ontologiesin different application areas, in order to stimulate an exchange of ideas andmethods between the subcommunities.

Pascal Hitzler, Carsten Lutz, Gerd StummeFOnt’05 organizers

Organization

Workshop Chairs

Pascal HitzlerAIFB, Universitat KarlsruheGermany

Carsten LutzDepartment of Computer Science, TU DresdenGermany

Gerd StummeDepartment of Mathematics and Computer Science, Universitat KasselGermany

Program Commitee

Jerome Euzenat, INRIA Rhone-Alpes, Grenoble, FranceBernhard Ganter, TU Dresden, GermanyJoseph Goguen, University of California at San Diego, California, USAWolfgang Hesse, Universitat Marburg, GermanyIan Horrocks, University of Manchester, UKYannis Kalfoglou, University of Southhampton, UKRobert E. Kent, Ontologos, Pullman, USAMarkus Krotzsch, Universitat Karlsruhe, GermanyClaudio Masolo, Institute of Cognitive Science and Technology, Trento, ItalyFabian Neuhaus, Universitat des Saarlandes, Saarbrucken, GermanyDaniel Oberle, Universitat Karlsruhe, GermanyUlrike Sattler, University of Manchester, UKMarco Schorlemmer, CSIC Barcelona, SpainAnthony K. Seda, University College Cork, IrelandSergio Tessaris, Universita di Bolzano, ItalyRudolf Wille, TU Darmstadt, GermanyGuo-Qiang Zhang, Case Western Reserve University, Cleveland, Ohio, USA

Programme

14:30–14:35 opening14.35–15.40 Keynote by Till Mossakowski

Heterogeneous Formal Ontologies15:40–16:00 Claudio Masolo and Stefano Borgo

Qualities in Formal Ontology16:00–16:30 coffee break16.30–16.50 Rainer Osswald

A Categorical Framework for Translating between ConceptualHierarchies

16:50–17:10 Winfried Schmitz-EsserOntology-based understanding of scientific natural language texts

17:10–17:30 Douglas FoxvogInstances of Instances via Higher-Order Classes

17.30–17.50 Sumit SenOntologies of One-way Roads

17.50–18:00 closing

Sponsor

Sponsored by the European Union Network of Excellence KnowledgeWebFP6-507482

Table of Contents

Keynote: Heterogeneous Formal Ontologies . . . . . . . . . . . . . . . . . . . . . . . . . . 1Till Mossakowski

Qualities in Formal Ontology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Claudio Masolo, Stefano Borgo

A Categorical Framework for Translating between Conceptual Hierarchies 17Rainer Osswald

Ontology-based understanding of scientific natural language texts . . . . . . . 34Winfried Schmitz-Esser

Instances of Instances Modeled via Higher-Order Classes . . . . . . . . . . . . . . 46douglas foxvog

Ontologies of One-way Roads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Sumit Sen

Keynote: Heterogeneous Formal Ontologies

Till Mossakowski

Department of Computer Science, Universitat Bremen, Germany

Formal ontologies are formalized as logical theories. Different formal ontolo-gies may involve different underlying logical systems (like equational logics, de-scription logics, first-order logics, higher-order logics, modal logics), and alsobecause different formalisms are in practical use (like KIF, Loom, RDF, OWL,EML).

There are different ways to deal with this diversity. One way is unification,like the recent initative for turning common logic (CL) into an ISO standard.Another way is to use a heterogeneous framework that allows for combinationof ontologies written in different formalisms and logical systems. We discuss thepros and cons of these two approaches and show when and how they can becomplementary.

We also discuss how mechanisms for structuring logical theories, like colimits,might be of use for ontology integration and mediation.

Qualities in Formal Ontology

Claudio Masolo and Stefano Borgo

Laboratory for Applied Ontology, ISTC-CNR, Trento, Italy

Abstract. We characterize and compare four different ways of representing qualitiesin formal ontology. Our goal is to discuss their ontological commitments and their ade-quacy in applications. We also show how the frameworks here presented relate to otherapproaches in ontology (trope theory), in cognitive science (conceptual spaces), and inphysics (International System of Units). The work we present focuses on ontologicalconstruction; we do not discuss issues specifically related to measurements, metrics andthe like.

1 Introduction

It is hard to conceive ontology and, more generally knowledge representation,without thinking about conceptualization and representation of endurants (ob-jects like chairs and cars) and perdurants (events like driving and sneezing).People differentiate these entities because of a variety of aspects and character-istics, hereafter called qualities, that can be recognized and classified like color,weight or duration. Nonetheless, when asked to list the qualities of an endurant,we cannot do more than listing a few of them (like shape, color, size, weight,temperature, duration, smell and so on). This observation is surprising if weconsider the crucial relevance qualities have in our life.

However, the lack of a set of qualities on which people agree upon is notthe only deficiency in this area and, perhaps, not even the most important. In-deed, the research community has not yet isolated a systematic and ontologicallysound framework to compare and analyze qualities. There are a few approaches(essentially based on the notions of individual quality, trope and property), buttheir systematic comparison from the formal and ontological perspective has notbeen carried out.

This paper gives a contribution in this area by formally presenting a set offrameworks and by discussing their relationships and ontological commitments.These frameworks build on a variety of entities like

– Individual qualities, e.g. “the weight of John“. Individuals qualities inhere inspecific individuals, that is, “the color of John” is different from “the colorof Mary“, and they can change through time since “the color of John” canmatch color red today and color rose tomorrow.

– Qualia, e.g. a specific color. These entities are obtained by abstracting indi-vidual qualities from time and from their hosts. If the color of John and thecolor of Mary match the same shade of red, then they have the same (color)

quale. In this sense qualia represent perfect and “objective“ similarity be-tween (aspects of) objects.

– Regions and spaces. These entities corresponds to different ways of “organiz-ing” qualia. They are motivated by “subjective“ (context dependent, qual-itative, etc.) similarity between (aspects of) objects. By means of spaces, astructure can be imposed on qualia (for example a geometry or a topology)and this makes it possible to differentiate several quantitative and qualitativedegrees of similarity.

We refer to [1] for a deeper discussion of the notions of quality and quale.Although the presentation there develops within the framework of the dolceontology (Descriptive Ontology for Linguistic and Cognitive Engineering), ourpaper does not commit (nor is limited to) that specific ontology.

Sections 3 to 6 present and discuss four different (yet related) approaches toquality representation. In section 7 we analyze the ontological nature of qualiaand we make explicit the link with approaches based on tropes and universals.We will see that all these approaches are comparable in expressive power butdiffer in their ontological commitment. It is then important to understand inwhich cases it is better to choose one approach rather than another one. Insections 8 and 9 we analyze the adequacy of these approaches with respect tothe theory of conceptual spaces [2] and the International System of Units1.

2 Focus and basic notions

In this work we concentrate only on qualities of endurants, i.e. qualities of entitiesthat are wholly present at any time they are present, e.g. a car, Einstein, theK2, a law, some gold, etc.2

The approaches we present are founded on the following (formal) basic no-tions and distinctions:

– Parthood. P(x, y) stands for “x is part of y”. We assume a classical extensionalmereology (CEM) (see [3, 4] for the axioms) defined only in restricted domainsthat we will make explicit in the following. In addition, the classical definitionof PP is considered.

– Endurants. ED(x) stands for “x is an endurant”.

– Time intervals or instants. T (x) stands for “x is a time interval or instant(briefly a time)”. Our presentation does not commit to a specific notion oftime even if we introduce the parthood relation in T .

1 See http://www.physics.nist.gov/cuu/Units/introduction.html2 We think that, with small changes, the frameworks introduced below can be applied also to perdu-

rants (entities that are only partially present at any time they are present like a process evolvingin time). We do not consider these changes in this paper.

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– Being present. PRE(x, t) stands for “x is present (exists) during the time t”.In the case of an endurant, we require that there exists a time during whichthe endurant is present.

(A1) PRE(x, t) → T (t)(A2) ED(x) → ∃t(PRE(x, t))

– We assume that all temporal relations we introduce enjoy the dissectivityproperty, that is, given relations A(x, t) and B(x, y, t), where t is a time, weassume

A(x, t) → ∀t′(P(t′, t) → A(x, t′))B(x, y, t) → ∀t′(P(t′, t) → B(x, y, t′))

in particular,

(A3) PRE(x, t) → ∀t′(P(t′, t) → PRE(x, t′))

– Finally, we assume that the categories/types/domains introduced (for exam-ple ED and T ) are all disjoint.

3 Endurants, Qualities, Qualia, and Spaces (EQQS)

We begin our analysis with the more sophisticated system we are going to con-sider. New classes of entities are introduced among which qualities and qualia.

– QT (x) stands for “x is a quality“. Qualities are partitioned into n non-emptysubtypes: QT1, . . . , QTn. Thus, given a quality x there is an index i such thatQTi(x).

– QL(x) stands for “x is a quale”. Qualia are also partitioned into n non-emptysubtypes: QL1, . . . , QLn.3 Analogously to the case of qualities, from QL(x)it follows that there is an index i such that QLi(x). We assume the CEM ineach QLi.

– For 1 ≤ i ≤ n, fix mi (mi ≥ 1) non-empty spaces that we indicate byS1

i , . . . , Smii . As working thypothesis, we assume CEM holds in each Sj

i , buta richer structure can be added (e.g. topological or geometrical properties).Indeed, some spaces may be atomic and others atomless; some may satisfycomplex mathematical properties and others just basic mereology.Entities in a space Sj

i are called regions. We write RG(x) to mean “thereare i, j such that x is in space Sj

i ”, in addition, spaces can be regrouped ingeneralized spaces, we write GSi(x) to mean “there is a j such that x is inspace Sj

i ”. Formally

(D1) RG(x) ,∨

i,j Sji (x)

3 Axiom (A7), see below, assures a correspondence between the QTi and the QLi.

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(D2) GSi(x) ,∨

j Sji (x)

The mereological sum is defined within each space Sji but not across spaces.

In particular, a region belongs to exactly one space Sji (for some i, j).

Now we look at the relationships among the entities so far introduced.We say that a quality x inheres in an endurant y, formally inh(x, y), if y is

the host of quality x. The relationship between qualities and qualia is dubbedabstract (abs). Expression abs(x, y, t) stands for “x is the quale of quality y attime t”. Abstraction is a ternary relation since the quality of endurants mayvary over time (e.g. a color might fade) and so the matching between qualityand qualia is time-dependent. Informally, abs captures a form of relative identityamong qualities in the sense that if both abs(x, y, t) and abs(x, y′, t) hold, thenqualities y and y′ (if different) can have only one distinguishing characteristic attime t: they inhere in different hosts. Finally, regions are interpreted as qualiapositions (posQL) in a space. We write posQL(x, y) to mean “x is a position ofthe quale y.”

Domain restrictions(A4) inh(x, y) → QT (x) ∧ ED(y)(A5) abs(x, y, t) → QL(x) ∧QT (y) ∧ T (t)(A6) posQL(x, y) → RG(x) ∧QL(y)Correspondences(A7) abs(x, y, t) →

∧i(QLi(x) ↔ QTi(y))

(A8) posQL(x, y) →∧

i(∨

j Sji (x) ↔ QLi(y))

Notation. Given a relation A and a predicate B, we will write A(x|B, y) forA(x, y) ∧B(x). If A is ternary, then A(x|B, y, z) stands for A(x, y, z) ∧B(x).

3.1 (Direct) Inherence

Each quality inheres in (has) a unique host that is an endurant (A12)+(A9),and it is present as long as its host is present (A11)4. In addition an endurantcannot have more than one quality of type i (A10).

(A9) inh(x, y) ∧ inh(x, y′) → y = y′

(A10) inh(x|QTi, y) ∧ inh(x′|QTi, y) → x = x′

(A11) inh(x, y) → ∀t(PRE(x, t) ↔ PRE(y, t))(A12) QT (x) → ∃y(inh(x, y))

3.2 Abstraction

Qualities are mapped to qualia only when they are present (A13), actually, theyare necessarily mapped to qualia when present (A15)5. Given a time t, a quality is

4 Here it is assumed that PRE is defined on qualities.5 The existential condition on t′ is introduced in order to avoid a commitment on temporal atoms.

This axiom (together with CEM and (A3)) assures that the temporal extension of a quality can be

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mapped to only one quale (A14). Also, recall that abs, being a temporal relation,is dissective. So far we have seen that relation PRE is defined over endurantsand qualities. We do not define it over qualia since qualia are atemporal.

(A13) abs(x, y, t) → PRE(y, t)(A14) abs(x, y, t) ∧ abs(x′, y, t) → x = x′

(A15) QT (x) ∧ PRE(x, t) → ∃y, t′(P(t′, t) ∧ abs(y, x, t′))

3.3 (Exact) Position

A quale is associated to at most one position (region) in one space (A16) andit has a position in every space associate to its quale type (A17). These spacesprovide a structure to compare and evaluate qualia and, indirectly, the corre-sponding qualities.

(A16) posQL(x|Sji , y) ∧ posQL(x

′|Sji , y) → x = x′

(A17) QLi(x) →∧

j(∃y(posQL(y|Sji , x)))

(T1) posQL(x, y) ∧ (PP(x′, x) ∨ PP(x, x′)) → ¬posQL(x′, y) (from (A16))

Refinement. On the basis of the relation posQL, one can define a refinementrelation (refin) between regions in different spaces (provided these refer to thesame quality type) and extend it to spaces themselves (refinS).

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(D3) refin(i, x, j, y, k) , Sji (x)∧Sk

i (y)∧ ∀z(posQL(x, z) → posQL(y, z)) ∧∀z(posQL(y, z) → ∃w(posQL(x,w)))

(region x is a refinement of region y)(D4) refinS(i, j, k) , ∀y(Sk

i (y) → ∃x(refin(i, x, j, y, k)))(space Sj

i is a refinement of space Ski )

(T2) refinS(i, j, k) → ¬∃x, y, y′, a, b(posQL(x|Sji , a) ∧ posQL(x|Sj

i , b) ∧posQL(y|Sk

i , a) ∧ posQL(y′|Sk

i , b) ∧ y 6= y′)(from (A16), (D3), and (D4))

4 Endurants, Qualities, and Spaces (EQtS)

We have seen that qualities depend on their host and, informally, capture onesingle aspect of the host like color, weight, shape and the like. Furthermore,two qualities are clustered together in a quale whenever their hosts are identicalrelatively to that aspect. Since this clustering of qualities into qualia is preservedin the spaces Sj

i , perhaps one can discharge qualia altogether without losing in-formation and expressive power. In this section we pursue this second approach.

divided into a number of times during which the quale associated to the quality does not change.No mereological assumption is imposed on these times.

6 If spaces are built out of atoms, relation refinS is a partial order.

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Let us consider again the categories ED, T ,QT ,QT1, . . . , QTn and the spacesS1

i , . . . , Smii (for 1 ≤ i ≤ n) as given before. The relation inh and the predicate

RG are characterized as in the previous section.

4.1 Temporalized position of qualities

In order to capture the changes in the qualities of endurants, the position relationinvolves now a temporal parameter. We call this the temporalized position ofqualities (posQT). Expression posQT(x, y, t) stands for “x is a position of thequality y at time t“. The axiomatization of posQT reflects in part that of relationsabs and posQL in the previous approach. In particular,

(A19) corresponds to (A8); (A20) to (A16); axioms (A21) to (A13); (A22)to (A15).

Domain restriction(A18) posQT(x, y, t) → RG(x) ∧QT (y) ∧ T (t)Correspondence(A19) posQT(x, y, t) →

∧i(∨

j Sji (x) ↔ QTi(y))

Other constraints(A20) posQT(x|Sj

i , y, t) ∧ posQT(x′|Sji , y, t) → x = x′

(A21) posQT(x, y, t) → PRE(y, t)(A22) QTi(x) ∧ PRE(x, t) →

∧j(∃y(posQT(y|Sj

i , x, t)))

Also recall that, being temporal, posQT has the dissective property.

4.2 Are qualia necessary?

We changed the EQQS system of section 3 in as much as needed to avoid theintroduction of qualia. At this point, it is natural to ask if the new system EQtSis somewhat weaker than EQQS, that is, if there is some situation that we cancapture in the latter system but not in the first.

In EQQS, a qualia identifies a sort of (temporary) equivalence relationamong qualities: qualities that are indistinguishable unless we refer to theirhosts, are all associated to the same quale. In a sense, qualia provide the onto-logical status of qualities: two endurants with qualities that match to the samequale are “the same” with respect to that quality type. Within this approach,two spaces Sj

i and Ski may be seen as different ways to organize the qualia in

subclass QLi and, indirectly, to represent coarse granularities on the qualities inQTi. This result is obtained by positioning different qualia in the same region.

But, beside identifying this ontological status of qualities, are qualia nec-essary to compare endurants or to evaluate them? Let us consider the case ofrefinement. Is it possible to define in EQtS a relation of refinement as done inEQQS? Taking into account the temporal parameter in posQT, the definition of

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refinement on regions can be formulated as follows (on spaces, it suffices to take(D4) with refin as given below):

(D5) refin(i, x, j, y, k) , Sji (x) ∧ Sk

i (y) ∧ ∀z, t(posQT(x, z, t) → posQT(y, z, t)) ∧∀z, t(posQT(y, z, t) → ∃w(posQT(x,w, t)))

As we have seen, in EQQS qualia provide the finest granularity in evaluatingthe category of qualities while spaces Sj

i add extra conditions by further groupingqualia and by furnishing ordering, topological, or even metric relations. In EQtSwe have only the spaces Sj

i but it is still possible to define a notion of “maximalgranularity” by imposing, for each quality type QTi, the existence of a spacethat, according to definition (D4), refines all the spaces associated to QTi. Let∃!k to mean “there exists a unique index k”, then

(A23)∧

i ∃!k(∧

j refinS(i, k, j))

Let us write S∗i for the space isolated by axiom (A23). One sees that, with the

introduction of S∗i , EQQS and EQtS are equivalent in expressive power.7 Thus,

an equivalence between the two systems can be established formally throughextra assumption (A23). Still, the two approaches differ ontologically as theextra category in EQQS shows. We will come back to this issue later.

