## Modules in Transition Conservativity, Composition, and Colimits

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@MISC{Kutz_modulesin,

author = {Oliver Kutz and Till Mossakowski},

title = {Modules in Transition Conservativity, Composition, and Colimits},

year = {}

}

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### Abstract

Abstract. Several modularity concepts for ontologies have been studied in the literature. Can they be brought to a common basis? We propose to use the language of category theory, in particular diagrams and their colimits, for answering this question. We outline a general approach for representing combinations of logical theories, or ontologies, through interfaces of various kinds, based on diagrams and the theory of institutions. In particular, we consider theory interpretations, language extensions, symbol identification, and conservative extensions. We study the problem of inheriting conservativity between sub-theories in a diagram to its colimit ontology. Finally, we apply this to the problem of conservativity when composing DDLs or E-connections. 1

### Citations

475 | Institutions: Abstract Model Theory for Specification and Programming - Goguen, Burstall - 1992 |

227 |
Abstract and concrete categories
- Adámek, Herrlich, et al.
- 1990
(Show Context)
Citation Context ... a morphism from each object involved in the diagram into the tip, such that the triangles arising from the morphisms in the diagram commute. A colimit is a universal, or minimal cocone. We refer to (=-=Adámek et al., 1990-=-) for formal details. In the sequel, we will assume that the signature category has all finite colimits, which is a rather mild assumption; in particular, it is true for all the examples above. Moreov... |

216 | Mathematical Logic - Shoenfield - 1967 |

112 | Distributed description logics: Assimilating information from peer sources - Borgida, Serafini - 2003 |

99 | A categorical manifesto - Goguen - 1991 |

95 | E-connections of abstract description systems
- Kutz, Lutz, et al.
(Show Context)
Citation Context .... (2006) forsontologies. Beyond imports, it allows for renaming, hiding and parameterisation. Other languages that have been introduced include DDLs (Borgida and Serafini, 2002, 2003), E-connections (=-=Kutz et al., 2004-=-), and P-DLs (Bao et al., 2006b,a), where, roughly speaking, more involved integration and modularisation mechanisms than plain imports are envisaged. We will use the formalism of colimits of diagrams... |

92 |
Using the universal modality: Gains and questions
- Goranko, Passy
- 1992
(Show Context)
Citation Context ...ation for QS5 is shown in Fine (1979). 7 But it holds for the quantified extension of K (Gabbay, 1972), and so does Craig-Robinson. Finally, the modal logic S4u has Craig-interpolation, 8 is compact (=-=Goranko and Passy, 1992-=-), and has implications (for M |= ϕ =⇒ M |= ψ, set χ = �ϕ → �ψ). Thus, S4u has Craig-Robinson interpolation. The amalgamation property (called ‘exactness’ in Diaconescu et al. (1993)) is a major techn... |

92 | Building specifications in an arbitrary institution - Sanella, Tarlecki - 1988 |

85 | Logical support for modularisation - Diaconescu, Goguen, et al. - 1991 |

83 | Did I damage my ontology? a case for conservative extensions in description logics
- Ghilardi, Lutz, et al.
- 2006
(Show Context)
Citation Context ...es not add new facts to the individual ontologies. Indeed, the notion of an ontology module of an ontology T has been defined as any “subontology T ′ such that T is a conservative extension of T ′ ” (=-=Ghilardi et al., 2006-=-)—this perfectly matches our notion of conservative diagram below. Definition 7. A theory morphism σ : T1 −→ T2 is proof-theoretically conservative, if T2 does not entail anything new w.r.t. T1, forma... |

83 | Properties of independently axiomatizable bimodal logics - Kracht, Wolter - 1991 |

82 | Many-Dimensional Modal Logic
- Venema
- 1992
(Show Context)
Citation Context ...ansfers to the fusion by a result of Kracht and Wolter (1991). However, since S4u is a Sahlqvist axiomatisable logic whose frame conditions are universal Horn, it also follows for S4u by a result of (=-=Marx and Venema, 1997-=-).sinstitution weakly semi-exact exact Craig-Robinson EL + - ? ALC ms + + + ALC + - + ALCO + - - ALCQO + - - SHOIN + - ? FOL ms + + + QS5 + - - Fig. 1. (Weak semi-) exactness and Craig-Robinson Theore... |

