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**Modalities for an allegorical conceptual data model.**
*(English)*
Zbl 1321.18002

The paper describes a conceptual data model, based on allegories, already introduced in a previous work, and extends it to support many world semantics and, thus, modalities. The work is motivated by database research.

Apart Section 4, which concludes the paper, where a few examples in the application to databases are illustrated, the work mainly focuses on the mathematical aspects.

Section 2 illustrates the preliminaries, while Section 3 contains the novel contribution of the article. While Sections 2.1 and 2.2 recall the basic definition of allegory and, particularly, of distributive and locally complete distributive allegory, which form the framework where the data model is interpreted, the core of this section is the last part, 2.3, where the conceptual data model is recalled.

Since allegories abstract over the relational algebra, which is the traditional mathematical context for databases, the conceptual data model is, in a way, what one should expect: a pair \((G,E)\) formed by a finite graph \(G\) and a finite set of equations \(E\) on terms in the allegorical language on \(G\). Then, a conceptual data model is interpreted in an allegory \(\mathcal{A}\) via a graph morphism \(G \to \mathcal{A}\) which respects the equations.

Section 3 starts by extending the definition of allegorical terms to accommodate structural modalities, intuitively, the box and the diamond operators. Then, a modal interpretation is defined as an abstract many world semantics, that associates to each object in the graph \(G\) the collection of all its possible values, abstracted as an object in the allegory, and a family of standard interpretations, one for each of the possible worlds. This ‘flat’ interpretation is then extended to include the modal operators, essentially mimicking the usual set-theoretic semantics of modal logics in the allegorical framework.

A pair of examples illustrate how the basic modal logic is rendered in the allegorical framework, specifically on the allegory \(\mathcal{R}\) of relations, and on the allegory \(\mathcal{R}[L]\) of locale-valued relations.

The section concludes showing how to introduce other modal operators in the framework, apart the structural ones: in particular, the temporal operators ‘until’ and ‘since’ are defined together with their interpretation.

Although the approach is very general, abstract, and thus promising, the rather difficult formalism, and the fact that the examples are limited to the canonical allegories, limit the direct use of the illustrated results.

Apart Section 4, which concludes the paper, where a few examples in the application to databases are illustrated, the work mainly focuses on the mathematical aspects.

Section 2 illustrates the preliminaries, while Section 3 contains the novel contribution of the article. While Sections 2.1 and 2.2 recall the basic definition of allegory and, particularly, of distributive and locally complete distributive allegory, which form the framework where the data model is interpreted, the core of this section is the last part, 2.3, where the conceptual data model is recalled.

Since allegories abstract over the relational algebra, which is the traditional mathematical context for databases, the conceptual data model is, in a way, what one should expect: a pair \((G,E)\) formed by a finite graph \(G\) and a finite set of equations \(E\) on terms in the allegorical language on \(G\). Then, a conceptual data model is interpreted in an allegory \(\mathcal{A}\) via a graph morphism \(G \to \mathcal{A}\) which respects the equations.

Section 3 starts by extending the definition of allegorical terms to accommodate structural modalities, intuitively, the box and the diamond operators. Then, a modal interpretation is defined as an abstract many world semantics, that associates to each object in the graph \(G\) the collection of all its possible values, abstracted as an object in the allegory, and a family of standard interpretations, one for each of the possible worlds. This ‘flat’ interpretation is then extended to include the modal operators, essentially mimicking the usual set-theoretic semantics of modal logics in the allegorical framework.

A pair of examples illustrate how the basic modal logic is rendered in the allegorical framework, specifically on the allegory \(\mathcal{R}\) of relations, and on the allegory \(\mathcal{R}[L]\) of locale-valued relations.

The section concludes showing how to introduce other modal operators in the framework, apart the structural ones: in particular, the temporal operators ‘until’ and ‘since’ are defined together with their interpretation.

Although the approach is very general, abstract, and thus promising, the rather difficult formalism, and the fact that the examples are limited to the canonical allegories, limit the direct use of the illustrated results.

Reviewer: Marco Benini (Buccinasco)

### MSC:

18B10 | Categories of spans/cospans, relations, or partial maps |

03B45 | Modal logic (including the logic of norms) |

18D20 | Enriched categories (over closed or monoidal categories) |

03B52 | Fuzzy logic; logic of vagueness |

68P15 | Database theory |

03B70 | Logic in computer science |

Full Text:
DOI

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