5 Endurants, Qualia, and Spaces (EQlS)

We have seen that, under some hypotheses, dropping the category of qualia onemaintains the same expressive power of EQQS. Do we reach the same resultremoving qualities instead of qualia?

Quality kinds are in a one-to-one correspondence with quale kinds, thereforeit seems quite natural to use this correspondence to “bypass” qualities by: (i)introducing a (temporalized) abstraction relation between endurants and qualia(absED), and (ii) maintaining the position relation posQL already introduced insection 3. The axioms characterizing absED follow closely those given for theother position relation, namely posQT (as before, we do not list dissectivity):

Domain restriction(A24) absED(x, y, t) → QL(x) ∧ ED(y) ∧ T (t)Other constraints(A25) absED(x|QLi, y, t) ∧ absED(x′|QLi, y, t) → x = x′

(A26) absED(x, y, t) → PRE(y, t)

In a sense, qualities carve up particular aspects of an object. “The weightof John“ is not comparable to a quale (the weight of John can change in time,

7 The careful reader might observe that space S∗i could present topological or geometrical propertiesthat cannot be present in the class QLi. This is irrelevant if space S∗i is present in both systems.

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the quale cannot; also it is specific of John while a quale can be contemporarilyassociated to several endurants). The quality combines John specificity and theweight kind (dimension), thus it can have properties that are not directly ascrib-able to a quale, nor to the positions of qualia in the spaces Sj

i . Let’s consider, forexample, the sentence “the weight of John is good now”. It can be interpreted inat least two different ways: (i) the weight-quale associated to John at this mo-ment is “good“; and (ii) the weight-quality of John is at this moment “good”.In the first case, “good“ is an absolute property since it applies to a quale, i.e.,to an atemporal entity. If it happens that the weight of Mary is mapped to thesame quale at the same time (or even at another time t), then necessarily “theweight of Mary is good now (at time t, respectively)” holds as well.

In the second case “good” is applied to the weight-aspect of John, which isspecific to John. With this interpretation the expression “the weight of Mary isgood now” may be false no matter where the weight of Mary is mapped now.Thus, one cannot capture the particular link between qualities and their hostsvia qualia unless introducing a class of qualia-properties like “John goodness“,“Mary goodness”, etc. which seems ontologically unpalatable. There is an al-ternative solution though. It is possible to reconstruct qualities of kind QTi ascouples8 〈e,Q|e|

i 〉, were e is a specific endurant, and Q|e|i is the set of all the qualia

of kind QLi that are linked to e:

(D6) Q|e|i = {q |QLi(q) ∧ ∃t(absED(x, e, t))}

Now, one can define inh and abs by taking 〈e,Q|e|i 〉 as argument of QTi (the

definition is given for quality subtypes; it is easily extended to the general case):

(D7) inh(〈e,Q|e|i 〉|QTi, e

′) iff e = e′

(D8) abs(q|QLi, 〈e,Q|e|i 〉, t) iff q ∈ Q|e|

i ∧ absED(q, e, t)

6 Endurants and Spaces (ES)

Can one reject qualities and qualia altogether? Here we take a step further andconsider a system that adopts the category of endurants and the spaces Sj

i only.In this approach, an endurant is positioned in the spaces which are associated todifferent aspects of that endurant. For example, an endurant for which color isdefined will be related by a position relation to spaces Sj

h where h is the index ofthe spaces classifying colors. In the previous systems, the quality types QTi (orthe quale types QLi) determined the different aspects of an endurant. Now weneed a new mechanism to capture this distinction. Typically, one would introducean additional level of classes: on the one hand regions are partitioned in spaces,

8 Another possibility is to introduce qualities as sums of an endurant and the qualia (of a specifickind) associated to it. In infinite domains this means to adopt general extensional mereology (GEM)in every QLi. Note that GEM is stronger than CEM [3, 4].

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on the other hand spaces are “clustered“ in generalized spaces (see definition(D2)) with each generalized space capturing an aspect of the endurants. Then,it is possible to set a correspondence between these generalized spaces and thequality kinds of EQtS (or the qualia kinds of EQlS). That is, given the qualitiestypes, let us say just color and length, one assumes that there exist the “colorspace” GS1, which is the union of all the color spaces Sj

1 (recall that all thespaces Sj

i are disjoint) and the “length space” GS2, the union of all the lengthspaces Sj

2. GS1 and GS2 are what we called the generalized spaces. In short,

– In EQtS(a) there are n non empty and disjoint sets of qualities QT1, . . . , QTn;(b) each QTi is associated to the spaces S1

i , . . . , Smii .

– In EQlS(a) there are n non empty and disjoint sets of qualia QL1, . . . , QLn;(b) each QLi is associated to the spaces S1

i , . . . , Smii .

– In ES(a) there are n non empty and disjoint generalized spaces GS1, . . . , GSn;(b) each GSi is partitioned into the spaces S1

i , . . . , Smii .

6.1 Temporalized position of endurants

To link endurants and regions we consider a temporalized position relation overendurants (posED). The axiomatization of posED is similar to the axiomatizationof posQT, the only major difference being axiom (A32) stating that an endurantpositioned in a space at t, has always a position in that space:

Domain restriction(A27) posED(x, y, t) → RG(x) ∧ ED(y) ∧ T (t)Other constraints(A28) posED(x|Sk

i , y, t) ∧ posED(x′|Ski , y, t) → x = x′

(A29) posED(x, y, t) → PRE(y, t)(A30) ED(x) ∧ PRE(x, t) →

∨i(∃y(posED(y|GSi, x, t)))

(A31) ED(x) ∧ ∃y(posED(y|GSi, x, t)) →∧

j(∃z(posED(z|Sji , x, t)))

(A32) ED(x) ∧ ∃y(posED(y|GSi, x, t)) ∧ PRE(x, t′) → ∃z(posED(z|GSi, x, t′))

6.2 Alternatives and expressive power

Instead of introducing generalized spacesGSi, it is possible to consider a (binary)similarity relation between regions. The intended meaning of such a relation is“regions x of Sj

i and y of Ski are similar if they are positions of the same aspect of

the same endurant”. With this relation at our disposal, it is easy to reconstructthe sets GSi.

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Another possibility is the introduction of an additional parameter in posED

that isolates the specific aspect we are considering. Technically this is not prob-lematic but we need to introduce this additional category of entities that isontologically obscure (there are no individual qualities but only “names“ or“reification of kinds” of qualities).

Finally, we can take advantage of our previous work to reconstruct in ESthe missing notions. In short, we can reconstruct qualia as regions in minimalspaces as done in EQtS, and following the strategy taken in EQlS we canreconstruct qualities (and the relations inh and abs) as couples 〈e,R|e|

i 〉, where

e is an endurant and R|e|i is the set of the positions that e has (in time) in the

minimal space R∗i .

7 The nature of qualia

7.1 Qualia and tropes

The distinction between qualia and tropes (see [5] for a review on tropes) isimportant in ontology since these entities provide different ways to representand explain qualitative change. In this paper we have adopted qualities as basicentities but this should not be considered as a rejection of tropes. Tropes supplya different view and can be formalized in a similar way. Here we discuss theiradvantages and drawbacks.

In trope theories, qualitative change is expressed in terms of substitution oftropes: when an endurant a changes in time (say, it is red at time t1 and yellowat time t2), this means that a trope inherent in a at t1 disappears and a newtrope is created. Thus, differently from qualities which are associated to differentqualia over time, tropes do not change. It is their coming out and disappearingin time that explains the changes we observe in endurants: endurants change byacquiring some tropes while losing others. Tropes represent the different prop-erties an endurant has at/during t and, at each time, an endurant can possessonly one trope for each property. We encountered a similar restriction on qualia:only one quale instance of a specific quale type is allowed at a time. Similarly,the tropes of the endurant at each time t need to be instances of different (kindsof) universals [6]. Thus, the different tropes that inhere in the same endurant aat time t must be related to different aspects of the endurant.

In the theories EQQS and EQtS, qualities persist through change, i.e. thecolor of an endurant survives to the change from red to yellow and no entitydisappears or is created. The change is represented by a relational change whichexplains why the aspects of the endurant vary in time.9 Sometimes it is claimedthat trope theory better explains changes in endurants since there is something

9 In EQlS and in ES change is explained analogously. In this cases, the involved relations (absED

and posED, respectively) apply directly to endurants.

11

happening that motivates the change: a trope that was in the endurant is sub-stituted by a new one (an explanation resembling a substitution of “parts”).

Note that, from the point of view of expressiveness, there is no real differencein adopting qualities or tropes. This is not so if we look at the ontological natureof these entities.

It is possible to formalize the trope approach by taking the class of tropesto be partitioned in n types10 TR1, . . . , TRn and by associating TRi to thenon-empty spaces S1

i , . . . , Smii . Then, it suffices to use the inherence and the

(non temporalized) position relations. For lack of space, we do not list axioms.Anyway, the formalization is quite similar to the one given for direct inherence(inh) and exact position (posQL) in the EQQS approach. An important differenceis about axiom (A10). For tropes, an additional condition constraining temporalcoincidence of tropes is necessary, i.e. it is possible that two different tropes ofthe same type inhere in one endurant, but only at different times:

(D9) x ∼T y , ∀t(PRE(x, t) ↔ PRE(y, t))(A33) inh(x|TRi, y) ∧ inh(x′|TRi, y) → (x ∼T y ↔ x = x′)

Note that, like qualia, two tropes with different positions in a space Sji could

be associated to the same position in another space Ski . Furthermore, tropes are

not dependent on spaces.Tropes are similar to qualia in as much as they provide the finest distinc-

tion on aspects of endurants but differently from qualia they are in time, andtheir temporal extension represents the time during which the property they“embody“ is valid.

Finally, note that tropes are often related to a realistic approach in ontol-ogy which results in more constrained systems while, as we have seen, qualitiesand qualia do not force such a strong stand and their systems are quite flex-ible allowing the reconstruction of several notions starting from different setsof categories. From a practical point of view, this very fact can make a systembased on tropes unfit for integration or interaction with systems that are lessontologically committed.

7.2 Qualia and their positions in spaces

The EQQS and EQtS approaches consider qualia as ontological markers ofaspect similarity. This means that two endurants with qualities mapped (at onespecific time) to the same quale are, at that time and limited to this quality type,ontologically indistinguishable. This is not a matter of empirical or epistemo-logical equivalence: independently form the instruments or cognitive processesused to analyze the endurants, these endurants are identical with respect to that

10 Formally, universals are here considered as predicates. One can reify universals in the domain andadd an “instance-of“ relation to capture the same notions [6].

12

particular aspect. As pointed out earlier, this is a case of relative identity. Onthe contrary, positions in spaces capture the empirical/epistemological level: thequalia can be organized, ordered, regrouped in very different ways depending onthe space structure. Therefore each space supplies a “point of view” on qualia.

From a metaphysical viewpoint the distinction is quite interesting, but itloses importance when applications are considered. In this case, the analysisof the endurants is always conducted at an empirical or theoretical level. Forexample, in the case of engineering (domain) models, the available informationon the domain and the available measurement instruments determine the spacesto consider. Any other (finer) distinction is irrelevant. This is also the case ofscientific theories that aim to describe “reality“: the spaces and their structuresdepend on the theories scientists are considering, and, to some extent, on themeasurement methods they employ. Analogously, in cognitive science the spacesSj

i are built according to the behavior of subjects in experiments. We will comeback to this in sections 8 and 9.

With or without qualia, the spaces Sji and their structure can depend on (i)

culture (e.g. people in different societies classify colors, shapes, etc. in differentways); (ii) instruments of investigation or scientific theories; (iii) interpretationsof experiments; etc. In general, they are created, adopted, and destroyed in timeby (communities of) intentional agents. It follows that the spaces with theirinternal structure (mereological, topological, geometrical, metrical etc.) have adefinite temporal extension, and therefore may or may not be present at a giventime. Here we do not analyze this aspect further.

When qualia are part of the ontological framework, spaces can assume amore abstract role. We can see them as different “structures” that apply to anyquale type along the lines of Klein’s notion of geometry (in this case, a spaceis identified to a set of transformations and each class QLi furnishes a domainof application). We will not pursue this approach in this paper. Also, note thatby introducing the category of qualia we gain uniformity across domains. If attime t the color qualities of two endurants map to the same quale, then at ttheir color qualities are indistinguishable in any domain, let it be psychology,astronomy, or linguistics. Neither a change of spaces Sj

i nor the introduction ofnew measurement methods can affect this basic fact. On the other hand, thismight be seen as a lack of flexibility in the system. Once the ontological levelof qualia is characterized, the resulting ontology is not compatible with otherontologies (or frameworks) that adopt a larger set of qualia.

8 Qualities and conceptual spaces

In [2], Peter Gardenfors models the “representations“ used in cognitive scienceby introducing the notion of conceptual space. Conceptual spaces are collections

13

of related domains each of which is a collection of (integral and separable) di-mensions like, for example, temperature, weight, pitch, and brightness.

The theory of conceptual spaces is based on the notion of similarity: “Judg-ments of similarity (. . . ) are central for a large number of cognitive processes.(. . . ) such judgments reveal the dimensions of our perceptions and their struc-tures” ([2], p.5). Dimensions correspond to “the different ways stimuli are judgedto be similar or different“ ([2], p.6), and, in this sense, they are taken to repre-sent the various qualities of endurants. A point in a dimension may represent,for example, a particular temperature. Then, the association of two endurantsto the same point represents the experimental fact that the two endurants arecompletely similar with respect to temperature. Points can be ordered (e.g. atone can be “low” or “high“) and it is generally assumed that each dimension isendowed with a mathematical structure: the level of similarity between stimuliis therefore embedded in the metric (or pseudo metric) relation defined on thedimensions. A set of dimensions is integral if an endurant that has a “position”inside one dimension, necessarily has a position inside all the other dimensions.For example, {hue, brightness} is integral because if an endurant has a partic-ular hue it necessarily has also a particular brightness (and viceversa). A set ofdimension is separable if it is not integral like {hue, size}.

In Gardenfors terminology, domains are maximal sets of integral dimensions.For example the three-color dimensions hue, chromaticness, and brightness forma domain because the set {hue, chromaticness, brightness} is integral, but hue,chromaticness and brightness are separable from any dimension that does notbelong to this set. Domains can be used to assign properties to endurants, i.e.,to classify endurants: a particular property corresponds to a region in a domain.The separability constraint allows to assign properties (regions in a domain)independently from other properties (regions in other domains). This capturesthe experimental fact that the weight of an endurant is independent from theendurant’s color.

Finally, conceptual spaces are defined as collections of one or more domainsand concepts are represented as complex regions in conceptual spaces. A pointin a conceptual space constraints the properties of endurants at the maximallevel of detail.

Clearly, conceptual spaces are thought to be theoretical entities and they are“static in the sense that they only describe the structure of representations“ ([2],p.31). Furthermore, they are often understood as part of a relativistic approach:their structure depends on the underlying culture, on measurement methodsand sensors (in the case of scientific conceptual spaces), or on interpretation ofthe behavior of subjects (in the case of phenomenal conceptual spaces). This isanother reason to conclude that conceptual spaces do not match the ontologicalimport of qualities nor that of qualia: ontologically the theory of conceptualspaces seems close to the ES approach.

14

9 Qualities and the International System of Units

The NIST guide to the International System of Units (SI)11 distinguishes be-tween:

– a quantity in the general sense: a property ascribed to phenomena, bodies,or substances that can be quantified for, or assigned to, a particular phe-nomenon, body, or substance (e.g. mass and electric charge); and

– a quantity in the particular sense: a quantifiable or assignable property as-cribed to a particular phenomenon, body, or substance (e.g. the mass of themoon and the electric charge of the proton).

Also, the SI introduces the notion of physical quantity as a quantity thatcan be used in the mathematical equations of science and technology. A unit isa particular physical quantity, defined and adopted by convention, with whichother particular quantities of the same kind are compared. The result of thiscomparison is a number. The value (or magnitude) of a physical quantity is the“product” of a number (the numerical value) and an unit. For instance, we canrepresent the fact that the tower of Pisa is 55 meter high and that its weightis 14.453 tonnes by: hPT = 55m;wPT = 14.453.000Kg. This representationexplicitly refers to the particular height (hPT ) and weight (wPT ) of the towerof Pisa. Of course, another building might have the “same“ height-quantity(in the SI general sense), i.e., hb = 55m. However, hb and hPT are differentquantities in the particular sense. They are different because ascribed to differentendurants and yet they have the same value: 55m. The identity hPT = hb in SIis understood as a shortcut for value(hPT ) = value(hb). In this sense, quantitiesin the SI particular sense are similar to our qualities. Furthermore, the notionof ascription is similar to that of inherence as captured by relation inh.

Unfortunately, SI does not give much information on how to interpret anexpression like 55m = 180 ft. That is, one could take 55m and 180 ft to be thesame position in a given space (including both units) or to be two distinct posi-tions in different spaces which are connected by some correspondence relation.The interpretation changes depending on the way one introduces spaces and,more specifically, on their relationship with units of measure. For instance, thetwo values 55m and 180 ft must be considered ontologically different entities ifwe take a value to be the combination of a numerical value (55 in one case, 180in the other) and a unit (rispectively, m and ft) since, clearly, their componentsare different. Alternatively, one can say that there is only one space (and so oneposition) and that the two units simply provide different ways to identify theunique position in that space.

The SI considers a set of base (general) physical quantities (length, mass,time, electric current, thermodynamic temperature, amount of substance, and

11 http://physics.nist.gov/cuu/index.html

15

luminous intensity) in terms of which all the other physical quantities can beexpressed by means of an equation12. (An similar equation is given for units.)For example, velocity is expressed in terms of length and time by the followingequation: v = l · t−1 (and the SI derived unit of velocity is m · s−1). Notethat these equations do not involve quantities in the particular sense but justquantities in the general sense. The goal is to state how the values of complexquantities can be reduced to the values of the base quantities. This is similarto the distinction between dimensions and domains in the Gardenfors approach(see section 8). For Gardenfors the link between the dimensions of a domainand the domain itself is directly coded into the structure of the space. Thisfact explains why, for instance, the color domain has the shape of a doublecone and not of a 3D cube. The existential dependence of complex quantitieswith respect to the base quantities in SI corresponds to the relationship betweendimensions and domains in Gardenfors’ approach. However, SI is very restrictivein combining basic quantities to obtain different spaces so that all possible spacesare essentially multi-dimensional cubes. In conclusion, the SI approach seemscloser to the EQlS system.

