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Organization and Management of Large Categorical Systems

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Abstract

This chapter surveys approaches to handling categorical systems of extensive size, spanning from semi-formal systems in terminology and classification sciences to formal logical approaches. In particular, we briefly review the transition from terminologies to ontologies that are formalized in logics, exemplified in the medical domain. The main part presents the state of the art of the modularization of logical theories with an ontology-related background. Since the field is still very young and active, an evaluation of these approaches results in a heterogeneous landscape of proposals and leaves perspectives for future research.

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Notes

  1. 1.

    Since 2007, SNOMED CT, the Systematized Nomenclature of Medicine – Clinical Terms, is maintained and further developed by the International Health Terminology Standards Development Organization (IHTSDO), see http://www.ihtsdo.org. The number originates from the page http://www.ihtsdo.org/snomed-ct/snomed-ct0/, accessed on 13.09.2008.

  2. 2.

    Most of the current terminological systems arrange concepts by subsumption relations, i.e., along their degree of generality. The broad use of the term “subsumption hierarchy” includes trees and directed acyclic graphs, also called polyhierarchies. The depth of such hierarchies refers to the path lengths between roots and leaves.

  3. 3.

    This mixture appears, e.g., in the description of first-, second- and third-generation terminologies; (cf. Rossi Mori, 1997; Spackman and Campbell, 1998; Straub, 2002).

  4. 4.

    GALEN and SNOMED CT have adopted weak description logics very early. Spackman and Campbell (1998) and Spackman (2001) report the use of a very restricted and therefore computationally well tractable description logic for an earlier version of SNOMED CT. The only concept constructors referred to are conjunction and existential restrictions, plus top and bottom symbols (apart from bottom, these constructors form nowadays the EL description logic (Baader, 2003a)). Such usage is also claimed for GRAIL, the language used for GALEN. However, according to Rector et al. (1997), the structure of GRAIL appears to be related to, but somewhat different from standard description logics; (cf. also Rector and Rogers, 2005, p. 12 f.).

  5. 5.

    Straub (2002) further proposes so-called multi-point models in order to allow for multiple values on one and the same dimension on a regulated basis, motivated by examples like a double fracture and given an analysis why extensions by another dimension would not solve that representation problem. Though we agree on the examples, some skepticism about this solution remains on our side, but cannot be elaborated here.

  6. 6.

    See also Dzbor and Motta (2008) for the navigation problem and related challenges from the perspective of human-computer interaction applied to ontological engineering.

  7. 7.

    On the technical level, they are most often delivered in the form of huge data files, with a technically oriented structure which is rather independent of the conceptual architecture.

  8. 8.

    Open Biomedical Ontologies: http://obo.sourceforge.net/

  9. 9.

    Terminologically, we separate the use of “ontology” from that of logical “theory”, generally adhering to more formal, logical vocabulary in this part of the chapter. In the same line the term “semantically” should now be read as referring to a formal, model-theoretic semantics.

  10. 10.

    Note that Section 3.6 provides a review of the selected approaches which should to a large extent be readable without detailed knowledge of the framework. However, the latter supports a unified view and collects recurrent properties required for modules in Section 3.5.

  11. 11.

    Note that partially similar issues are treated in Kutz and Mossakowski (2008) at a comparable level of generality, exposed in more sophisticated terms on the basis of category theory; cf. Section 3.7.1 for more details about this and related works.

  12. 12.

    We consider only closed formulas, i.e., formulas without free variables (in languages with variables).

  13. 13.

    One may consider first-order logic as a prototypical case for those notions; (cf. Ebbinghaus et al., 1994, ch. III).

  14. 14.

    This requirement is a real restriction compared to replacing a bidirectional interface with an arbitrary pair of an import and an export interface.

  15. 15.

    In the sequel, following common conventions and despite actually distinguishing theories and deductively closed theories, we abbreviate this composition operation as “union” or “set-theoretical union”, denoting it as ∪.

  16. 16.

    Concerning the usage structure among basic modules, there is some arbitrariness between two options: either to consider all interfaces connected to each other, or to see connections only if there are non-empty intersection languages among the Mi. We chose \(Ifm{\textrm{((M}}_{\textrm{i}} {\textrm{)}}_{{\textrm{i}} \in {\textrm{I}}} {\textrm{)}} \times Ifm{\textrm{((M}}_{\textrm{i}} {\textrm{)}}_{{\textrm{i}} \in {\textrm{I}}} {\textrm{)}}\) because even with empty intersection languages some interaction may occur among the Mi, e.g., creating inconsistency based on logical sentences restricting the universe to different cardinalities. However, for FOL the second option is in effect equivalent to this Amir and McIlraith (2005).

  17. 17.

    Interpretation constraints are a special kind of “compatibility constraints” as mentioned in Section 3.6.5. Their proof-theoretic counterparts share the same intuition and are called “bridge rules”, a new type of inference rule in DFOL.

  18. 18.

    In FOL with equality, the equals symbol must be noticed and can be seen as commonly shared. For example, contradictory equality sentences may produce contradictions when joining seemingly signature-disjoint theories.

  19. 19.

    The field emerged around the mid of the 1990s, exemplified by dedicated publications and events such as the workshop series “Frontiers of Combining Systems”; (cf. Caleiro et al., 2005).

  20. 20.

    SHOIQ is a slightly more expressive description logic than SHOIN, the description logic underlying OWL-DL. See Cuenca Grau (2005, Section 2) for an in-depth discussion.

  21. 21.

    This property has an effect which may be problematic in some cases. Given a domain ontology D modularizable according to Cuenca Grau et al. (2006c), the use of an ontology with more general categories G to integrate D-categories by means of subsumption causes all modules to collapse into one. This has practically been observed in tests with the GALEN ontology (Cuenca Grau, 2005, p. 150 f.), and will prevent the use of foundational ontologies in modular fashion according to this approach.

  22. 22.

    Actually, Bao et al. (2006d) discusses three types of semantics: a local semantics per module, a global semantics, in which all model domains are united and domain correspondences are merged, and a distributed semantics which includes a central package and all of its imports.

  23. 23.

    It is tempting to view foreign terms as defining import interfaces instead of the bidirectional identity interfaces of basic modules. This assumption is also supported by the propagation of subsumption relations among terms to modules importing all atomic term components. However, conservativity properties are not generally satisfied by P-DL, and accordingly, bidirectional interfaces appear more appropriate than import interfaces.

  24. 24.

    Information Flow Framework: http://suo.ieee.org/IFF/

  25. 25.

    Standard Upper Ontology: http://suo.ieee.org/

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Acknowledgements

We would like to thank Heinrich Herre, Frank Wolter, Denis Ponomaryov, Alexander Heußner, and Roberto Poli for valuable comments on earlier versions of the manuscript, and we gratefully acknowledge the members of the research group Onto-Med at the University of Leipzig for maintaining a lively and inspiring atmosphere.

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  1. Department of Computer Science, University of Leipzig, Leipzig, Germany

    Frank Loebe

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    Roberto Poli

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    Michael Healy

  3. Computer Technology Institute, Hellenic Open University &, N. Kazantzaki Str., Patras, 265 00, Greece

    Achilles Kameas

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Loebe, F. (2010). Organization and Management of Large Categorical Systems. In: Poli, R., Healy, M., Kameas, A. (eds) Theory and Applications of Ontology: Computer Applications. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-8847-5_3

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