Nested Conceptual Graphs

Abstract

The nested conceptual graph model presented in this chapter is a directextension of basic or simple conceptual graphs able to represent notionssuch as internal and external information, zooming, partial descriptionof an entity, or specific contexts. This model also allows reasoningwhile taking a tree hierarchical structuring of knowledge into account.Nestings are represented by boxes. A box is an SG and, more generally, abox is a typed SG. In full conceptual graphs, a box represents thenegation of the graph inside the box. Thus, for differentiating thesenegation boxes from the boxes used in this chapter, these boxes areusually called ``positive{''} boxes. Nevertheless, since the only kindof boxes considered hereafter are positive boxes, we omit the term``positive.{''}In Sect. 9.1 different notions representable by nested conceptual graphsare presented. In Sect. 9.2, we introduce Nested Basic Conceptual Graphs(NBGs), whose boxes consist of BGs. Nested Conceptual Graphs (NGs),which extend NBGs with coreference links, are presented in Sect. 9.3.Coreference links can relate concept nodes of the same box (thus boxesbecome SGs) but also of different boxes. Coreference links in nestedgraphs are more difficult to manage than in simple graphs. Indeed, sinceboxes can represent contexts, it is generally irrelevant to merge allnodes of a coreference class into a single node. In Sect. 9.4, we definegraph types, typed SGs, which are SGs with a graph type, and NestedTyped Conceptual Graphs (NTGs), which generalize NBGs by typing theboxes. All of these nested graphs classes are provided withhomomorphism. The FOL semantics Phi introduced for SGs is generalized toNTGs in Sect. 9.5 and a homomorphism soundness and completeness theoremis stated. As this semantics is a formula of the positive, conjunctiveand existential fragment of FOL, nested and non-nested CGs are somewhatequivalent. Finally, we build a mapping ng2bg from nested to non-nestedCGs which preserves homomorphisms. This mapping ng2bg shows, in anotherway than through logical semantics, that nested and non-nested CGs havethe same descriptive power. It is easy to implement ng2bg and thisavoids the construction of specific nested graph homomorphismalgorithms.Nevertheless, from a user viewpoint, NTGs are interesting wheneverknowledge is intrinsically hierarchical, and when reasonings must followthe hierarchical structure, because in ail NTG the hierarchy isexplicitly and graphically represented. Nested graphs can also beinteresting whenever large graphs have to be manually constructed, asthe separation of levels of reasoning increases efficiency and claritywhen extracting information.