References

1. Masolo, C., Borgo, S., Gangemi, A., Guarino, N., Oltramari, A.: Wonderweb deliverable d18:Ontology library. Technical report, ISTC-CNR (2003)

2. Gardenfors, P.: Conceptual Spaces: the Geometry of Thought. MIT Press, Cambridge, Massachus-setts (2000)

3. Casati, R., Varzi, A.: Parts and Places. The Structure of Spatial Representation. MIT Press,Cambridge, MA (1999)

4. Simons, P.: Parts: a Study in Ontology. Clarendon Press, Oxford (1987)5. Daly, C.: Tropes. In Mellor, D., Oliver, A., eds.: Properties. Oxford University Press, Oxford

(1997) 140–1596. Neuhaus, F., Grenon, P., Smith, B.: A formal theory of substances, qualities, and universals. In

Varzi, A., Vieu, L., eds.: Formal Ontology in Information Systems (FOIS’04), Turin, Italy, IOSPress (2004) 49–59

12 This equations has the form Q = lα ·mβ · tγ · Iδ · T ε · nζ · Iηv ·

KPk=1

ak.

16

A Categorical Framework for Translating

between Conceptual Hierarchies

Rainer Osswald

Department of Computer Science, FernUniversitat in Hagen, Germany

Abstract. Any classification of objects by attributes naturally determines a concep-tual (or ontological) hierarchy consisting of the attribute sets that are closed under allimplications holding in the classification. In this paper, we present a thorough cate-gorical treatment of translations between implications over different sets of attributesand study in detail how such a translation is reflected on the side of the associatedconceptual hierarchies.

1 Introduction

If the elements of a given domain of discourse are classified with respect to acertain set Σ of attributes then these data naturally give rise to a hierarchy offormal entities that can be interpreted both conceptually and ontologically. Awell-known example of this sort of construction is provided by Formal ConceptAnalysis (FCA) [10], where classifications are called formal contexts and theresulting hierarchies are lattices of (formal) concepts. The formal concepts ofFCA can be identified with those subsets of Σ that are closed under all conjunc-tive implicational statements holding in the given context or classification. Thiskind of systematic relationship between statements holding in a classificationand conceptual hierarchies consisting of closed attribute sets can be generalizedfrom purely conjunctive implications to those also involving disjunction, truth,and falsity, that is, to implications where arbitrary positive or affirmative termsmay occur as premise and conclusion [9, 25]. The resulting hierarchies includeall finite partial orders and have once been dubbed information domains [8].

The present paper is concerned with translations between different sets ofattributes. We present a categorical framework for theories, i.e., sets of state-ments, which makes use of a very natural and explicit notion of translation, andstudy in detail how translations of theories are reflected on the side of the cor-responding information domains.1 In particular, we give equivalence criteria fortheories with identical information domain and we present general constructionschemes both for theories and information domains.

1 Our exposition presumes some background knowledge of the standard definitions of category, lat-tice, order, and domain theory, all of which can be found in [2, 18], [6], and [1], respectively.

Overview. Section 2 introduces basic notions like interpretation, theory, andmodel, as well as the specialization relation given by an interpretation. We thendefine the information domain of a theory as the ordered universe of a universalmodel of that theory and review an alternative construction via the Linden-baum algebra of positive terms determined by the theory, which is a boundeddistributive lattice. Section 3 explores the relation between theories over differ-ent base vocabulary; in particular, we are interested in criteria for equivalence.To this end, we define an appropriate notion of translation between positiveterms and study the effect such a translation has on the respective informationdomains. Categorically speaking, we study the information domain functor fromthe category of theories to that of (directed-complete) ordered sets. Section 4 isconcerned with various ways of constructing theories and with the correspondingconstructions of information domains. The most general construction we con-sider is the quasi-colimit of theories, which is a slight variant of the standardcategory-theoretic notion of colimit. The information domain functor is shownto take quasi-colimits of theories to limits of information domains. This gives usa characterization of information domains as well as possible ways to constructthem.

Related Approaches. There is a significant overlap with the extensive work onsequent structures (consequence systems, information systems, entailment rela-tions) [8, 29, 3, 7, 5], which are essentially normal forms of the theories consideredin this paper. The focus of the present paper differs insofar as sequent structuresare typically seen as suitable representations for domains, whereas for us a giventheory is the primary object of interest. It seems that the category of theoriesintroduced in this paper has not been studied in depth before. For instance,in [7], only primitive translations, i.e., translations of attributes by (primitive)attributes, are taken into account, whereas [5] makes use of the more generalnotion of an approximable relation. Categorical treatments of conjunctive theo-ries and their relation to FCA can be found in [14, 13]. It is furthermore worthmentioning that [20, 19] take the Lindenbaum algebra of positive terms insteadof the information domain as an appropriate representation of the conceptualhierarchy determined by a theory. As mentioned above, the Lindenbaum algebracanonically determines the information domain of a theory (and vice versa inthe finite case).

2 Information Domains

2.1 Theories, Models, and Positive Terms

Let Σ be a set of (primitive) attributes. We introduce two special attributes Vand Λ which respectively hold of everything and nothing in any universe of dis-course. To build compound attributes, we employ the usual Boolean connectives

18

∧, ∨, and ¬. Let B[Σ] be the term algebra of (compound) Boolean attributesinductively defined that way. The meaning of these compound attributes, whenapplied to classify the elements of a certain domain of discourse U , is the obviousone: Let � be a binary relation from U to Σ that expresses the satisfaction of(primitive) attributes by elements of U . Then x ∈ U satisfies φ∧ψ iff x satisfiesφ and ψ; similarly, x satisfies ¬φ iff x does not satisfy φ; etc. The satisfactionrelation � can thus be inductively extended to a relation from U to B[Σ]. Asusual, we write φ→ ψ and φ↔ ψ for ¬φ∨ψ and φ→ ψ ∧ψ → φ, respectively.

A natural way to reformulate the notions introduced so far within a standardlogical setting is to regard attributes as monadic predicates (see also [25]). Asatisfaction relation � from U to Σ is then essentially the same as a (set valued)interpretation function M from Σ to ℘(U), with M(p) = {x ∈ U | x � p}, sincewithin (first-order) predicate logic, monadic predicates are interpreted by subsetsof a universe U . Moreover, the interpretation function M can be inductivelyextended to a function M from B[Σ] to ℘(U) such that M(φ) = {x ∈ U |x � φ}.We refer to M(φ) as the extent of φ.

In order to formulate statements that hold or are true with respect to a giveninterpretation (or classification), we need to quantify over the terms in B[Σ].In this paper, only universal quantification is taken into account, which coversthe approaches mentioned in the introduction. That is, we consider universalstatements of the form ∀x(φx), abbreviated by ∀φ, with φ ∈ B[Σ], and take atheory over Σ to be a set of universal statements of this form. The followingnotions of truth and model are those of standard first-order logic: A statement∀φ is true with respect to an interpretation if φ is satisfied by all elements ofthe universe. A model of a theory Γ is an interpretation with respect to whichall statements of Γ are true.

A Boolean term φ ∈ B[Σ] is called positive or affirmative if it is free of¬. Let T [Σ] be the term algebra of positive terms over Σ. The special role ofthe positive terms reveals itself in connection with the specialization relation aninterpretation defines on its universe:

Definition 1 (Specialization). Given an interpretation of Σ with universe Uand x, y ∈ U , then x is specialized by y, in symbols, x v y, if y satisfies everyelement of Σ that is satisfied by x.

The specialization relation v is reflexive and transitive, i.e. a preorder. By thedefinition of v and straightforward term induction over T [Σ], it follows thatx v y iff ∀φ ∈ T [Σ] (x � φ → y � φ). That is, positive terms are persistentwith respect to specialization, whereas Boolean terms in general are not (seee.g. [24]).

Given two theories Γ and Γ ′ over Σ, we say that Γ entails Γ ′, in symbols,Γ ` Γ ′, if every model of Γ is also a model of Γ ′. The theories Γ and Γ ′ are saidto be equivalent if they entail each other. A theory over Σ has conditional (or

19

biconditional) form if its statements are of the form ∀(φ → ψ) (or ∀(φ ↔ ψ)),with φ and ψ positive. The conditional form is normal, if φ is purely conjunctive(or V ) and ψ is purely disjunctive (or Λ). It is a standard exercise in elementarylogic to verify that every theory is equivalent to a theory in conditional (nor-mal) form and to one in biconditional form. In the following, this fact will befrequently used to the effect that a theory is implicitly assumed to have condi-tional or biconditional normal form when appropriate. For convenience, let usintroduce two binary term operators � and ≡ such that φ � ψ and φ ≡ ψ are∀(φ→ ψ) and ∀(φ↔ ψ), respectively.

Given an interpretation M of Σ with universe U , the set {M(φ) | φ ∈ T [Σ]}of extents of positive terms forms a distributive lattice with respect to ∩ and∪, which is bounded by M(V ) = U and M(Λ) = ∅. More generally, we canconsider interpretations and models in arbitrary bounded distributive lattices.We speak of such lattices briefly as algebras, taking them as algebras of type〈2, 2, 0, 0〉.

Definition 2 (Algebraic Interpretation/Model). An interpretation m ofΣ in an algebra A, or A-valued interpretation, is a function from Σ to A; m isan A-valued model of a theory Γ over Σ iff m(φ) = m(ψ) whenever Γ entailsφ ≡ ψ. Let Mod(Γ,A) (or ModA(Γ )) be the set of A-valued models of Γ .

2.2 Information Domains

There is a standard way to associate with each theory Γ over Σ a canonicalmodel. Let the canonical interpretation of Σ in ℘(Σ) take p ∈ Σ to {X ⊆Σ | p ∈ X}, that is, X � p iff p ∈ X. Now let C (Γ ) be the set of all X ⊆ Σwhich, under the canonical interpretation, satisfy φ for every statement ∀φ of Γ .The canonical model of Γ takes p ∈ Σ to {X ∈ C (Γ ) | p ∈ X}. The canonicalmodel of Γ has the universal property that the statements true in it are preciselythose entailed by Γ .2

Specialization on C (Γ ) is set inclusion and hence a partial order. We referto the elements of C (Γ ) as the consistently Γ -closed subsets of Σ. Adapting theterminology of [8], we say that any ordered set order-isomorphic to C (Γ ) “is”or represents the information domain of Γ ; notation: D(Γ ). It is not difficultto see that the information domain of a theory is directed-complete, i.e., closedwith respect to suprema of (upwards) directed subsets.

Example 1. Let Γ be the theory over {a, b, c, d} with statements a ∧ b � c ∨ d,c ∧ d � Λ, c � a, d � a ∧ b. Then C (Γ ) is as depicted on the left of Figure 1.

The representation of the information domain by consistently closed sets isespecially useful for practical purposes because of its concreteness. For theoret-

2 In [9], C(Γ ) is referred to as the free extent ; in the terminology of [4], the canonical model isessentially the same as the generated classification; see [25] for a comparison of terminologies.

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C (Γ )

∅

{a} {b}

{a, c}

{a, b, c} {a, b, d}

L(Γ )[V ]

[a ∨ b]

[b ∨ c][a]

[b][c ∨ d]

[a ∧ b][c]

[d][b ∧ c]

[Λ]

Fig. 1. Information domain and Lindenbaum algebra of Γ

ical purposes, on the other hand, a more abstract representation of the infor-mation domain as the prime spectrum of the Lindenbaum algebra is often moreappropriate. The idea that underlies the Lindenbaum construction is to abstractaway from syntactical differences between positive terms that are equivalent withrespect to a given theory:

Definition 3 (Lindenbaum Algebra). The Lindenbaum algebra L(Γ ) of atheory Γ over Σ is the quotient T [Σ]/'Γ , with φ 'Γ ψ iff Γ ` φ ≡ ψ.

Clearly the Lindenbaum algebra of positive terms is an algebra in the senseintroduced at the close of Section 2.1, i.e., a bounded distributive lattice, with0 = [Λ], [φ] ∧ [ψ] = [φ ∧ ψ], etc. If Γ is the theory of Example 1 then L(Γ ) is asdepicted on the right of Figure 1. (The shaded circles in the diagram correspondto the ∨-irreducible elements of L(Γ ), which stand in an order-reversing one-to-one correspondence to the elements of C (Γ ).) Notice that every algebra A isisomorphic to the Lindenbaum algebra of a theory over (the carrier set of) A.For let Th(A) be the theory over A consisting of all statements φ ≡ ψ such thatidA(φ) = idA(ψ). Then L(Th(A)) ' A.

Let 2 be the algebra {0, 1}. There is a one-to-one correspondence betweenC (Γ ) and Mod2(Γ ) (cf. Definition 2), where X ∈ C (Γ ) corresponds to thecharacteristic function χX of X. Consequently:

Proposition 1. Mod2(Γ ) represents the information domain of Γ .

The Lindenbaum construction provides a universal model in the sense thatthe function mΓ from Σ to L(Γ ) that takes p to [p]≡Γ

has the following universalproperty:

Proposition 2. Every model of Γ in an algebra A factors uniquely through mΓ

by a homomorphism from L(Γ ) to A. Hence Mod(Γ,A) ' Hom(L(Γ ), A).

21

Definition 4 (Spectrum). The spectrum P (A) of an algebra A is the set ofall prime filters of A ordered by set inclusion.

Since h ∈ Hom(A,2) iff h−1(1) ∈ P (A), and F ∈ P (A) iff χF ∈ Hom(A,2):

Proposition 3. P (A) is order-isomorphic to Hom(A,2) (ordered pointwise).

Together with Propositions 1 and 2, this gives us the promised result thatthe information domain of a theory Γ can be represented by the spectrum of itsLindenbaum algebra L(Γ ).

3 Translations and Equivalences

3.1 Translations of Theories

Let Γ and Γ ′ be theories over Σ and Σ ′, respectively. Roughly speaking, atranslation of theories is a truth-preserving translation of positive terms. Thatis, if a statement is true in all models of Γ , its translation is required to be truein all models of Γ ′:

Definition 5 (Theory Translation). A translation of theories from 〈Σ,Γ 〉to 〈Σ ′, Γ ′〉 is a function µ from Σ to T [Σ ′] such that Γ ′ ` µ(Γ ). The translationµ is primitive if µ(Σ) ⊆ Σ ′.

Example 2 (Theory extensions). Suppose Σ ⊆ Σ ′ and Γ ⊆ Γ ′. The inclusionfunction ε from Σ to Σ ′ is a primitive translation of theories from Γ to Γ ′, calledan extension. We then also say that Γ ′ is an extension of Γ .

Let the composite of two theory translations µ from Γ to Γ ′ and µ′ from Γ ′

to Γ ′′ be the composite function µ′ ◦ µ from Σ to T [Σ ′′], which is easily seen tobe a theory translation from Γ to Γ ′′, and let the identity translation ıΓ of Γ bethe canonical inclusion function of Σ into T [Σ]. One straightforwardly verifiesthat these data define a category of theories Th.

The notion of isomorphism in Th is too restrictive to adequately character-ize the equivalence of theories. To give a simple example, let Γ be the emptytheory over Σ = {a} and let Γ ′ be the theory {b ≡ c} over Σ ′ = {b, c}. That is,within 〈Σ ′, Γ ′〉, we have precisely two primitive terms or attributes, which aresynonymous in the sense that they have the same extent, and there are no fur-ther constraints. Within 〈Σ,Γ 〉, on the other hand, we have only one primitiveattribute and no constraints. Clearly, these two theories should count as equiva-lent under any sensible notion of equivalence. But if µ(µ′(b)) = b then µ′(b) = a,because µ takes non-primitive terms to non-primitive terms. Similarly µ′(c) = aand thus µ(µ′(c)) = b. So there is no isomorphism between these theories in Th.

In order to get an appropriate definition of equivalence between theories weneed to relax the notion of an inverse morphism to that of a quasi-inverse, whichmeans that the composite morphisms are not required to be identical but onlyto be equivalent to identity in the following sense:

22

Definition 6 (Equivalence of Translations). Let µ and ν be theory trans-lations from 〈Σ,Γ 〉 to 〈Σ ′, Γ ′〉. Then µ is equivalent to ν, notation: µ ∼ ν, iff∀p ∈ Σ (Γ ′ ` µ(p) ≡ ν(p)).

Definition 7 (Quasi-Inverse). Let µ be a theory translation from Γ to Γ ′. Atranslation ν from Γ ′ to Γ is quasi-inverse to µ iff ν ◦ µ ∼ ıΓ and µ ◦ ν ∼ ıΓ ′.

We say that Γ and Γ ′ are equivalent, notation: Γ ∼ Γ ′, if there is a translationfrom Γ and Γ ′ that has a quasi-inverse. The existence of a quasi-inverse turnsout be equivalent to the following two conditions:

Definition 8 (Conservative/Essentially Surjective). A translation µ fromΓ to Γ ′ is conservative iff, for all statements α over Σ, Γ ′ ` µ(α) only if Γ ` α;µ is essentially surjective iff ∀p ∈ Σ ′ ∃φ ∈ T [Σ] (Γ ′ ` p ≡ µ(φ)).

Proposition 4. A theory translation µ has a quasi-inverse iff µ is conservativeand essentially surjective.

