Basic Conceptual Graphs

This chapter presents the kernel of the knowledge representation language, namely basic conceptual graphs. A basic conceptual graph (BG) has no meaning independently from a vocabulary, and both are defined in Sect. 2.1. The fundamental notion for reasoning with BGs is the subsumption relation. Subsumption is first defined by a homomorphism: Given two BGs G and H, G is said to subsume H if there is such a BG homomorphism from G to H. Section 2.2 defines the BG homomorphism, as well as the particular case of BG isomomorphism. BGs and subsumption provide a basic query-answering mechanism, as shown at the end of this section. The sub-sumption properties are studied in Sect. 2.3. Subsumption is not an order as there may be nonisomorphic equivalent BGs, i.e., BGs that subsume each other. However, by suppressing redundant parts, any BG can be transformed into an equivalent ir-redundant BG. Irredundant BGs are unique representatives of all equivalent BGs. Finally, the subsumption relation restricted to irredundant BGs is not only an order but also a lattice. Section 2.4 introduces another way of defining the subsumption relation by sets of elementary graph operations. There is a set of generalization operations and the inverse set of specialization operations. Given two BGs G and H, G subsumes H if and only if G can be obtained from H by a sequence of generalization operations (or equivalently, H can be obtained from G by a sequence of specialization operations). Section 2.5 introduces the issue of equality, which will be a central topic of the next chapter (Simple Conceptual Graphs). A BG is said to be normal if it does not possess two nodes representing the same entity. Normal BGs form the kernel of reasoning based on homomorphism.

In Sect. 2.6, the complexity of BG fundamental problems is studied. In particular, it is proven that checking whether there is a homomorphism between two BGs is an NP-complete problem.