44 | Distributed description logics: Directed domain correspondences in federated information sources
- Borgida, Serafini
- 2002
(Show Context)
Citation Context ... Initiative), 2004) has been used in Lüttich et al. (2006) forsontologies. Beyond imports, it allows for renaming, hiding and parameterisation. Other languages that have been introduced include DDLs (=-=Borgida and Serafini, 2002-=-, 2003), E-connections (Kutz et al., 2004), and P-DLs (Bao et al., 2006b,a), where, roughly speaking, more involved integration and modularisation mechanisms than plain imports are envisaged. We will ... |

41 | Institution-independent Model Theory - Diaconescu - 2008 |

35 | Hybrid logics - Areces, Cate - 2006 |

35 | Logical systems for structured specifications - Borzyszkowski |

31 | Institution morphisms. Formal Aspects of Computing 13:274–307 - Goguen, Rosu - 2002 |

30 | V.: On the semantics of linking and importing in modular ontologies
- Bao, Caragea, et al.
(Show Context)
Citation Context ... imports, it allows for renaming, hiding and parameterisation. Other languages that have been introduced include DDLs (Borgida and Serafini, 2002, 2003), E-connections (Kutz et al., 2004), and P-DLs (=-=Bao et al., 2006-=-b,a), where, roughly speaking, more involved integration and modularisation mechanisms than plain imports are envisaged. We will use the formalism of colimits of diagrams as a common semantic backbone... |

30 | The Heterogeneous Tool Set - Mossakowski, Maeder, et al. - 2007 |

29 | Conservative extensions in the lightweight description logic EL
- Lutz, Wolter
(Show Context)
Citation Context ... is easy to show that conservative theory morphisms compose. Moreover, Proposition 8. Model-theoretic implies proof-theoretic conservativity. However, the converse is not true in general: Example 9. (=-=Lutz and Wolter, 2007-=-) Consider the following two EL TBoxes: Γ1 = {Human ⊑ ∃eats.⊤, Plant ⊑ ∃grows in.Area, Vegetarian ⊑ Healthy} Γ2 = {Human ⊑ ∃eats.Food, Food ⊓ Plant ⊑ Vegetarian} It is easily seen that Γ1 ∪ Γ2 is a pr... |

25 | Grothendieck institutions - Diaconescu - 1994 |

20 | Comorphism-based grothendieck logics - Mossakowski - 2002 |

18 | Connecting many-sorted theories - Baader, Ghilardi - 2005 |

15 |
Modular ontology languages revisited
- Grau, Kutz
- 2007
(Show Context)
Citation Context ...0 9 We consider here only DDL in its most basic form, comprising into and onto rules, but no individual correspondences. DDL in this form can be seen as a sub-formalism of one-way E-connections, see (=-=Cuenca-Grau and Kutz, 2007-=-). 10 But can be weakened to ALC ms or the link language of DDLs, or strengthened to more expressive many-sorted DLs such as ALCQI ms . T2sSuch many-sorted theories can easily be represented in a diag... |

13 | Failure of interpolation in combined modal logics
- Marx, Areces
- 1998
(Show Context)
Citation Context ...s some χ that only uses symbols occurring in both ϕ and ψ, with ϕ |= χ and χ |= ψ. TIP follows from AIP in the presence of a deduction theorem, and in this case, a further generalisation follows (see =-=Areces and Marx, 1998-=-): SIP. (Splitting Interpolation) If ϕ0, ϕ1 |= ψ, then there exists some χ that only uses symbols occurring in both ψ and the union of the symbols of ϕ0 and ϕ1, with ϕ0 |= χ and ϕ1, χ |= ψ. We now fur... |

12 | V.: Modular ontologies - a formal investigation of semantics and expressivity
- Bao, Caragea, et al.
- 2006
(Show Context)
Citation Context ... imports, it allows for renaming, hiding and parameterisation. Other languages that have been introduced include DDLs (Borgida and Serafini, 2002, 2003), E-connections (Kutz et al., 2004), and P-DLs (=-=Bao et al., 2006-=-b,a), where, roughly speaking, more involved integration and modularisation mechanisms than plain imports are envisaged. We will use the formalism of colimits of diagrams as a common semantic backbone... |

12 | Institutionalising Ontology-Based Semantic Integration - Schorlemmer, Kalfoglou |

11 | Failures of the interpolation lemma in quantified modal logic - Fine - 1979 |