Proof. Suppose ν is quasi-inverse to µ. Then Γ ′ ` φ′ ≡ µ(ν(φ′)), for every φ′ ∈ T [Σ′], which showsthat µ is essentially surjective. If Γ ′ ` µ(φ) ≡ µ(ψ) then Γ ` ν(µ(φ)) ≡ ν(µ(ψ)), because ν is amorphism. Moreover, Γ entails φ ≡ ν(µ(φ)) and ψ ≡ ν(µ(ψ)). Hence Γ ` φ ≡ ψ; so, µ is conservative.Conversely, assume µ is conservative and essentially surjective. Then, by the axiom of choice, one canchoose ν(p′) ∈ T [Σ], for every p′ ∈ Σ′, such that Γ ′ ` p′ ≡ µ(ν(p′)). Thus µ ◦ ν ∼ ıΓ ′ . In particular,it holds that Γ ′ ` µ(p) ≡ µ(ν(µ(p))), for every p ∈ Σ. Hence Γ ` p ≡ ν(µ(p)); that is, ν ◦ µ ∼ ιΓ .

Example 3. Let Σ be {a0, a1, . . . ak} ∪ {b0, b1, . . . bk}, with k finite, and let Γ bethe theory over Σ with statements an ≡ an+1∨bn+1 (0 6 n < k) and an∧bn ≡ Λ(0 6 n 6 k). Then C (Γ ) = {∅, {a0, a1, . . . , ak}}∪{{a0, a1, . . . , an−1, bn} |n 6 k}.So the information domain of Γ is flat and thus order-isomorphic to the informa-tion domain of the theory Γ ′ = {cm∧cn ≡ Λ |m 6= n} overΣ ′ = {c0, c1, . . . , ck+1}.Now consider the function µ from Σ to T [Σ ′] with µ(an) = cn+1 ∨ . . . ∨ ck+1

and µ(bn) = cn (0 6 n 6 k). Then Γ ′ ` µ(Γ ), that is, µ is a theory translationfrom Γ to Γ ′. Moreover, it is not difficult to see that the function ν from Σ ′ toT [Σ], with ν(ck+1) = ak and ν(cn) = bn for every n 6 k, is a translation fromΓ ′ to Γ which is quasi-inverse to µ. The case k = 1 is illustrated by Figure 2 interms of Lindenbaum algebras and information domains (decorated with extentsof primitives).

In the foregoing example, the equivalence translation µ from Γ to Γ ′ inducesan isomorphism from L(Γ ) to L(Γ ′). We shall see in the next section that thisholds in general. Moreover, finite theories turn out to be equivalent wheneverthey have order-isomorphic information domains.

Remark 1 (Theory Morphisms in Institution Theory). It is tempting to sub-sume our category Th under the framework of Institution Theory (see [12, 11]for background). The sets Σ of primitives can be taken as the signatures ofthe underlying institution, the sentences associated with Σ are our statements

23

L(Γ )1

b0 ∨ a0

a0 = a1 ∨ b1

b0 b1a1

0

L(Γ ′)1

c1 ∨ c2

c0 c1 c2

0

D(Γ )

b0 b1a1

a0

D(Γ ′)

c0 c1 c2

Fig. 2. The case k = 1 for the theory of Example 3

over Σ. A theory is then again a pair 〈Σ,Γ 〉, where Γ is a set of statementsover Σ. In order to get theory morphisms, we still need to specify the signaturemorphisms of the institution. But here a problem arises insofar as the standardapplication of institution theory to logic takes signature morphisms simply asfunctions from signature to signature. This would mean to restrict ourselves toprimitive translations, which is not adequate as it has been argued above. Apossible way out would be to define a signature morphism from Σ to Σ ′ asa function from Σ to T [Σ ′]. On the other hand, it seems to violate the basicdivision of labor in Institution Theory if term forming operations are alreadyinvolved at the level of signature morphisms.

3.2 Functors and Equivalences

Our primary goal is to study the information domain functor that takes theo-ries to their information domains. Recall from Section 2.2 that the informationdomain can be identified with the spectrum of the Lindenbaum algebra, whichallows us to factor the information domain functor into the Lindenbaum functorand the spectrum functor. The gain is that we can employ results from universalalgebra and lattice theory.

The Lindenbaum Functor. Let µ be a theory translation from Γ to Γ ′. ThenmΓ ′ ◦ µ is a model of Γ in L(Γ ′). Since Mod(Γ, L(Γ ′)) ' Hom(L(Γ ), L(Γ ′)), byProposition 2, µ gives rise to a homomorphism L(µ) from L(Γ ) to L(Γ ′) suchthat mΓ ′ ◦ µ = L(µ) ◦mΓ , and this assignment is functorial.

Proposition 5. Two theory translations µ and ν from Γ to Γ ′ are equivalentiff L(µ) = L(ν). In particular, L(Γ ) ' L(Γ ′) iff Γ ∼ Γ ′.

24

Proof. By definition, µ ∼ ν iff, for every φ ∈ T [Σ], µ(φ) ∼=Γ ′ ν(φ), that is, iff L(µ) ◦mΓ = L(ν) ◦mΓ .By Proposition 2, the latter condition implies that L(µ) = L(ν).

Proposition 6. A theory translation µ is conservative iff L(µ) is one-to-one;µ is essentially surjective iff L(µ) is onto.

Proof. (i) L(µ) is one-to-one iff ∼=Γ is the congruence kernel of mΓ ′ ◦ µ iff ∀φ, ψ ∈ T [Σ](µ(φ) ∼=Γ ′

µ(ψ) → φ ∼=Γ ψ) iff µ is conservative. (ii) L(µ) is onto iff bmΓ ′ ◦ µ is onto iff ∀φ′ ∈ T [Σ′]∃φ ∈T [Σ](φ′ ∼=Γ ′ µ(φ)) iff µ is essentially surjective.

Let L be the category of bounded distributive lattices viewed as algebras oftype 〈2, 2, 0, 0〉.

Proposition 7. The functor L from Th to L is full and every object A of L isof the form L(Γ ) for some object Γ of Th.

Proof. Let h be a homomorphism from L(Γ ) to L(Γ ′). For every p ∈ Σ choose µ(p) ∈ T [Σ′] suchthat [µ(p)]∼=Γ ′ = h([p]∼=Γ ). One easily shows that Γ ′ ` µ(Γ ). Hence µ is a translation with h = Γ (µ);so L is full. Moreover, A ' L(Th(A)).

By Proposition 5, ∼ is a congruence relation with respect to composition.Hence we can switch to the quotient category Th/∼ of Th by ∼, which hasthe same objects as Th whereas its morphisms are the equivalence classes ofmorphisms of Th modulo ∼. The functor L factors uniquely by the quotientfunctor from Th to Th/∼ and a faithful functor from Th/∼ to L (see [18, Sect.II.8]). Together with Proposition 7, we have:

Theorem 1. The categories Th/∼ and L are equivalent.

Let Thp be the subcategory of Th whose objects are those of Th and whosemorphisms are the primitive theory translations (cf. Definition 5). Thp corre-sponds to the standard category of presentations by generators and relations.Notice that Th (see Section 2.2) can be naturally extended to a functor from Lto Thp. The following fact is folklore (e.g. [20, pp. 182f]):

Proposition 8. The functor L from Thp to L is left adjoint to Th.

Remark 2 (Kleisli Construction). Another method to define an appropriate no-tion of theory morphism makes use of the so-called Kleisli construction:3 Theadjunction 〈L, Th〉 from Thp to L gives rise to the monad T = L◦Th on the cat-egory Thp, which in turn has the associated Kleisli category K(T ) whose objectsare those of Thp and whose morphisms from Γ to Γ ′ are the Thp-morphismsfrom Γ to Th(L(Γ ′)). Since there is a one-to-one correspondence between mor-phisms in K(T ) and equivalence classes of morphisms in Th, it follows that thecategories K(T ) and Th/∼ are equivalent. In particular, the Kleisli comparisonfunctor from K(T ) to L is an equivalence of categories (cf. [18, p. 144]).

3 See [15, p. 130f] for a similar use of the Kleisli category in the context of categorical logic.

25

The Spectrum Functor. Every algebra homomorphism h from A to B givesrise to a function P (h) from P (B) to P (A) such that P (h)(F ) = h−1(F ). Thefunction P (h) is Scott-continuous, i.e., P (h) is order-preserving and preservessuprema of directed sets. Thus P is a contravariant functor from L to the cate-gory Dcpo of directed-complete ordered sets (dcpos) and Scott-continuous func-tions. Notice that P is naturally isomorphic to the contravariant Hom-functorHom2; cf. Proposition 3. The following proposition reformulates a well-knowresult from the theory of distributive lattices (cf. [6, p. 265]).

Proposition 9. A homomorphism h of algebras is one-to-one iff P (h) is onto;h is onto iff P (h) is an order embedding.

The Information Domain Functor. By Proposition 1, the information do-main of a theory Γ is represented by Mod2(Γ ) (ordered pointwise). For eachtranslation µ from Γ to Γ ′, let Mod2(µ) be the function from Mod2(Γ

′) toMod2(Γ ) that takes m to m◦µ. We define “the” information domain functor Dto be any contravariant functor from Th to Dcpo that is naturally isomorphicto Mod2. In particular, Hom2 ◦ L and P ◦ L are information domain functors.Moreover, C can be extended to an information domain functor as well. Its effecton theory translations can be described as follows:

Proposition 10. Let µ be a morphism of theories from 〈Σ,Γ 〉 to 〈Σ ′, Γ ′〉 andsuppose Y ∈ C (Γ ′). Then C (µ)(Y ) = {p ∈ Σ | Y � µ(p)}. In case µ is anextension of theories then C (µ)(Y ) = Y ∩Σ.

According to Birkhoff’s representation theorem for finite distributive lattices([6]), the functor P is a dual equivalence between the category of finite algebrasand the category of finite ordered sets. Hence, by Theorem 1:

Proposition 11. The functor D induces a dual equivalence between the quotientcategory of finite theories modulo ∼ and the category of finite ordered sets.

In other words, the information domain of a finite theory (i.e., a theory withfinite Σ) represents the theory up to equivalence. For infinite Σ, however, thisis not necessarily the case:

Example 4. Consider the theory 〈Σ,Γ 〉 with Σ = {a0, a1, . . .} ∪ {b0, b1, . . .} andΓ = {an ∧ bn ≡ Λ, an ≡ an+1 ∨ bn+1 | n > 0}. Its information domain is shownon the left of Figure 3, with extents of primitives added. Since D(Γ ) is flat,it is isomorphic to the information domain of the theory 〈Σ ′, Γ ′〉, with Σ ′ ={c0, c1, . . .} ∪ {cω} and Γ ′ = {cm ∧ cn ≡ Λ | m 6= n}; see Figure 3. However, itcan be shown that L(Γ ) has a non-principal prime filter whereas all prime filtersof L(Γ ′) are principal. So L(Γ ) is not isomorphic to L(Γ ′) and thus Γ is notequivalent to Γ ′, by Proposition 5. Compare this result with Example 3.

26

D(Γ )

. . .

b0 b1 b2

a0

a1

D(Γ ′)

. . .

c0 c1 c2 cω

Fig. 3. Non-equivalent theories with isomorphic information domains

Remark 3. The dual equivalence of Proposition 11 can be generalized to thecategory of finitistic theories ([23, 22]) and the category of coherent algebraicdomains with continuous functions restricted appropriately; see also [28, 1].

We conclude this section by combining Propositions 6 and 9:

Proposition 12. A translation µ of theories is conservative iff D(µ) is onto; µis essentially surjective iff D(µ) is an order embedding.

4 Constructions

4.1 Quasi-Colimits

Many universal constructions in category theory can be seen as limits or colimits.Colimits, for instance, cover coproducts, pushouts, and inductive limits, besidesothers. Since in Th, we are more interested in characterizing theories up toequivalence than up to isomorphism, the notion of a quasi-colimit proves morefruitful than that of a colimit.

Let G be directed graph with vertex set I, edge set E, and two functionss and t from E to I, where s(e) and t(e) are respectively source and targetof e ∈ E. A diagram D of shape G in a category C is a pair consisting of afamily 〈Di〉i∈I of C-objects and a family 〈fe〉e∈E of C-morphisms such that fe

is a morphism from Ds(e) to Dt(e).

Definition 9 (Quasi-Cocone/-Colimit). A quasi-cocone 〈Γ, 〈µi〉i〉 of a dia-gram 〈〈Γi〉i∈I , 〈µe〉e∈E〉 of theories is a theory Γ and a family of translations µi

from Γi to Γ such that µs(e) ∼ µt(e) ◦ µe. A quasi-cocone 〈Γ, 〈µi〉i〉 is a quasi-colimit if for every other quasi-cocone 〈Γ ′, 〈µ′i〉i〉 of the diagram there is a trans-lation µ from Γ to Γ ′, which is unique up to equivalence, such that µ′i ∼ µ ◦ µi.

Quasi-Colimits can be shown to exist for all diagrams in Th by a fairlystandard construction. In the following, let ιi be the canonical injection of Σi

into the disjoint union⊎

iΣi of a family 〈Σi〉i∈I of sets.

Proposition 13. If 〈〈〈Σi, Γi〉〉i∈I , 〈µe〉e∈E〉 is a diagram of theories, the theory⋃i ιi(Γi) ∪ {ιs(e)(p) ≡ ιt(e)(µe(p)) | e ∈ E ∧ p ∈ Σs(e)}

27

over⊎

iΣi, together with the family 〈ιi〉i of injections, is a quasi-colimit of thediagram.

Proof. Let 〈Σ,Γ 〉 be the theory defined in the proposition. Clearly 〈Γ, 〈ιi〉i〉 is a quasi-cocone since,by definition, ιs(e) ∼ ιt(e) ◦ µe. Suppose 〈〈Σ′, Γ ′〉, 〈µi〉i〉 is a quasi-cocone of the given diagram, i.e.µs(e) ∼ µt(e) ◦ µe for all e ∈ E. We claim that there is a translation µ from Γ to Γ ′ with µi = µ ◦ ιi.Since every element of

Ui Σi is of the form ιi(p), where i and p ∈ Σi are uniquely determined, the

only function µ fromU

i Σi to T [Σ] satisfying the desired condition takes ιi(p) to µi(p). Then µ is atranslation, since Γ ′ entails µi(α) and thus µ(ιi(α)), for all α ∈ Γi. This leaves us to check that Γ ′

entails µ(ιs(e)(p)) ≡ µ(ιt(e)(µe(p))), i.e. µs(e)(p) ≡ µt(e)(µe(p)), for every p ∈ Σs(e). But this is justthe assumption that µs(e) ∼ µt(e) ◦µe; so µ is a translation from Γ to Γ ′. It remains to show that if νis another translation from Γ to Γ ′, with µi ∼ ν ◦ ιi, then µ ∼ ν. Since µ ◦ ιi = µi ∼ ν ◦ ιi, it followsthat Γ ′ ` µ(ιi(p)) ≡ ν(ιi(p)) for all p ∈ Σi. Hence µ ∼ ν, by definition.

Example 5 (Coproducts). Let 〈〈Σi, Γi〉〉i∈I be a family of theories and let⊎

i Γi

be 〈⊎

iΣi,⋃

i ιi(Γi)〉. Then 〈⊎

i Γi, 〈ιi〉i〉 is a coproduct of 〈〈Σi, Γi〉〉i∈I in Th.

Since, by definition, the quotient functor from Th to Th/∼ takes quasi-colimits to colimits, we have by Theorem 1:

Proposition 14. The functor L takes quasi-colimits in Th to colimits in L.

By Proposition 8, the restriction of L to a functor from Thp to L has a rightadjoint and hence preserves colimits ([18, Sect. V.5]). Colimits need not exist forall diagrams in Thp (because Thp has not enough morphisms). If, however, thecolimit of a diagram in Thp exists then it is also a quasi-colimit of that diagramin Th.

Proposition 15. Colimits in Thp are quasi-colimits in Th.

Proof. Suppose a diagram D in Thp has a colimit 〈Γ, 〈µi〉i〉 in Thp. Then 〈L(Γ ), 〈L(µi)〉i〉 is acolimit of the diagram L(D) in L. By Proposition 13, D has a quasi-colimit 〈Γ ′, 〈µ′i〉i〉 in Th, which,by Proposition 14, is taken to another colimit of L(D). Hence L(Γ ) ' L(Γ ′) and thus Γ ∼ Γ ′, byProposition 5. So 〈Γ, 〈µi〉i〉 is a quasi-colimit of D in Th.

4.2 The Information Domain in the Limit

We now show that the information domain functor takes quasi-colimits in Thto limits in Dcpo. In other words, the information domain of the quasi-colimitof a diagram of theories is the limit of the corresponding diagram of informationdomains. Limits in Dcpo can be constructed as canonical limits in Set, i.e., assubsets of Cartesian products, with elements ordered coordinatewise (cf. [1, p.45]). The following lemma gives us a criterion in which cases a limit of a diagramof dcpos (under the forgetful functor) in Set is actually a limit in Dcpo.4

Lemma 1. Suppose 〈D, 〈fi〉i∈I〉 is a cone of a diagram D in Dcpo which istaken to a limit cone of D in Set by the forgetful functor. Then the cone is alimit of D in Dcpo iff, for every two x, y ∈ D, it holds that x 6 y wheneverfi(x) 6 fi(y) for all i ∈ I.4 The argument is essentially the same as that used in [16, p. 249].

28

Proof. Let 〈D′, 〈pi〉i〉 be the canonical limit cone in Dcpo of D. Then there is a unique Scott-continuous function f from D to D′ such that fi = pi ◦ f . By assumption, f is one-to-one because〈D′, 〈pi〉i〉, under the forgetful functor, is a limit cone of D in Set. Since f is Scott-continuous andone-to-one, it is enough to check that f is an order embedding to make sure that f is an isomorphismin Dcpo. Suppose f(x) 6 f(y), for x, y ∈ D. Then fi(x) = pi(f(x)) 6 pi(f(y)) = fi(y) for every i,since f and pi are order preserving. Hence x 6 y, by assumption.

Theorem 2. The functor Hom2 takes colimits in L to limits in Dcpo.

