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A Categorical Approach to Networks of Aligned Ontologies

  • Original Article
  • Published:
Journal on Data Semantics

Abstract

Ontology matching and alignment are key mechanism for linking the diverse datasets and ontologies arising in the Semantic Web and other application areas for formalised ontologies. We show that category theory provides the powerful abstractions needed for a uniform treatment of ontology alignment at various levels: semantics, language design, reasoning and tools. The general representation and reasoning framework that we propose includes: (1) an abstract notion of logical system, consisting of a logic syntax and a model theory, based on an extension of institutions with additional features specific to alignments, (2) a declarative language to specify networks of ontologies and alignments, with independent control over specifying local ontologies and complex alignment relations, based on and improving the Distributed Ontology, Model and Specification Language DOL, (3) the possibility to align logically heterogeneous ontologies, and (4) the provision of generic proof support for global reasoning over networks of aligned ontologies, employing different semantics. In particular, we show how the three semantics of Zimmermann and Euzenat can be uniformly and faithfully represented using \(\mathsf {DOL}\) language constructs, by refining them into four different kinds of semantics: simple, integrated (general and inclusive), and contextualised. Finally, we discuss the implementation of the \(\mathsf {DOL}\) alignment features in the Ontohub/Hets tool system.

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Notes

  1. This paper is an extended version of [14]. While in [14] we present the semantics of alignments for the particular case of \(\mathsf {OWL}\), in the present paper we introduce a general construction, independent of the underlying logical formalism. This means that the results of [14] can be obtained as an instantiation of the results of this paper. Note however that we needed to enhance the formalisation of the notion of logical system in order to achieve this goal.

  2. We do not claim here that the reasoning methods we provide outperform more specialised alignment reasoning methods, say for DDL, or alignment debugging: our main contribution is the provision of a unifying framework that works simultaneously at the various levels.

  3. Note that this paper does not cover search for alignments.

  4. \(\mathsf {DOL}\) has been adopted as a standard by the Object Management Group (OMG), see http://www.omg.org/spec/DOL/ and http://dol-omg.org.

  5. Since we split integrated semantics into two versions, we have four semantics, and not just three as in [80].

  6. Referring to the intuition of categories of graphs, a functor is just a graph homomorphism. However note that also the monoidal composition of arrows must be preserved.

  7. A functor is faithful if it is injective when restricted to each set of morphisms that have a given source and target.

  8. According to the \(\mathsf {OWL}\)  2 specification, names of \(\mathsf {OWL}\) symbols should be IRIs; see https://www.w3.org/TR/owl2-syntax/.

  9. See http://www.w3.org/TR/owl2-overview/. Note, however, that OWL also includes datatypes, which further enrich the logic. We omit these because they are inessential for the presentation of the logical framework presented in this paper and would unnecessarily complicate exposition.

  10. An approach overcoming this limitation could easily be used to reason about \(\mathsf {OWL}\) ontologies featuring, e.g. irreflexive transitive roles, which is however undecidable.

  11. Note that \(\rightarrow \) associates to the right, i.e. we have \(=_t:t\rightarrow (t\rightarrow Bool)\).

  12. Note that \(=_t\) suffices to define the other constants.

  13. Strictly speaking, \(\mathbb {C}at\) is not a category but only a so-called quasicategory, which is a category that lives in a higher set-theoretic universe [33].

  14. Model reducts are known from model theory. If \(\varphi \) is an inclusion of a subsignature \(\Sigma _1\) into a larger signature \(\Sigma _2\), then \(M^2|_{\varphi }\) is the restriction of model \(M^2\) to \(\Sigma _1\), forgetting those model components interpreting symbols in \(\Sigma _2{\setminus }\Sigma _1\).

  15. Normally, \(\mathsf {MSFOL}\) models do not feature such a universe. However, for our technical results, having a universe is needed, and it does not change the logic essentially: given a standard many-sorted model, it is always possible to let \(M_U\) be the union of all carrier sets.

  16. Normally, \(\mathsf {HOL}\) models do not feature such a universe, but the same remark as for \(\mathsf {MSFOL}\) applies.

  17. That is, with the same objects as the original category.

  18. That is, for each family of signatures \(\mathbb {S} \in |\mathbf {Sign}|\), we have that \(\mathbf {Sen}(\bigcap \mathbb {S}) = \bigcap _{\Sigma \in \mathbb {S}} \mathbf {Sen}(\Sigma )\).

  19. Actually, here we only need the existence of an initial (“empty”) signature.

  20. Recall that a functor is similar to a graph homomorphism.

  21. Given two functors \(F,G:\mathbf {A}\rightarrow \mathbf {B}\), a natural transformation \(\eta :F\rightarrow G\) compares the image of F with that of G by linking them with suitable arrows. More precisely, a natural transformation \(\eta :F\rightarrow G\) consists of a family of arrows

    $$\begin{aligned} (\eta _A:FA\rightarrow GA)_{A\in |\mathbf {A}|}, \end{aligned}$$

    such that for any \(f:A_1\rightarrow A_2\) in \(\mathbf {A}\), the following diagram commutes (i.e. \(\eta _{A_1};Gf=Ff;\eta _{A_2}\)):

    figure b

  22. Although the intuitions behind relativisations are stable, and for specific logics there usually is a standard form of canonical relativisation, the specifics vary with the syntactic variations of a given logic, and this notion can therefore not be stated fully logic-independently.

  23. For example, in Example 19, instead of a unary top predicate, one could introduce a binary one, but only use it in the form top(x, x).

  24. \(\mathsf {DOL}\), in accordance with the Alignment API, has further syntax for cardinality of alignments, which however is not relevant here.