9 |
Combining OWL Ontologies Using EConnections
- Cuenca-Grau, Parsia, et al.
(Show Context)
Citation Context ...ave been discussed in recent years, including the introduction of various so-called ‘modular ontology languages’. The module system of the Web Ontology Language OWL itself is as simple as inadequate (=-=Cuenca-Grau et al., 2006-=-b): it allows for importing other ontologies, including cyclic imports. The language Casl (see Bidoit and Mosses, 2004; CoFI (The Common Framework Initiative), 2004) has been used in Lüttich et al. (2... |

9 |
Interpolation and Definability: Modal and Intuitionistic Logic. Number 46 in Oxford Logic Guides
- Gabbay, Maksimova
- 2005
(Show Context)
Citation Context ...raig interpolation for L implies Craig-Robinson interpolation. Example 15 (Interpolation). The description logic ALC can be conceived as a syntactic variant of multi-modal K, for which (Gabbay, 1972; =-=Gabbay and Maksimova, 2006-=-) show Craig interpolation. K does not have implication, but satisfies the assumptions of Prop. 14. Hence, ALC has Craig-Robinson interpolation. The situation for DLs with nominals is less positive, i... |

9 |
Craig’s interpolation theorem for modal logics
- Gabbay
- 1972
(Show Context)
Citation Context ...frames. Then Craig interpolation for L implies Craig-Robinson interpolation. Example 15 (Interpolation). The description logic ALC can be conceived as a syntactic variant of multi-modal K, for which (=-=Gabbay, 1972-=-; Gabbay and Maksimova, 2006) show Craig interpolation. K does not have implication, but satisfies the assumptions of Prop. 14. Hence, ALC has Craig-Robinson interpolation. The situation for DLs with ... |

8 |
E-Connections and logics of distance
- Kutz
- 2004
(Show Context)
Citation Context ...inson interpolation. The situation for DLs with nominals is less positive, in fact, the presence of nominals generally destroys (standardly formulated) Craig interpolation (compare the discussion in (=-=Kutz, 2004-=-, Chapter 3.4), and Areces and ten Cate (2006)) but can sometimes be restored, for instance, by treating nominals as logical constants, i.e., by freely reusing them. Here is a counterexample formulate... |

7 | Institutional 2-cells and grothendieck institutions. In Essays Dedicated to - Mossakowski - 2006 |

6 | Interpolation for First Order S5
- Fitting
- 2002
(Show Context)
Citation Context ...owing propositions are folklore in institutional model theory, see Diaconescu (2007). 7 Craig interpolation for QS5 can be restored, however, by extending the language with propositional quantifiers (=-=Fitting, 2002-=-) or nominals and @-operator (Areces et al., 2003). 8 S4u can be thought of as the independent fusion of the modal logics S4 and S5, which both have interpolation, plus the containment axiom �ϕ → �ϕ. ... |

6 | Composition of Default Specifications - Guerra - 2001 |

5 |
available at http://www.cofi.info
- Bidoit, Mosses
- 2004
(Show Context)
Citation Context ...odule system of the Web Ontology Language OWL itself is as simple as inadequate (Cuenca-Grau et al., 2006b): it allows for importing other ontologies, including cyclic imports. The language Casl (see =-=Bidoit and Mosses, 2004-=-; CoFI (The Common Framework Initiative), 2004) has been used in Lüttich et al. (2006) forsontologies. Beyond imports, it allows for renaming, hiding and parameterisation. Other languages that have be... |

4 | Three uses of the Herbrand-Genzen theorem in relating model theory and proof theory - Craig - 1957 |

4 | Ontology reuse: Better safe than sorry, 2007. Available at http://www.cs.man.ac.uk/ ∼ bcg/extended.pdf - Grau, Horrocks, et al. |

4 | Development of Modular Ontologies in CASL - Lüttich, Masolo, et al. - 2006 |

3 | my Ontologies Fit Together - Cuenca-Grau, Horrocks, et al. - 2006 |

2 | On Importing Knowledge from DL Ontologies: some - Borgida - 2007 |

1 |
Repairing the Interpolation Theorem
- Areces, Blackburn, et al.
- 2003
(Show Context)
Citation Context ...ional model theory, see Diaconescu (2007). 7 Craig interpolation for QS5 can be restored, however, by extending the language with propositional quantifiers (Fitting, 2002) or nominals and @-operator (=-=Areces et al., 2003-=-). 8 S4u can be thought of as the independent fusion of the modal logics S4 and S5, which both have interpolation, plus the containment axiom �ϕ → �ϕ. The interpolation property transfers to the fusio... |

1 | On a generalised modularisation theorem - Dimitrakos, Maibaum - 2000 |