Proof. Let 〈A, 〈hi〉i〉 be a colimit of a diagram in L. Hom2, as a contravariant Hom-functor, takescolimits in L to limits in Set. Suppose v, w ∈ Hom2(A) with Hom2(hi)(v) 6 Hom2(hi)(w) for all i.Then v(hi(a)) 6 w(hi(a)) for all i and a ∈ Ai. Since A is generated by

S{hi(Ai) | i ∈ I}, it follows

inductively that v(a) 6 w(a) for every a ∈ A; so v 6 w. Now apply Lemma 1.

Corollary 1. The functor D takes quasi-colimits in Th to limits in Dcpo.

Inductive Limits of Extensions. Let I be a directed ordered set and suppose〈〈Σi, Γi〉〉i∈I is an inductive system of theory extensions over I with extensionsεij, for i 6 j. Then the theory

⋃i Γi over

⋃iΣi together with the extensions εi

from Γi to⋃

i Γi is a quasi-cocone of the inductive system.

Proposition 16. Let 〈Γi〉i be an inductive system of theory extensions. Then〈⋃

i Γi, 〈εi〉i〉 is an inductive quasi-limit of that system.

Proof. According to Proposition 15, it suffices to show that 〈S

i Γi, 〈εi〉i〉 is an inductive limit of〈Γi〉i in Thp. Suppose there are primitive translations µi from 〈Σi, Γi〉 to a theory 〈Σ,Γ 〉 such thatµi = µj ◦ εij for i 6 j. Let µ be the function from

Si Σi to Σ that takes p ∈ Σi to µi(p); µ is well

defined because if p ∈ Σi ∩Σj , there is a k with i, j 6 k and thus p ∈ Σk; so µi(p) = µk(p) = µj(p).Notice that µ is the only function satisfying µi = µ ◦ εi for all i. It remains to check that µ is atranslation, i.e. that Γ ` µ(α) for every α ∈

Si Γi. But if α ∈ Γi then µ(α) = µi(α), and µi is a

translation from Γi to Γ .

This result has the following straightforward but useful application. Sup-pose 〈Σ,Γ 〉 is a theory. For S ⊆ Σ let Γ |S be the set of all statements of Γwhose primitives belong to S. Let F be the directed set of finite subsets of Σ.Then the family 〈〈F, Γ |F 〉〉F∈F together with the extensions from Γ |F to Γ |F ′ ,whenever F ⊆ F ′, is an inductive system, whose inductive limit is 〈Σ,Γ 〉, byProposition 16. Therefore:

Corollary 2. Every theory is an inductive quasi-limit of finite theories.

Suppose Σ is countable, i.e. Σ = {p0, p1, p2, . . .}. Let Σi be {p0, p1, . . . , pi}and Γi be Γ |Σi

. Then 〈Σ,Γ 〉 is the inductive limit of the inductive system〈〈Σi, Γi〉〉i∈ω, with extensions from Γi to Γj for all i 6 j. Hence:

Corollary 3. Every theory over a countable set of primitives is an inductivequasi-limit of a sequence of finite theories.

Another consequence, in combination with Proposition 11, is Speed’s [27]characterization of spectra as profinite ordered sets, where profinite means tobe the projective limit of a projective system of finite ordered sets:

Corollary 4. Every information domain is profinite and every profinite orderedset is an information domain.

29

∅

{c}

×

∅

{d}' ;

∅

{c} {d}

{c, d}

∅

{c} {d}

∅

{c} {d}

×

∅

{a}' ;

∅

{c} {d} {a}

{a, c} {a, d}

∅

{a} {d}

{a, c} {a, d}

∅

{a} {d}

{a, c} {a, d}

×

∅

{b}' ;

∅

{a} {d}

{a, c} {a, d}

{b}

{a, b} {b, d}

{a, b, c} {a, b, d}

∅

{a} {b}

{a, c}

{a, b, c} {a, b, d}

Fig. 4. Step-by-step construction of the information domain of Example 1

4.3 Applications

A Generic Construction Scheme. Suppose 〈Γ,Σ〉 is a finite theory. LetΣ0, Σ1, . . . , Σn be a strictly increasing sequence of sets, with Σ0 = ∅ and Σn =Σ, and let Γi be Γ |Σi

. The extension from 〈Σi, Γi〉 to 〈Σi+1, Γi+1〉 factors throughan extension from 〈Σi, Γi〉 to 〈Σi+1, Γi〉 and one from 〈Σi+1, Γi〉 to 〈Σi+1, Γi+1〉.Since 〈Σi+1, Γi〉 = 〈Σi, Γi〉 ] 〈Σi+1 \Σi,∅〉, we get C (Γi+1) by deleting fromC (Γi) × ℘(Σi+1 \ Σi) all elements that are not consistently closed with respectto Γi+1 \ Γi.

Example 6. Consider the theory Γ over Σ = {a, b, c, d} introduced in Example 1.Let Σ1 = {c}, Σ2 = {c, d}, Σ3 = {a, c, d}, and Σ4 = Σ. Then Γ1 = ∅, Γ2 \Γ1 ={c ∧ d � Λ}, Γ3 \ Γ2 = {c � a}, and Γ4 \ Γ3 = {a ∧ b � c ∨ d, d � a ∧ b}. Theconstruction of C (Γi+1) from C (Γi) is depicted by the i-th row of Figure 4, wherethe framed elements of C (Γi) × ℘(Σi+1 \ Σi) are not consistently closed withrespect to Γi+1 \ Γi and thus subject to deletion.

Figure 5 presents an algorithmic formulation of this generic constructionscheme, with F and Σ ′ for Σi+1 \ Σi and Σi+1, respectively. Notice that thealgorithm says nothing about how to choose F . The proper choice of the par-tition of Σ into the sets Σi+1 \ Σi is of course essential for keeping |C (Γi)|small during the construction process, since calculating C (Γi+1) requires to check|C (Γi)| · 2|Σi+1\Σi| sets against |Γi+1 \ Γi| statements. To develop good heuristicsfor choosing such a partition is part of future research.

30

function C (Σ: set; Γ : theory): system of sets

if not cc? (∅, Γ |∅) then C := ∅else

C := {∅}; Σ′ := ∅while Σ 6= ∅ and C 6= ∅ do

F := any nonempty subset of ΣΣ′ := Σ′ ∪ F ; Γ ′ := Γ |Σ′ ; C′ := ∅foreach X ∈ C, Y ⊆ F do

X ′ := X ∪ Yif cc? (X ′, Γ ′) then C′ := C′ ∪ {X ′}

Σ := Σ \ F ; Γ := Γ \ Γ ′; C := C′

function cc? (X: set; Γ : theory): boolean{ true if X is consistently Γ -closed, false otherwise }foreach (φ � ψ) ∈ Γ do

if X � φ and X 2 ψ then return (false)return (true)

Fig. 5. A generic algorithm for constructing information domains

Merging. Suppose a given domain of discourse is described by two theories〈Γ1, Σ1〉 and 〈Γ2, Σ2〉 whose vocabulary turns out to be partly equivalent in thesense that a set ∆ of statements of the form φ1 ≡ φ2 is held to be true, withφ1 ∈ T [Σ1] and φ2 ∈ T [Σ2]. For ease of exposition, let us assume that Σ1 and Σ2

are disjoint. Merging these two theories in a way that respects the equivalencesin ∆ then simply means to take the theory Γ1 q∆ Γ2 = 〈Σ1 ∪Σ2, Γ1 ∪ Γ2 ∪∆〉.Now let Σ∆ be the set of all pairs 〈φ1, φ2〉 such that φ1 ≡ φ2 belongs to ∆and let µ1, µ2 be the projections from Σ∆ to T [Σ1] and T [Σ2], respectively.Then Γ1 q∆ Γ2 is the (quasi-)pushout of µ1, µ2 seen as theory morphisms from〈Σ∆,∅〉 to 〈Σ1, Γ1〉 and 〈Σ2, Γ2〉, respectively. By Corollary 1, it follows thatthe information domain of Γ1 q∆ Γ2 is the pullback of D(µ1), D(µ2).

Example 7. Let Γ1 be the theory {a ∧ b � Λ, a � c} over Σ1 = {a, b, c}, let Γ2 bethe theory {d ∧ e � Λ} overΣ2 = {d, e}, and suppose∆ = {b ∧ c ≡ d, b ≡ d ∨ e}.Figure 6 shows the information domain of Γ1 q∆ Γ2 as the pullback of C (µ1),C (µ2), where µ1 and µ2 are as introduced before and ι1, ι2 are the canonical ex-tensions. In order to determine the effect of C (µi) observe that C (µi)(X) ={〈φ1, φ2〉 ∈ Σ∆ | X � φi}, by definition of µi and Proposition 10. So C (µ1)takes ∅, {c}, and {a, c} to ∅, whereas {b} is taken to {〈b, d ∨ e〉} and {b, c}to {〈b ∧ c, d〉, 〈b, d ∨ e〉}. And C (µ2) takes ∅ to ∅, {d} to {〈b ∧ c, d〉, 〈b, d ∨ e〉},and {e} to {〈b, d ∨ e〉}.

Remark 4. Since the information domain of a theory can be viewed as its genericontology ([22]), the described construction can be regarded as a case of ontologymerging via pullbacks. In [17], in contrast, it is argued that ontology merg-ing is best captured by pushouts. Although a thorough discussion of this issueis beyond the scope of the present paper, a few brief remarks may be in or-

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C (Γ1 q∆ Γ2)

∅

{b, e}{c}

{b, c, d}{a, c}

C (Γ1)

∅

{b}{c}

{b, c}{a, c}

C (Γ2)

∅

{d} {e}

∅

{〈b ∧ c, d〉} {〈b, d ∨ e〉}

{〈b ∧ c, d〉,〈b, d ∨ e〉}

C (ι1) C (ι2)

C (µ1) C (µ2)

Fig. 6. Pullback of C(µ1), C(µ2)

der. Consider the base case of merging two ontological or conceptual hierarchieswithout any equivalence constraints. Then the pullback construction leads to thedirect product whereas pushouts are given as disjoint unions, i.e., coproducts.If, for instance, a typical zoological taxonomy of vertebrates is merged with oneof birds, the disjoint union would probably be the first choice (maybe extendedby an additional most general concept). Now notice that this construction im-plicitly takes the concepts of the two taxonomies as pairwise incompatible. If,however, concepts are conceived as sets of attributes, as in the present paper,then combining attributes from different conceptual hierarchies should be pos-sible without necessarily requiring them to be equivalent. In fact, our mergingapproach can be easily extended by allowing explicit incompatibility constraints.

5 Conclusion

We have presented a natural and explicit notion of translation between classifi-cations, seen as theories, over different base vocabularies. The resulting categoryof theories, together with the information domain functor, provides a nice cat-egorical framework for studying the effect a translation of classifications hason the associated conceptual hierarchies as well as for constructing conceptualhierarchies by means of categorical constructions on the level of theories.

A major topic for future research would be an appropriate conception oftranslation, and thus of morphism, for more expressive logical frameworks, espe-cially those involving attributive descriptions, like description logics and featurelogics ([26, 21]).

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15. Bart Jacobs. Categorical Logic and Type Theory. North-Holland, Amsterdam, 1999.16. Peter T. Johnstone. Stone Spaces. Cambridge University Press, Cambridge, 1982.17. Markus Krotzsch, Pascal Hitzler, Marc Ehrig, and York Sure. What is ontology merging? In

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Ontology-based understanding of scientific natural language texts

Winfried Schmitz-Esser

University of Applied Sciences, Hamburg

Abstract: Foundational aspects of an onomasiological approach in constructing and maintaining terminology-based ontologies are discussed, as well as a transfer oriented mode in establishing cross-language implementation. Starting out from the concepts rather than from lexical primitives enables best choices as to the representational quality of the ontology while remaining open to any accepted linguistic way of expression encountered in the different languages. Means of how to ease usual restrictions as to expressiveness, consensuality, and conceptual orientation are focussed on, as well as questions of reliability, diachronic stability, and editorial engagement.

Introduction Few linguistic ontologies have been built so far on the basis of grammatical units that are bigger than words, and where pluri-lingual ontologies exist, they usually focus on a one-to-one mapping of small lexical units [1]. In knowledge modelling, this state of the art sets narrow boundaries as to expressiveness, consensuality, reuse, and knowledge sharing and considerably reduces an ontology’s potential of grasping the desired meaning in real world natural language texts. An onomasiological approach in ontology construction, update and maintenance might be the answer to such problems, combined with a transfer approach on cross-language implementations. The layout and logical structure of the paper presented is shown in Fig.1.

1. What a software tool (or human interpreter) needs in order to understand natural language text 2. How the ontology must look like to allow statements (knowledge) to be modelled and formalized

2.1.Arguments 2.1.1.Concepts 2.1.2.Instances 2.1.3.Foundational aspects

2.2 Semantic relations 2.2.1. Foundational aspects

3. Modelling, and the functions of models 4. Foundations and principles of stipulating semantic statements

Fig. 1. Layout and logical structure of paper presented 1. What a software tool (or human interpreter) needs in order to understand natural language text Assuming that some knowledge machine is capable of understanding scientific natural language text by means of a knowledge base constructed as an ontology, then what should we expect to be the foundations of such an ontology, and what are its principles of construction, update, and maintenance?

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Spontaneously, one might prompt: Well, in the first instance, the ontology must know the specific language that is used in common communication between the members of a community on subjects of interest relating to a specific domain. This seems simple, but it is not simple at all. Every single element in the equation is soft. What is a <specific domain>? Evidently, domains exist. But what are the frontiers? What is a <domain’s specific language> as opposed to the specific languages of other domains, and how does this relate to those parts of some general language, authors would use anyway in their natural language texts? What are <subjects of interest> in a given domain, to-day, yesterday? And what may they come to be tomorrow? What is <common communication>? And what does it mean that somebody, or, in this case, a machine, < must know the language>? Obviously, this is not a matter concerned solely with dictionaries. And it is not a matter of just a few search words on the Internet. We have to plunge into the deeper waters of terminological communication [2] [3]. To understand the meaning of a message expressed in terms of a natural language (NL), a thorough knowledge of the respective World segment, what it is about, and how it works, is required on the part of the receiver, as well as an equally thorough knowledge of how all this usually is, or may be, addressed in terms of the respective NL. Much of this knowledge will then consist of tacit knowledge that comes from experience, observation, reasoning, inference, etc. If we want a machine to reason and to understand the semantics of NL text, then we have to provide it with the knowledge required for that purpose. We have to do it by means of our natural language, and in an utterly explicit way [4]. This is what occurs in an ontology of our times. Such an ontology explicitly describes, or models, the knowledge with respect to a specific segment of the world. A suitable, machine-readable format, mechanism and repository are required for that purpose. Since the condition is natural language, such modelling will find its limits within the possibilitiy of being expressed by means of natural language. We are on the well articulated, sometimes ambiguous, but practical grounds of first order logic, proven in centuries. In models of this kind, we are not talking about variables with their much higher degree of exactness. The main condition is conceptualization. It is not possible to describe, or model some piece of knowledge by means of NL without having the possibility to express it by means of concepts that we expect to be known by other members of the language community. This touches the focal question of the representational capacity of the spoken language. In an ontology conceived as a model of some section of the world, no authentic knowledge modelling is possible unless full, unequivocal representation of the meaning in terms of NL can be assured [5]. I do not want to enter the discussion about terminological representation here. My view in this dispute is a constructivist [6] view. Yes, such representation is possible, within, however, the limits of some serious constraints and conditions.

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2. How the ontology must look like to allow statements (knowledge) to be modelled and formalized Let us assume that the ontologic model can be based on a binary semantic relation of the type Argument 1 - Relational Operator - Argument 2 The relational operator then is normally presented in two forms, as RelOp(a) and RelOp(b), expressing which of the two arguments stands on the active side and which one is on the passive side. In NL texts of the different single languages, however, the semantics of such a statement may be encountered as expressed in various other ways, e.g. as nouns or integrated as parts of nouns, or as modifiers to nouns or verbs. This is a major part of the problem (Fig. 2). short-circuit - Cause(s) - black-out black-out - Is/AreCausedBy - short-circuit In English, equivalent expressions of the argument may range from “short circuit” to totally different constructs like: “induction”, or “lower resistance electric connection”. Likewise, a relation saying that something is, or may be “CausedBy” another something, is equivalent to a number of other expressions in English NL texts, like “brings about”, “effectuates”, “leads to”, etc. This, still, is not exhaustive. Expressions may appear in other syntactic forms like: “a short-cut induced black-out”, or “thanks to a short-cut in our old transformer station we had a black-out yesterday”. Fig. 2. A statement (enouncement formalized) Having said this, and searching for the foundations of such an ontologic model, let us take a closer look on this kind of enouncements, and of its constituent elements, (1) the arguments and (2) the relations. 2.1.1 Arguments - Concepts Maybe it is true “that languages affect the way we conceptualize the world”, as Hjörland. B. et Nicolaisen J. (2004) state [7]. Differences in word meanings among the different languages, (like wood [English] - Holz [German] - trae [Danish] -bois [French] etc.) are quoted here as obstacles to safe inter-language communication. This however holds true only on the assumption that conceptualized communication functions on the basis of what in early information retrieval was called a uniterm basis. But this is far from what we see in reality. A string of terms is much more precise than a uniterm. Besides, a string of words usually makes a much better discriminator in searches. To model our conception of the world, we have to enlarge our conception of a concept. To take the above example: The single word Holz is considered a concept, it is a uniterm with its particular, valid meaning only in German. We cannot expect a uniterm with this same meaning to exist in English as well, or in Danish. When talking of concepts, we have to separate the idea of what we mean - the essence of the concept -, from the word or words that express it, in a particular language. Such expressions may be manifold for a given object. What holds us back of treating any universal idea, or topic, simple or complex as it may be, as a concept in our ontology, irrespective of how it may be expressed in a particular language,