  25. See http://www.loa.istc.cnr.it/DOLCE.html.

  26. See http://www.ifomis.org/bfo/.

  27. We use the following notations: \(r_{21}(C) = \{ x\in D_1 \mid (y,x)\in r_{21} \text { for some } y\in C \}\) if \(C\subseteq D_2\) and \(r_{21}(R) = \{ (x,y) \mid x,y\in D_1, \exists x',y' \in D_2 \text { with } (x',y')\in R \text { and } (x',x), (y',y)\in r_{21} \}\) if R is a relation on \(D_2\).

  28. The last four axioms involve general concept inclusions, which can be expressed in OWL, but not in Manchester syntax. We have taken the liberty to keep using Manchester syntax for them.

  29. Again, this is not valid Manchester syntax, but expressible in \(\mathcal {SROIQ}\) (general concept inclusion).

  30. Actually, integrated semantics was originally motivated by the need for heterogeneous alignments (J. Euzenat, personal communication).

  31. We make the simplifying assumption that the logics \(J_i\) of the theories that internalise the semantics of all correspondences are the same. In practice, when this is not the case, there exists a logic J and logic translations \(\gamma _i:J_i\rightarrow J\), for each i, and the theory obtained by translating \((\Sigma _i,\Delta _i)\) along \(\gamma _i\) also internalises the semantics of \(c_i\).

  32. For example, the first theory is the abbreviation of \(\forall x, y~.~(\exists z_1, z_2~.~r_S(x, z_1) \wedge r_1(z_1, z_2) \wedge r_S(y, z_2)) \iff (\exists w_1, w_2~.~r_T(x, w_1) \wedge r_2(w_1, w_2) \wedge r_T(y, w_2))\).

  33. http://www.meteck.org/files/ontopartssup/MereoTopoD.owl.

  34. As described in http://www.loa.istc.cnr.it/old/Papers/D18.pdf.

  35. https://ontohub.org/hets-lib/Ontology/Dolce/DolceSimpl.het.

  36. Empty bridges, corresponding to empty alignments, can be removed as long as the diagram stays connected.

  37. See [61] for the formalisation of the monotonic part of F-logic as an institution.

  38. The \(\mathsf {DOL}\) syntax is: entailment e = i in S entails { \(\varphi \) }, where e is some name for the entailment.

  39. The proposed \(\mathsf {DOL}\) syntax is: entailment e = i,j in S entails c, where e is some name for the entailment.

  40. Note that \(\Sigma ^b\) is \(\Sigma _B\), whose definition depends on the choice of sem and was introduced in the corresponding subsection of Sect. 5.

  41. With these, typed symbols (like in many-sorted FOL) can easily be realised.

  42. That is, a quotient of a disjoint union, as done by the colimit. However note that the problem already arises with simple unions.

  43. http://bioportal.bioontology.org/mappings.

  44. https://ontohub.org/repositories/bioportal_mappings.

  45. The proposed \(\mathsf {DOL}\) syntax for a module of a network S is extract S.

  46. Full support for all four semantics is currently in progress, the implementation of simple semantics for alignments in Hets is stable.

  47. The names of the nodes in the diagram of the alignment, as well as the names and for the relativisation of ontologies are generated during the analysis of the alignment .

  48. These are extensively studied in the Workshops on Modular Ontologies (WoMO), see http://iaoa.org/womo.

  49. See [60] for an institution independent treatment of terms.

  50. ADS were first introduced and extensively used in [2] to prove transfer results for fusions of modal and description logics.

  51. See https://foust.inf.unibz.it.

  52. To give some more detail, DOLCE is available in various DL versions, in FOL, and in an extension using quantified modal logic (see http://www.loa.istc.cnr.it/old/DOLCE.html); GFO is currently in FOL, BFO versions exist in OWL DL, OBO, Isabelle, and CLIF (a syntax for Common Logic [58]) (see https://github.com/BFO-ontology/BFO), GUM exists in OWL DL and CASL (see http://www.ontospace.uni-bremen.de/ontology/gum.html), UFO is written in higher-order logic (Coq/Gallina), and YAMATO in FOL and OWL.

  53. Note that the distributed nature is limited: DRAGO “can answer consistency and satisfiability queries at one peer” [20]. And for DRAOn, “cross-ontology correspondences are limited to concept subsumption or disjointness, and role subsumption (so, role disjointness is not supported)” [82].

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Acknowledgements

We would like to thank Jérôme Euzenat and Fabian Neuhaus for extensive discussions of ideas found in this paper. We also thank the anonymous reviewers for their substantial feedback and for suggesting a number of improvements, both on a technical level and regarding the presentation of results.

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Authors and Affiliations

  1. KRDB, Free University of Bozen-Bolzano, Piazza Domenicani 3, 39100, Bozen-Bolzano, Italy

    Mihai Codescu & Oliver Kutz

  2. IKS, Otto-von-Guericke University of Magdeburg, Universitätsplatz 2, 39106, Magdeburg, Germany

    Till Mossakowski

Authors
  1. Mihai Codescu
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  2. Till Mossakowski
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  3. Oliver Kutz
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Corresponding author

Correspondence to Oliver Kutz.

Additional information

We gratefully acknowledge the financial support of the Future and Emerging Technologies (FET) programme within the Seventh Framework Programme for Research of the European Commission, under FET-Open Grant Number: 611553, project COINVENT.

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Codescu, M., Mossakowski, T. & Kutz, O. A Categorical Approach to Networks of Aligned Ontologies. J Data Semant 6, 155–197 (2017). https://doi.org/10.1007/s13740-017-0080-0

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  • DOI: https://doi.org/10.1007/s13740-017-0080-0

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