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and irrespective of the number of single words of this language that may be in use to express its meaning? The only limits I see is practical feasibility in the face of an exploding complexity, and limited resources needed for an immense task. But that may change as the usefulness of ontologic modelling becomes more obvious, and the need for better ontologies arises. The key to a valid ontology lies in how we construct it. We have to do it the onomasiologic way, as linguists are used to say. We do not start out from a given term asking for what it stands (the semasiologic way), but rather from the idea of what we want to express, of what we find expressed in a text. Then we have to ask how we find this particular idea/topic/universal expressed in other heterogeneous, valid, textual sources. This is the onomasiologic way. In a recent critical analysis of the 17,000 concepts strong Thesaurus of the National Cancer Institute (NCI), which is the achievement of a spectacular attempt to homogenize different terminological resources, Ceusters, W. et al. (2005) [8] found many mistakes and inconsistencies, denouncing weaknesses and deficiencies that are seemingly unavoidable (and difficult to correct) in this and other semasiologic approaches. The onomasiologic approach, in contrast, forces us to find the many variants of expressions used in communication for a given subject, including the most used paraphrases. As we find them in authoritative source texts, we see them as Authoritative Expressions (AE) that convey the sense. So, we are well advised to include them in the ontology. In an Equivalence Chain of Expressions (ECE) yielded from such a procedure we can separate those AEs which are (A) clear, univoque expressions for the subject from those (B) with other meanings (polysems in a wider sense). Among the univoque expressions of the A-type, the curator/peer should then be able to stipulate the most fitting one to function as a proxy for the other univoke A-types which in my papers [18] are named “Additional Access Expressions - (AAE)”. The proxy here is made up as a descriptor (DESC) very much along the traditional lines. In the ontologic model, the DESC represents the meaning. It is singular as ID, univoque and context-independent. The polysemic B-types in the ECE are the MULTIs (Fig. 3). They can be handled and used in other, appropriate ways. In source texts used for an ontology on public finances the following expressions for “public budget deficit” were encountered: 1) “public deficit“”, (2) “scarce public means”, (3)“hole in the federal budget”, (4) “public deficit for a central government in a country”, (5) “lack of public money””, (6) “federal budget deficit”, (7) “gap in the federal budget”, (8) “state budget deficit”, (9) “budget gap” (10) “gap in the budget”, (11) public budget deficit The curator/peer chose/grouped/ stipulated: (4) as DESC (1), (3), (6), (7), (11) as AAEs (2), (5), (9), (10) as MULTIs, if at all (8) in a special way Fig. 3. Three different types of elements in an Equivalence Chain of Expressions (ECE) To the practitioner it is no surprise that perfect identity between any two words in a language is rare, and that on a level of phrases it is almost non-existent. Margotti, F. W. (2004) [9], in a

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recent study, demonstrates the variety of algorithms modern languages offer to construct possible paraphrases, and that “the choice of one or the other paraphrase structure is never aleatoric”. While identity of meaning is the exception, Margotti states, identity is possible in the sense of equivalence, “and equivalence is always gradual” (ibid). If this is true, the occurrence of paraphrases for a given utterance in heterogeneous, valid source texts has something to say. We can use it in a statistical sense in ontology construction. We would have to accept the phrases, or chunks of phrases, as they come provided they occur in authoritative texts. Equivalences within an argument encountered in English, French, and German authoritative texts, could then be formalized for use as arguments in the ontology the following way. (Fig. 4), presented in a redundant format, (MULTIs suppressed) 202, 0111, DESC, EN, breast cancer cell proliferation 202, 0111, AAE, EN, proliferating breast cancer cells 202, 0111, AAE, EN, motility of breast cancer cells 202, 0111, AAE, EN, proliferation of breast cancer 202, 0111, AAE, EN, breast cancer spreading 202, 0111, AAE, EN, breast cancer cells migrating 202, 0111, DESC, FR, prolifération de cellules cancereuses mammaires 202, 0111, AAE, FR, prolifération de la tumeur mammaire 202, 0111, AAE, FR, prolifération de cellules de la tumeur mammaire 202, 0111, AAE, FR, prolifération du cancer mammaire 202, 0111, AAE, FR, motilité du cancer mammaire 202, 0111, AAE, FR, prolifération de cellules de cancer du sein 202, 0111, AAE, FR, prolifération de cellules des tumeurs du sein 202. 0111, AAE, FR, motilité des cellules des tumeurs du sein 202, 0111, AAE, FR, prolifération des tumeurs du sein 202, 0111, DESC, GE, Proliferation von Brustkrebszellen 202, 0111, AAE, GE, Brustkrebsmotilität 202, 0111, AAE, GE, Motilität von Brustkrebszellen 202, 0111, AAE, GE, Proliferation der Brustkrebszellen 202, 0111, AAE, GE, Ausbreitung der Brustkrebszellen 202, 0111, AAE, GE, Brustkrebszellen wandern 202, 0111, AAE, GE, Entstehung neuer Brustkrebsherde 202, 0111, AAE, GE, Zellen des Mammakarzinoms verbreitet Fig. 4. An example: Argument No. 202, an English, French, German set ------------ Legend 202 = individual ID number of what is meant in the given ontologic model, whereby the first digit indicates the quality of the meaning as a (universally understandable) concept (as opposed, e.g. to an instantiation) 0111 = in humans (01), in vivo (11) DESC = Descriptor, preferential expression in an Equivalence Chain of Expressions in a specific language AAE = Additional Access Expression, equivalent in meaning to the DESC in a specific language EN = Expression used in English NL texts FR = Expression used in French NL texts GE = Expression used in German NL texts Last element in line = NL expression as found to actually occur in a valid, scientific NL text, unabridged, full forms Set in a Boolean OR position, the NL fragments of each of the three language sets can be used for effective browsing in the respective NL text collections.

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2.1.2 Arguments - Instances With some minor modifications, the underlying construction can be used to build instantiations, i. e. descriptions of instances (individuals) which are phenomena singular in space and time: bodies by law or agreement, trade marks, events, the meteorologic highs and lows, typhoons, in short: unique cases of a class/category of universals, which in normal life all are given an individual name. Like concepts, the names of instances obey the linguistic laws of their respective languages. 2.1.3 Arguments - Foundational Aspects To find out a multiple language set of the type shown above is more than a good translator’s job. Both first class domain knowledge and an intimate command of the languages implemented on the ontology are required. The rules are those of a researcher and an interpreter combined, and the method is the one known among translators as “Transfer”, or “transposition”. This fits nicely into the onomasiological conception of the approach. The choice of scientific texts that contain the knowledge for the knowledge base is subject to dispute in no other way than this is usual in traditional scientific reading and processing. The same applies to the choice of the single arguments (universals/concepts/categories and instantiations/individuals). Enforcement by standard procedure should make sure that the extraction and transfer process is fully documented on all stages. So, what are the foundations of such identification and transfer?

• Pragmatism, skill, and editor’s responsibility. The editor of the ontology is responsible for the work of his team of curators/peers. It is their task to keep up the ontology, to control the choice of concepts and instances stipulated, to keep the ontology clear of idiosyncrasies, to warrant that the relevant part of the knowledge is modelled and organized in the ontology, and to execute this in conformity with the principles of good practice in the handling of terminology [10]. Any shades of meaning encountered as an equivalent or near to equivalent of the DESC should be considered a valid paraphrase as long as the meaning remains true (ILARI, R. e GERALDI, W. 1985) [11]. Pragmatism prevails in the handling of words borrowed from other languages: So, EN expressions found in German source texts will be labelled as GE expressions.

• Linguistics and Representation. Any full, unabridged, unaltered term string used by

an author of a relevant source text to describe a relevant subject in a particular NL is considered an Authoritative Expression (AE). All AEs encountered are to be transferred 1:1 and included in their respective argument sets. Each act of transfer is to be controlled and fully documented in a scientific way. Within the Ontology, no bias or limitation is admitted from any lingware or other linguistic inference.

• Probability. Probabilistic approaches are admitted insofar as (1) the representational

aspect of the sample source text collection needed in the beginning is concerned, and/or (2) the validity of the choice of current, new source texts needed for update, enhancement and extension of the ontology. Also (3) occurrence of structural forms

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encountered in the source texts. No probabilistic approaches admitted on matters of meaning.

2.2 Semantic Relations The semantic relations in focus are “Essential Relations” i. e. relations that exist on an inter-language, universal level [12]. They can be expressed in any language. As to their meaning, they are independent of a particular language, like “Cause/Effect” in the sample above. Such relations may be of a general type like the 13 relations proposed for general use in the nineties by the German Committee on Classification and Thesaurus Research - KTF [13]. Others must be more specific to meet a corresponding need, such as in “Insulin-like growth factors IBFs Regulate(s) breast cancer cell proliferation”. As an example, the definition of such a specific, universal relation is given in Fig. 5, plus an insight on the respective semantic/linguistic toolbox: Fig. 5.1. - What the Semantic Relation “Enhancement” means RelClass2-Type 33: Enhancement

In OntoStatements, the enhancing relationship expresses that Argument A which is the enhancer, enhances, or may enhance, Argument B which is the enhanced. Roles of Arguments: A is the Enhancer B is the Enhanced Meaning of enhance as modelled in this Ontology: The enhancer (A) enhances, or may enhance, the enhanced (B), makes B greater as in value, activity, performance, desirability, but also as in negative values like fear, budget deficit, morbidity.

Fig. 5.2. - How the Semantic Relation is expressed in the different languages >English NL Relational Expressions English NL expressions for RelClass2-Type 33: Enhancement to be activated for searches in English NL text collections: Phrases**A enhance(s) Phrases B Phrases A activate(s) Phrases B Phrases A increase(s) Phrases B Phrases A accelerate(s) Phrases B Phrases A intensify/intensifies Phrases B Phrases A promote(s) Phrases B (and all reverse) Open for inclusion of more candidates when encountered in valid, real world texts ------ **)Phrases are either (1) given, defined, “known” NL denominations of the argument (universals or instances) inferred from the Ontology, or (2) searched, “unknown” NL character strings of variable length expected to occur in NL text corpora when searching.< >French NL Relational Expressions French NL expressions for RelClass2-Type 33: Enhancement to be activated for searches in French NL text collections: Phrases A augmente(nt) Phrases B Phrases A agrandit/agrandissent Phrases B Phrases A renforce(nt) Phrases B Phrases A fait hausser/font hausser Phrases B Phrases A stimule(nt) Phrases B Phrases A intensifie(nt) Phrases B Phrases A accellère(nt) Phrases B (and all reverse) Open for inclusion of more candidates when encountered in valid, real world texts<

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>German NL Relational Expressions German NL expressions for RelClass2-Type 33: Enhancement to be activated in searches in German NL text collections: Phrases A erhöht/erhöhen Phrases B Phrases A vergrößert/vergrößern Phrases B Phrases A verstärkt/verstärken Phrases B Phrases A intensiviert/intensivieren Phrases B Phrases A beschleunigt/beschleunigen Phrases B Phrases A stimuliert/stimulieren Phrases B (and all reverse) Open for inclusion of more candidates when encountered in valid, real world texts< Fig. 5.3. - How Semantic Relations inter-relate

As to: RelClass2-Type 8: Beneficial RelClass2-Type 33: Enhancement Is/AreNarrowerConceptOf RelClass2-Type 8: Beneficial RelClass2-Type 8: Beneficial Is/AreBroaderConceptOf RelClass2-Type 33; Enhancement As to: RelClass2-Type 9: Detrimental RelClass2-Type 33: Enhancement Is/AreNarrowerConceptOf RelClass2-Type 9: Detrimental RelClass2-Type 9: Detrimental Is/AreBroaderConceptOf RelClass2-Type 33; Enhancement.<

------------- Remarks On a higher level of abstraction, ENHANCEMENT OF something may be judged to be BENEFICIAL or DETRIMENTAL, respectively, depending on the respective value attributed to it. Enhancement is value-free. This suggests that the same type of interrelations may happen to occur between both Beneficial and Detrimental.

Fig. 5. Striving to grasp as many NL expressions for the “Enhancing” relation as possible and its potentially related subjects or objects 2.2.1 Foundations and Principles of Relation Stipulation So, what are the foundations, and principles, of stipulating semantic relations? Pragmatism again. The set of semantic relations admitted to the ontology must fit the purpose. There are no established theories about how to find out, and define, task or domain-related inventories of semantic relations. Wherever workable sets of them have been tackled, we see them as a result of trial and error. Pragmatism prevails. In a way, the task resembles that of a translator, i. e. insofar as true, valid, understandable expression is the target of what a relation of relevance means. But the meaning of a particular relation must be defined in a general way, and it must be semantically related in a widely intersection-free manner with respect to the meaning of each of the other relations of the set. This is a task of a linguist and information scientist. It is obvious that ideally the inventory of semantic relations should be ready at the beginning of semantic modelling. Practice, however, shows that a given set must always be open for further adjustment. Some lack of exactness seems unavoidable. All in all, the choice of the relations admitted should guarantee fair coverage of the knowledge organized/to be organized.

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A clear-cut, widely intersection-free definition of each relation is of prime importance, as well as a reasonable restriction in number, right from the beginning. This is important to ease handling and to maintain a sharp consensual profile. Insofar as specific, task-oriented relationships are admitted, it is wishful that they are put in a formal relationship to a restricted number of known, general relations. This, then, may help to obtain higher recall in searches if wanted in given cases. Experience from two major projects (EXPO 2000 [14], SERUBA [15]) shows that with only half a dozen of well-chosen, general relationships recall as well as precision can be improved. 3. Modelling, and the Functions of Models The instruments outlined allow the modelling, i. e. the construction of a Statement, saying that Insulin-like growth factor IGF-I enhance(s) breast cancer cell proliferation the same as (produced by algorithm) breast cancer cell proliferation Is/AreEnhancedBy Insulin-like growth factor IGF-I and to express this in all three languages, in all paraphrased shades detected as occurring in valid source documents, and stipulated in what can be assumed are its most used/accepted, relevant structural forms. In reuse, queries like e.g. “What Is/AreEnhancedBy IGF-I? can then be prompted right away from the internal Basic Semantic Reference Structure (BSRS) of the ontology. Easy- to-understand NL answers to the query can be generated in EN, FR and GE. They will be given on the basis of their respective DESCs, in their canonic forms (which are supposed to be applicable free of any other context, by definition), without their paraphrased AAEs, however, since some contextual pollution would be unavoidable as shown in Example 3 above. . The other part of the search result would go beyond that. It would consist of a number of uncontrolled text strings detected in NL texts in either of the three browsed text collections. These should be strings of variable length that either precede “EnhancedBy”, or follow “enhance” or “enhances” (this covering all stipulated shades of meaning) in the three languages. That type of search one day may allow to detect tacit, henceforth unknown knowledge in NL texts. On top of that, such a query could be extended by taking an upposted, broader view , like: “What Is/AreFavouredBy IGF-I”?, which is a specific shade of the “Beneficial” relation as recommended for use by the former KTF mentioned above. All this is possible also on a cross-language basis without further formal restraints. An information seeker may put the query in German and receive texts from divergent sources written in English, French or any other language implemented in the ontology (Cross-Language Information Retrieval - CLIR). The same applies for reused knowledge streaming directly from the reasoning within the BSRS. This, then, would be something really new: Cross-Language Knowledge Reuse - CLKR. The consensuality command poses restraints for sharing: Another pragmatic aspect is that of sharing. Since the most specific and most valuable part of knowledge organized in ontologies is drawn from the statements, and since the main element of the statements are semantic relations, the semantics of semantic statements are of crucial importance in sharing

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knowledge among different ontologies. In any attempt to share knowledge, consensuality potentially is at stake. To illustrate this: If in ontology A the enhancing relation is admitted and used as explained in Chart 5 above, and in ontology B nothing more specific exists than the beneficial relation, then a sharing of knowledge is possible between the two ontologies. This is due to the inter-relational facility foreseen in ontology A, but it is possible only at a lower level of expressiveness which is dictated by ontology B. If B would force A further down to a still broader level, say, a loose, undefined isness relation, then the usefulness of sharing statements would be drastically reduced. This leads us to the following recommendation: Stipulate relations in accordance with the required exactitude of expressiveness or level of exploitation (which in the end is coming to the same) while purveying interrelated relations down to the more common, broad levels! Provide the possibility of segmentation according to the criteria of such relational hierarchy! In all future sharing operations, you will then find yourself on the strong side. 4. Foundations and principles of stipulating semantic statements Pragmatism also, but… The stipulation of semantic statements will always be governed by a pragmatic, target-oriented approach. The act of transfer and transposition of enouncements from the source texts can be made fully transparent and widely controlled. Curators/peers are supposed to be bilingual (at least) and experts in their proper disciplines. Machine-aided support must be provided in many ways, and at different points of the stipulation process. The ontology is transparent, not open to uncontrolled content, the sources are well documented, the BSRS and the vocabulary is under full machine-aided, but in last instance, intellectual control, consensuality among curators/peers is high and can be enforced, the editor’s responsibility for the message is established. So, what is the problem? Are there any? There are some [16]. Just to mention a few: Representation. No single ontology will ever represent a domain in full. This may be in contrast to what users expect from ontologies. Exact sampling of source texts will be crucial, especially so in the early stages of ontology development. A more process-oriented approach may take over as the ontology matures. The explosion of the demand for constant, daily update alone will quickly impose serious economic constraints with gaps and rebates on reliability as a possible consequence. Task ontologies are better off inasmuch as tasks can be exhaustively defined beforehand. Epistemology. Transferring statements and terminology from heterogeneous textual sources on to an axiomatic model under strict, general, defined conditions, but of limited expressiveness, forces the ontology constructor to take an epistemologic stance that cannot always be expected to be up to what the author of the source wants to express. The model at best is an abbreviation, not the original. What finally is transferred may lag behind an author’s message, it may not wholly meet his message, e. g. because it cannot be expressed by the instruments the ontology provides, or for lack of understanding on the side of the curator/peer, or, on the other hand, it may be the ontology that poses a demand for clear-cut expression which the source text cannot fulfil, because the author hesitates to be clear-cut in the matter, etc., etc.

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Logic, and Conclusion. It is often postulated that knowledge representation systems must strictly follow the rules of logic, must be contradiction-free, and insofar must be consistent [17]. Such postulate cannot apply to that part of the ontology that deals with AAEs and MULTIs. It is easy to demonstrate that NL language often is all but logic. Isolating AAEs and MULTIs corresponds to that need. Principles of logic and truth, and the demand for consistency, however, do apply to what is represented by the descriptors in terms of words, and is stipulated as DESC. Whether we can go beyond postulating logic and truth to be the underlying principles when dealing with semantic relations, and thus with entire statements, I am not that sure. As Margotti [9] puts it: “Equivalence is always gradual” This seems to be especially true of the relations between relations. However, can we expect a model to enable a better match than does its prototype? References

1. GÓMEZ-PÉREZ, A. et al., 2004: Ontological Engineering. Springer, 2004

2. BENVENISTE, E., 1966: Problèmes de linguistique générale. Paris, Gallimard

3. MARTINET; A., 1985: Syntaxe générale. Armand Colin, Paris 1985

4. GIBBON, D., et al., 1997: Handbook of standards and resources for spoken language systems. Mouton de Gruyter, 1997

5. JACKENDOFF, R., 1996: How language helps us think. Pragmatics and cognition. 4/1. Amsterdam,

Benjamins, 1996, 1-34

6. FUCHS, C., 1997: Diversité des représentations linguistiques : Quels enjeux pour la cognition ? In : Diversité des langues et représentations cognitives, Paris, Ophrys, 1997, 5-24

7. HJÖRLAND, B., NICOLAISEN, J., 2004: The epistomological lifeboat. http://www.db.dk/jni/lifeboat/

8. CEUSTERS, W. et al., 2005: A terminological and ontological analysis of the NCI Thesaurus. In:

Methods of Information in Medicine, 4/2005. http://ontology.buffalo.edu/medo/NCIT.pdf

9. MARGOTTI, F. W., 2003: Sinonímia e paráfrase: Algumas consideracões a partir de dados do Atlas Lingüístico-etnográfico da Região Sul-ALERS. In: Linguagem em discurso, 3/2, Tubarão, Brasil

10. SCHMITZ-ESSER, W. 2000: How to cope with dynamism in ontologies. In: Dynamism and Stability

in Knowledge Organization. Proc. 6th Int. ISKO Conf., Toronto, Canada. Ergon. 2000, 83-89

11. ILARI, R., GERALDI, W., 1985: Semântica. Sao Paulo, Atica 1985

12. SCHMITZ-ESSER, W., 2003: Meaning, understanding, and the organization of knowledge in a multilingual world - New tools for new tasks: Ontologies. In: Linguistic cultural identity and international communication. Vielberth, J., Drexel, G., Eds., AQ-Verlag, 2003, 149-171

13. SCHMITZ-ESSER, W., 1999: Thesaurus and Beyond: An advanced formula for linguistic engineering

and information retrieval. Knowl. Org. 26 (1999) No. 1, 10-22

14. SCHMITZ-ESSER, W., 2000: EXPO 2000 - INFO 2000. Visuelles Besucherinformationssystem für Weltausstellungen. Springer 2000.

15. SCHMITZ-ESSER, W., 2000: SERUBA - A New Search and Learning Technology for the Internet and

Intranets. Proc. 11th ASIS&T SIG/CR Classification Research Workshop. Chicago, IL., Nov. 12, 2000, 91-102

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16. SCHMITZ-ESSER, W., 2005: Ontologien als Herausforderung an Publizisten und Wissensorganisatoren. Elbe E-Lectures, Broadcasting &Events, Hamburg, 2005 www.elbe-studios.de/list.php?pers=101

17. DE BRUIJN, J. 2003: Using ontologies. DERI Technical Report. DERI-2003-10-29. Innsbruck,

Galway, 2003. http://www.deri.at/publications/teachpapers/documents/DERI-TR-2003-10-29.pdf

18. SCHMITZ-ESSER, W. 2002: www.schmitz-esser.de

Instances of Instances Modeled via

Higher-Order Classes

douglas foxvog

Digital Enterprise Research Institute (DERI), National University of Ireland, Galway, Ireland

Abstract. In many languages used for expressing ontologies a strict division betweenclasses and instances of classes is enforced. Other languages permit instances of instanceswithout end. In both cases, ontological meaning is often mistakenly assumed from thesepurely syntactic features. A rigorous set of definitions for different levels of classes ispresented to enable concepts to be unambiguously defined.

1 Introduction

Ontology languages normally distinguish types of objects (also called class, con-cept, collection, ...) from things that are not types (individuals, instances, ...).An issue that has often been recognized is that types of classes exist, thus mak-ing instances of these meta-classes classes themselves. Some systems address thisissue by disallowing meta-classes, others allow meta-classes without restriction,and others do not distinguish between classes and individuals. In order to pro-vide for rigor in this field, an ontology of levels of meta-classes was created forthe Cyc Project and is presented here.

Section 2 presents terminology used in this paper; Section 3 presents relatedwork; Section 4 presents the system of levels of meta-classes; Section 5 describesobject ontology order; and Section 6 provides some rules concerning classes ofvarious levels.

2 Terminology

In this document, the word class is used to denote a type for which instances arepermissible in an ontology, whether or not the instances are actually includedin the ontology. Various ontology languages use the words ”concept“, ”type“, or”collection“ for this meaning or a subset of this meaning. Outside the field ofcomputerized ontologies the word ”category“ is also used. Class is distinct fromgroup in that a class has no physical properties, while a group of objects mayhave mass, temporal duration, location, ownership, and other such properties1.Class is distinct from set in that a class may have different instances at different

1 Note that groups may have properties in addition to those derived from their members: a group ofpeople may own a piece of land even though none of its members individually owns even a pieceof the land.

times or in different contexts2, while a set’s elements are fixed3. Different classesmay have the same extent (including an empty extent) in a given context, but bedifferent classes, while two sets with the same members are the same set. A class’sinstances typically have one or more features or attributes in common, while a setmay have arbitrarily selected members - however, ontology languages typicallyalso allow for extensionally defined classes. A class is similar to an intensionallydescribed set (one whose members are specified by rules as opposed to by arecounting of its members) and the distinction between the two concepts is notmade in some ontology languages in which the context for the description isconstant.

The term individual is used here to represent any component of an ontologythat is an element or instance of a class but is not itself a class. Certain ontologycomponents, such as datatype values (e.g. numbers and character strings), rela-tions, and functions are included as ”individuals“ (and types of them as classes)in some systems but not in others. In systems that do not permit a class to be aninstance of another class, the word ”instance“ is frequently used for all elementsof a class, even if the real-world referent is a type, such as a dish offered on amenu.

The word ”instance“ is also used for all instances of classes in systems thatallow meta-classes but do not require every element of the ontology to be aninstance of some class. In systems in which every term is an instance of someclass, the word is not used in an unqualified manner. The term instance isused in this document only to refer to the relationship which one element of anontology has with respect to a class in the same ontology.

The term meta-class is defined as the class of all classes each of whoseinstances is necessarily a class. Since each instance of meta-class is by definition aclass, meta-class is an instance of itself. An ontology of meta-classes is presentedin Section 4, with the terms being defined as they are presented.

Two classes are disjoint if and only if they necessarily have no elements incommon.

3 Related work

3.1 Russell’s Simple Theory of Types

Russell’s Simple Theory of Types [13], developed in the early 20th Century,defines and describes different categories of sets, starting with sets of individuals,sets of sets of individuals, and so on. Allowing sets of more than one level to be

2 A context is a domain of applicability within which a set of statements are true, for example thepoint of view of a political party, the background situation of a work of fiction, or a hypotheticalsituation. For a detailed description of contexts see [9].

3 A computer variable which refers to a set may reference a different set at different times. The setsthemselves do not change.

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members of the same set opened the way for the paradoxical ”set of all sets thatdo not contain themselves“, and so were disallowed.

Russell and other philosophers developed numerous variations of the originalSimple Theory of Types.

Russell (for obvious reasons) did not apply his various levels or orders of setsto computerized ontologies.

3.2 KIF

KIF [6] distinguishes class and individual and allows for classes of classes. KIFdoes not further distinguish different types of meta-classes. In order to avoidRussell’s paradox, KIF does not permit a class to have an unbounded element,with a bounded element defined as either a non-set or a set all of whose elementsare bounded. Using this definition, no class may contain itself, because any suchclass would be unbounded.

3.3 UML

UML has a four-level architecture in which each component at one level is aninstance of exactly one component at the next highest level [1]. Each componentat the highest level is an instance of some component (possibly itself) at thissame level. As such, each level is used to define the next lower level, with thelowest level (M0) being the ”user object“ level, the next level (M1) being the”user model“ level, the next level (M2) being the ”metamodel“ or ontologylanguage level used for defining terms at the M1 level, and the highest level(M3) being the ”metametamodel“ level used for defining the ontology language.

Some items may appear at different levels and others may be instances ofcomponents at different levels. For example, Class is a component of both M2and M3, and HenryThoreau from the M0 level is an instance of both Person atthe M1 level and Instance at the M2 level.

Since only items at the lowest two levels are part of the user-created ontology,user ontologies can not contain concepts which are classes of classes.

3.4 RDFS(FA)

Pan and Horrocks argue in their proposal for RDFS(FA) (RDFS with Fixedmetamodeling Architecture) that at least two levels of class are needed andrelay a finding that ”in practice, it has not been found useful to have more thantwo class primitives in the metamodeling architecture,“ so they find it reasonableto define only two class primitives [12]. They do not consider a possible desireby the ontology builder to use multiple levels of classes in the user ontology.

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3.5 Other Ontology Languages

Other ontology languages either do not distinguish individuals from classes,allowing any instance of a class to have its own instances (e.g., F-Logic [2],RDF-S [8], TRIPLE [14]), or do not permit classes of classes (e.g., DAML+OIL[16], OWL [11], OXML [5], Power-Loom [3], WSML [4]).

4 Ontology of Levels of Meta-Classes

Although the Cyc Project [7] had distinguished meta-classes for years, in 2002,the lack of an ontology of levels of meta-classes became obvious to ontologi-cal architects at Cycorp. Since every constant in CycL is an instance of someclass, the following system was designed (using the term ”Collection“ insteadof ”Class“ in conformance with standard CycL terminology) to provide cleandefinitions and to prevent occasional misuse of meta-classes during initial ontol-ogization of a new area.

Classes are distinguished by their ”order“ - the number of iterations of in-stantiations that are necessary in order to obtain individuals. The order of emptyintentionally-defined classes can be determined from their defining rule.

4.1 First-Order Class

First-Order Class is defined as the meta-class of all subclasses of Individual. Eachinstance of First-Order Class is a class, each of whose instances is necessarilyan individual. This is the most common type of class, having instances such asPerson, Computer, and Ontology. Hobbit would also be an instance of First-Order Class since, although it has no instances in the real world, in the contextof The Lord of the Rings [15] it has many instances, all of which are necessarilyindividuals. The class Individual is an instance of First-Order Class since, bydefinition, all of its instances are individuals.

4.2 Second-Order Class

Second-Order Class is defined as the meta-class of all subclasses of First-OrderClass. Each instance of Second-Order Class is a class, each of whose instances isa First-Order Class. Typical Second-Order classes include Car-Brand (with in-stances such as VolkswagenCar and HondaCar), AnimalSpecies (with instancessuch as GreyWolf and Dodo), Occupation, and USArmyRank. First-Order Classis an instance of Second-Order Class since, by definition, all of its instances areFirst-Order classes.

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4.3 Third-Order Class

Third-Order Class is defined as the meta-class of all subclasses of Second-OrderClass. Each instance of Third-Order Class is a class, each of whose instancesis a Second-Order Class. Instances of Third-Order Class are rare in OpenCyc.A few typical instances are BiologicalTaxonType, having instances such asBiologicalSpecies and BiologicalGenus, and MilitaryRankType, having in-stances such as USArmyRank and RussianArmyRank. Second-Order Class is aninstance of Third-Order class since, by definition, all of its instances are Second-Order classes.

4.4 Fourth- and Higher Order Classes

Fourth-Order Class is defined as the meta-class of all subclasses of Third-OrderClass. Higher order meta-classes could be similarly defined; however, the utilityof implementing such classes is questionable. The only Fourth-Order Class inOpenCyc [10] is Third-Order Class (called ThirdOrderCollection), which isan instance since, by definition, all of its instances are Third-Order classes.Similarly, Fourth-Order Class would likely become the only instance of Fifth-Order Class, and so on, ad infinitum.

4.5 Fixed-Order Class

First-Order Class, Second-Order Class, Third-Order Class, and Individual aremutually disjoint classes, which each have the property that every one of theirinstances has the same order, i.e. it takes a fixed number of iterations of instan-tiation to reach an Individual. [For a formal definition of ontology element order,see Section 5.]

Thus, every instance of Individual is an Individual - this could be con-sidered zero-order. No instance of First-Order Class is an Individual, but everyinstance of every instance of it must be. No instance or instance of an instanceof Second-Order Class is an Individual, but every instance of an instance of aninstance of it must be. Third- and Fourth-Order Class are similarly one and twosteps more removed.

Fixed-Order Class is defined as the meta-class of all classes with this property.First-Order Class, Second-Order Class, Third-Order Class, and Fourth-OrderClass are not only instances of Fixed-Order Class; they are subclasses of it aswell, which means that all of their instances are transitively instances of Fixed-Order Class. Individual is an instance, but not a subclass of Fixed-Order Class.

4.6 Variable-Order Class

Not every class is a Fixed-Order Class. Fixed-Order Class, itself, has classes ofdifferent orders as instances. Thus, it is an instance of Variable-Order Class.

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Fig. 1. Instance-Of Relations in Meta-Class Ontology

Variable-Order Class is defined at the class of all classes which (1) may haveinstances of more than one order, (2) are instances of themselves, or (3) haveany instances which are variable-order classes. Every class is therefore eitherfixed-order or variable order, but not both.

Variable-Order Class is an instance of both itself and Meta-Class. Class andMeta-Class are instances of Variable-Order Class. The universal class (calledThing in CycL and RDF) is an instance of Variable-Order Class. Variable-OrderClass is not a subclass of Meta-Class since some of its instances, e.g. Thing, haveinstances that are Individuals as well as instances that are Classes.

4.7 At Least Nth-Order Class

The class Class is equivalent to ”at least first-order class.“ The class Meta-Classis equivalent to ”at least second-order class.“ Meta-Class Type, equivalent to ”atleast third-order class,“ has been created in OpenCyc (as CollectionTypeType),but is little used. At-least Nth-Order class for N higher than three has not beenfound useful, no instances having been ontologized other than Third-Order Classand Fourth-Order Class.

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4.8 At Most Nth-Order Class

We found no need for a class such as ”at-most second-order class“, although suchclasses were considered. Such a class would have been a subclass of Fixed-OrderClass in that none of its instances would be variable order classes. It would havebeen an instance of Variable Order Class.

First-Order class would be equivalent to ”at-most first-order class, and thuscreation of such a class would be redundant. “At-most third order class” wouldinclude all classes except Fourth-Order Class, Third-Order Class, Fixed-OrderClass, and those classes which include variable-order classes as instances, andwas not deemed useful.

4.9 Self-Including Class

Although Cyc has several classes that include themselves as instances, a meta-class of such classes was not created. This class would be an instance and subclassof Variable-Order Class. Classes in OpenCyc that would be instances of this classinclude Thing, VariableOrderCollection, CollectionType (meta-class), andCollectionTypeType (meta-class type).

However, since the class of all classes that are not instances of this class isparadoxically both an instance of this class and not (see Russell’s paradox), thisclass is also problematic and was not included in the ontology. 4

4.10 Self-Excluding Class

This class corresponds to the paradoxical set referred to in Russell’s Paradoxwhich is neither an member of itself nor not a member of itself. It was not createdas part of this ontology.

4 Note that if this class existed, it would be an element of itself, the argument being reductio adabsurdum:

– Define Self-Including Class (SIC) as being the class that has as instances those, and only those,classes which have themselves as instances.

– Assume this class is not an element of itself.

– Add to that class the single class necessary to generate a class that contained itself. E.g., if SIC

were {Thing, Meta-Class, VariableOrderClass, ...}, SIC2 would be {SIC2, Thing, Meta-Class,VariableOrderClass, ...}.

– This new class would contain all the self-containing classes (including itself) and only self-containing classes.

– Thus, this new class would be the Self-Including Class and our original assumption (that Self-Including Class does not contain itself) is proved wrong.

The third step in this argument would not be permitted for sets in modern Set Theory – anotherreason to shun the creation of such a class, even though classes are not sets.

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5 Ontology Element Order

Let Individuals be zeroth-order elements of an ontology. A class is defined asbeing of the Nth order if every one of its instances is necessarily of (N-1)storder. Any class that cannot be assigned an order under this rule is deemed avariable-order class.

Note that empty classes may still have an order. The class Dodo is first-orderin 2005 even though there have been no instances of the class for hundreds ofyears. Similarly, the class AnimalSpecies was second-order a billion years agoeven though there were no animals at the time.

6 Rules

The following rules relating to class order hold:

– Any instance of a First-Order Class is an Individual.– Any instance of an Nth-Order Class (for N > 1) is an N-1st Order Class.– Any instance of Nth-Order Class (for N > 1) is a subclass of N-1st Order

Class.– Every Class is either a Fixed-Order Class or a Variable-Order Class.– No Fixed-Order Class has an instance of the same order as itself.– If a Variable Order Class has an Nth-Order Class as an instance, it also has

an instance that is not an Nth-Order Class (in some context, even if not inthe present context).

– If N does not equal M, Nth-Order class and Mth-Order class are disjoint.– Meta-Class is disjoint with First-Order Class.– Meta-Class is disjoint with Individual.

7 Acknowledgment

This work was developed as part of the Cyc project at Cycorp of Austin, Texas.

References

1. Alvarez, J. et al.: MML and the Metamodel Architecture. WTUML: Workshop on Transformationin UML (2001).

2. Balaban, M.: The F-Logic Approach for Description Languages. Annals of Mathematics andArtificial Intelligence 15 (1995) 19–60

3. Chalupsky, H., et al.: PowerLoom Manual. Information Sciences Institute, USC. Downloaded fromhttp://www.isi.edu/isd/LOOM/PowerLoom/documentation/manual/manual.pdf 15/5/2005.

4. de Bruijn, J., et al.: The Web Service Modeling Language WSML. DERI Technical ReportD16.1v0.2 (2005).

5. Erdmann, M.: OXML 2.0 Reference manual for users and developers of OntoEdit’sXML-based Ontology Representation language, Version 0.92. (2001). Downloaded fromhttp://www.cs.ait.ac.th/ waralak/oxml2.0.pdf 15/5/2005.

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6. Genesereth, Michael R., Fikes, R. E.: Knowledge Interchange Format Volume 3.0 Reference Man-ual (1992).

7. Lenat, D.: Mapping Ontologies into Cyc. Ontologies and the Semantic Web, Technical ReportWS-02-11. AAAI Press. (2002) 1–6.

8. Manola, F., Miller, E.: RDF Primer. (2004) Downloaded from http://www.w3.org/TR/rdf-primer15/5/2005.

9. McCarthy, J.: Notes on Formalizing Context. IJCAI ‘93. (1993) 555–560.10. OpenCyc: OpenCyc website. http://www.opencyc.org, downloaded 15/5/2005.11. Patel-Schneider, P. F., et al., eds.: OWL Web Ontology Language: Semantics and Abstract Syntax.

Downloaded from http://www.w3.org/TR/2004/REC-owl-semantics-20040210/ on 15/5/2005(2004).

12. Pan, J. Z., Horrocks, I.: Metamodeling architecture of web ontology languages. Proceedings ofthe Semantic Web Working Symposium. (2001) 131–149.

13. Russell, B.: Principia Mathematica (1910).14. Sintek, M., Decker, S.: TRIPLE - An RDF Query, Inference, and Transformation Language.

DDLP’2001, Japan (2001).15. Tolkein, J.R.R.: The Lord of the Rings (1954).16. van Harmelen, F.; et al.: Reference description of the DAML+OIL (March 2001) ontology markup

language. Downloaded from http://www.daml.org/2001/03/reference 15/5/2005 (2001).

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Ontologies of One-way Roads

Sumit Sen

Institute for Geoinformatics, University of Münster, Germany

Abstract. This paper attempts to contrast two approaches of specifying ontolo-gies; a set based Description Logic (DL) approach in comparison to an alge-braic approach. The algebraic approach which assumes a category theory view of objects can be argued to adequately represent the structures that constitute the concept category. The reasoning about concepts expressed in such ontolo-gies would thus be based on behavioral equivalence as compared to subsump-tion reasoning in DL based specifications. A case study of road ontology speci-fication illustrates the differences in the two approaches. An analysis of the two approaches presents us the complexities associated with such specifications.

1 Introduction

Ontologies are gaining significance in geographic information sciences and the shar-ing of geospatial information. Geographic database schemas, like any other database schema, correspond to conceptual models which are often expressed in the form of ontologies. Gruber [1] defines ontology as “an explicit specification of a conceptuali-zation”. This definition provides the flexibility of specifying concepts using different logic systems and approaches. The approach and methodology used is often related to the problem at hand and scope of the task. The Semantic web community uses De-scription Logic (DL) based specifications whereas KIF has been a long preferred option for agent communication and knowledge sharing in the Artificial Intelligence community. Category theory has often been referred to, in the context of algebraic speci-fication of ontologies. The possible flexibility in such specifications combined with sound mathematical foundations is beyond the simplistic view in which concepts are seen as sets and membership to such sets are defined by necessary and sufficient properties. Categories are defined on the basis of sets of objects, the set of morphisms between them and further requirements like identity and composition (these are dis-cussed later in this paper). Correspondences between concepts that occur in cognitive sciences and categories in mathematical literature have only been explored briefly. This illustrates the motivation in this paper to attempt an algebraic specification of a road ontology and discuss the reasoning of such specification based on behavioral proofs and executablity. Section 2 of this paper provides a brief background to the notion of categories as concepts while section 3 discusses the case study of road on-tologies. A comparison of the reasoning mechanisms using description logics and the

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behavioral proofs is the focus of section 4. Finally some conclusions and future ex-plorations are discussed in the last section.

2 Categories and concepts

The word “category” has been understood in different ways, by different people, at different times. Right from Aristotle’s notion of categories which considers categories as kinds of being (based on substance, quantity, quality, relation, place, time, posi-tion, state, actions or affection) to Rosch’s principles of categorization [2] there has been a claim of a conceptual basis of such categories. Psychological and philosophi-cal division of thought on categorization has thus allowed certain ambiguity about the usage of the term “category”. This is in contrast to the mathematical definition of categories in category theory as used by Goguen [3] and Hitzler et al [4]

A category C consists of a class1, denoted ⎪C⎪, of objects, for each pair A, B of objects a set C(A,B) of morphisms from A to B, for each object A, a morphism 1A in C(A,A), called the identity at A, and for each three objects A, B, C an operation called composition, C(A,B) × C(B,C) → C(A,C) denoted “,” such that f ; (g ; h) = ( f ; g ) ; h and f ; 1A = f and 1A ; g = g whenever these compositions are defined. Write f : A → B when f ∈ C(A,B) and call A the source and B the target of f.

This definition already introduces the importance of morphisms between objects in a given set as criteria for the existence of a category. Also the notion of projection maps in category theory is a useful one which enforces a view on functions to have a domain and codomain in which arguments and results of such functions, respectively are meaningful. For example, a category of roads consists of all the members of this collection of roads along with the morphisms such as connected_to (we can assume that this morphism the domain and codomain are both members of the collection of roads; will hold for identity and composition axioms defined earlier.) The importance of the domain and codomain of any relation is imposed in the definition of the rela-tions. This is reflected, for example, in the definition of an institution [4] in the use of signatures and signature morphisms as below.

An institution consists of an abstract category Sign, the objects of which are signa-tures, a functor for sentences Sen : Sign → Set, and a functor for models Mod : Signop → Set with a satisfaction condition for the ╞∑ relation between models of ∑ and sentences of ∑ that for any signature morphism f : ∑ → ∑’ , any ∑’ model M’ and any ∑ - sentence e

M’ ╞∑’ f(e) iff f (M’) ╞∑ e The above statement is only a mathematical statement of Tarski’s semantic defini-

tion of truth and its invariance under change of notation. Any further discussion of this is not relevant to the objective of this paper and it is indeed to highlight the in-

1 A class is also used in different senses and the usual DL sense of a class as a set of objects is

not applicable here. The term class simply represents a collection of objects.

57

herent differences in category theory and set theory based approaches. The difference being in specifying operations and sorts together as signatures which could be seen as type signatures which form a critical part of the algebraic specifications discussed further.

2.1 Algebraic specifications

We have introduced the notion of institutions and signatures with an assumption of familiarity with notations of algebraic specifications. However for clarity we state the following components of (classical) algebraic specifications:

• Signature Σ = (S, OP) (where S is a set2 of sorts and OP is a set of opera-

tion symbols op : s1, . . . , sn → s), • signature morphism σ : Σ → Σ’ • (total) Σ-algebra A = ( (A s)s∈S , (opA)op∈OP)

It is important to note that there have been many algebraic specification languages

like CASL [5] and OBJ [6]. We choose to use Haskell based specifications (which can be argued to be specifications as well although Haskell itself is a programming language. Verification requirement of such specifications can be linked to the type checking of the Hugs compiler [7] and shall be discussed in a later section.) A typical specification for a surface object is shown below. It assumes three operations with type signatures:

class Surface surface object where putOn :: object -> surface -> surface takeOff :: object -> surface -> surface

2.2 Types, inheritance and hierarchies

The notion of a class specification in Haskell as shown above is dependent on the type signatures which are an essential part of an operation definition and hence the class definition. We extend this type checking possibility to ontology specifications. Geospatial concepts are assumed to be set2 of sorts and operations related to such sorts are specified together as its signature.

Any sub class relations between the two concepts would mean an inheritance of the operations and hence a signature morphism between the two concepts. Haskell additionally provides type classes which can be seen as collection of types. It is also possible to introduce (parameterized) multiple inheritance by using instances that inherit class behavior to a type. This can lead to a nested hierarchy of concepts in an ontology specification. By using type algebra one could also attempt higher order

2 As before the use of the term set only signifies the sense of a collection of objects.

58

specifications as well. Abstract data types and named types are the basis for such possibilities.

3 A Case Study: Roads

It is common to have different conceptual views on geographic entities and this of-ten results in different data models. Roads can be seen as such entities and different conceptual models of Roads such as GDF (Geographic Data File) and ATKIS [8] have been discussed by Rüther et al [9] Data producers have handled these aspects successfully and data standards for describing road networks and transferring the data have been developed, such as the international standard GDF [10]. These standards typically focus on the relevant geographic entities, their attributes, and relationships, but do not relate them directly to actions and related decisions that the data can sup-port. Such actions can be linked to operations that are linked to certain collection of objects and hence together with the specifications of the actions the sort signatures are the artifacts of our algebraic specifications.

Roads are often designated to be One way and although this can be easily

categorized as a social role we shall not engage in splitting the properties or opera-tions based on physical and social levels. Instead the argument that will be used here would be if the operation of driving on a road can be sub divided to One-way-driving and Two-way-driving. The other alternative which is to assume that Roads can have the property isOneWay which could be true for some roads and false for some others. One could also argue that OneWayRoads and TwoWayRoads exists as subclasses of Roads.

3.1 Description Logics

As an initial exercise we try to model the above description in description logics. Consider the case where there are roads, cars and humans. The set of properties of the system are drives, drivenOn, drivenOneWayOn, isOneWay, hasName. Based on the assumption that OneWayRoad and BothWayRoad are sub-concepts of Road, we can assume a taxonomical hierarchy as shown in figure 1. The drivenOn property thus holds on both sub-concepts of Road (specification 1).

Now consider the case where we drop the two sub-concepts and assume that there

Fig. 1. Specification 1: Concepts and properties

OneWayRoad

TwoWayRoad

Car

Human Road

drives

isA

drivenOn

non-isA

Name

hasName

59

is a sub-property of drivenOn, which is DrivenOneWayOn, which has range restric-tion that it only applies to Roads that have isOneWay property equal to true (hence it can be said drivenOn subsumes DrivenOneWayOn)). This leads to another specifica-tion with a smaller taxonomical hierarchy (specification 2) shown below.

Let us now reason about the relationship between the specification 1 Car drivenOn OneWayRoad and Car from specification 2, which has the behavior drivenOne-WayOn. Logically speaking they are substitutable but are not equivalent unless fur-ther information of mapping of the two sub-concepts of Road with specification 2 is available. The merge process of specification 1 to specification 2 is shown as below. (Road(isOneWay)) => OneWayRoad (1) (Road(¬isOneWay)) => BothWayRoad (2) ...given mappings Range(drivenOneWayOn) ≅ OneWayRoad Domain(drivenOneWayOn) ≅ Domain(drivenOn) ¬Road(isOneWay) => (Range(drivenOn)) ≅ (¬ Range (drivenOneWayOn)) (3) Road(isOneWay ) => (Range (drivenOn)) ≅ Range(drivenOneWayOn) (4)

Since the range of drivenOneWayOn is a subset of drivenOn the subsumption of

sub-property (drivenOneWayOn) is still maintained. If we consider the instance merged from specification 2 to 1, the information required would be: is there any Road in specification 2, which has non-singular values for isOneWay property. Such a case can be seen in one of the instances (as shown in figure 3). The foxroad can have both true and false values. This is classified as Road under specification 1 and not further classified under OneWayRoad or BothWayRoad. The issues arise during consistency checks because it cannot be determined if foxroad allows drivenOne-WayOn. The practical solution to the problem would be to consider only such proper-ties whose range is fully satisfied by this Road and hence it is not possible to classify either as OneWayRoad or BothWayRoad.

Car

Human Road

drives

isA

drivenOn

non-isA

Name

hasName

Boolean

isOneWay

Fig. 2. Specification 2: Concepts and properties

60

nicecar

mike

newroad

drives

drivenOn

false

isOneWay

oldcar

drives

true

drivenOn

foxroad

drivenOneWayOn

isOneWay?? isOneWay??

Therefore foxroad can have the property of drivenOn but not drivenOneWayOn. However, common sense advises to assume one-way drive as the worst-case scenario in case of such indeterminacy. So is drivenOn of specification 1 really a super-property of drivenOneWayOn? Can the Road of specification 2 be called the super-concept of OneWayRoad and BothWayRoad of specification 1?

3.2 Algebraic Approach

As a part of the algebraic approach that has been used by Rüther et al [9] and Raubal and Kuhn [11] we use a multi level view of a road network and assume the existence of a graph theoretical building block consisting of nodes and edges. The basic level of operations on a graph namely Link and move are given below (spec 1 assumes one-way move and spec 2 assumes bothway move.

Using (importing) either spec1 or spec2 class definitions above, one could now define nodes and edges with respect to roads and cars. This is shown below. It is important to note that for the following specification to work.

--Node data Node = Node Name deriving (Eq, Show) -- Edge: "A sequence of line segments with nodes at each end" data Edge = Edge Node Node deriving Show

Fig. 3. Specification 1 & 2: Combining Instances

61

-- Car data Car = Car Node deriving (Eq, Show) type Cars = [Car] -- a Node is a named object instance Named Node Name where name (Node n) = n -- an Edge as Link between Nodes instance Link Edge Node where from (Edge node1 node2) = node1 to (Edge node1 node2) = node2

Here the edge is drawn as directional graph and thus a One-Way-Road would

have only one instance of an edge where as a Both-WayRoad will have two in-stances.

-- a Car located at a Node instance LocatedAt Car Node where location (Car node) = node

Further we may specify the next level of concepts which include concepts about

the relation of Edges to RoadElements. It is interesting to note that based on the choice of Spec1 or Spec2 we would obtain OneWayRoads or TwoWayRoads.

-- Road Element data RoadElement = RoadElement Edge deriving Show class (Path path conveyance ) => Conveyance conveyance path object where transport :: path -> conveyance -> object -> object -- RoadElements as Links between Nodes instance Link RoadElement Node where from (RoadElement edge) = from edge to (RoadElement edge) = to edge -- RoadElements as Paths for Cars instance Path RoadElement Car where move (RoadElement edge) (Car node) = Car (other edge node)

The above specification takes the view that the move and link operations are inten-

tionally defined and that if one choose to, either definitions could be used by the instance. Further axioms could be applied to the derived behavior.

62

4 Approaches to reasoning

The primary purpose of formal specifications is to be able to make deductions about the behavior of any implementation that results from a specification that is made. To reason about conceptualizations expressed in such specifications is naturally the pur-pose of ontology specifications. Consistency and completeness are also to be evalu-ated.

While DL based approach mainly rely on subsumption/equivalence reason-

ing (and also negation reasoning in a local closed world assumption) Consistency is usually related to A Box reasoning which concerns instance membership to a class to which it is found consistent. Subsumption of any concept by another ensures an en-tailment relation between them. On the other hand algebraic specifications are checked for type consistency and equivalence could be evaluated on the basis of observational (or behavioral) equivalence. Algebraic specifications usually depend on (automatic) theorem provers for verification of specifications. For our case however type checking and executablity form the basis of verification of our specifications in Haskell. This is done by executing test scripts such as

start = Node "start" end = Node "end" theEdge = Edge start end theCar = Car start theRoadElement = RoadElement theEdge t1 = location (move theRoadElement theCar) == end

Thus t1 will be true if spec 1 is used and false if spec2 is used deciding if the road

specification is of a one way road.

4.1 One Way Roads are Roads?

As discussed earlier, subsumption relations form one of the important inferences in DL and this can also be seen as a particular form of ISA relation or the inheritance relation. By recollecting the questions raised in section 3.1 we can now attempt to answer the question if One Way Roads can be considered as a specialization of Roads. Formal ontology principles would not allow this because one could argue that being one way is related to a role and hence an anti-rigid property [12]. Although one can say that if a One Way Road always remains One Way this property can hardly be called dynamic. However by assuming the ground level and social level of categori-zation [13], we can say that they are anti rigid. Beyond this we could also debate if drivingOneWay as a concept is a sub concept of driving. The DL based approach would be to ascertain if the domain of the two are equivalent and if the range of the drivingOneWay is a subset of driving.

63

If driving was taken as an action, its relationship to oneWayDriving would be based on a functor that maps all axioms (Sentences & Models) of the signatures of driving to that of oneWayDriving. This provides a more convincing and flexible alternative as compared to the rigid notions of ISA relationships.

5 Conclusions and future work

We have explored two views of ontology specifications which differ in the way they specify concepts, namely, a DL based view which considers concepts as sets of ob-jects with common properties and an algebraic specification approach which relies on type signatures3. The later can thus claim to have a category theoretic basis of its specifications.

Category theory has been increasingly looked upon for information flow integra-tion based on the notion of institutions and efforts to formalize ontology merging [4] and relations between ontologies based on category theory are already being ex-plored. Algebraic specifications are also being integrated to other ontologies [15]. It remains to be seen if these efforts will lead to a wider and more productive use of algebraic specifications in ontologies is to be seen in future. Nevertheless the work in this paper is only a start in this direction some of the aspects in regard to such specifi-cations which will have attention in future include

• Formalization of behavioral inheritance in algebraic specifications (which

would include paramertized multiple inheritance). Formalization of behavior would involve specifications of states which is a well discussed topic.

• Exploring Theorem provers for proving equivalence between the concepts specified in ontologies. This would enable commonly used subsumption rea-soning although expressing disjoint relations is a bit problematic.

• The use of institutions to map ontologies to & from other specification ap-proaches is also an important area of concern. The HetCASL based approach (Lüttich et al, 2005) using Grothendieck institutions would be a promising area to explore.

Acknowledgements

The ideas presented have been shaped by discussions with members of the seman-tic interoperability research group at the Institute for Geoinformatics, University of Münster (http://musil.uni-muenster.de).

3 The complete OWL and Haskell specifications of the case study can be found at

http://ifgi.uni-muenster.de/~sumitsen/ontozip.zip

64

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