Simple Conceptual Graphs

Concept nodes representing the same entity are said to be coreferent nodes. In basic conceptual graphs (BGs) only individual concept nodes may be coreferent. Simple Conceptual Graphs (SGs) enrich BGs with unrestricted coreference. The introduction of this chapter develops the discussion concerning equality started in the previous chapter and presents the conjunctive type and coreference notions. A new definition of a vocabulary extending the previous one with conjunctive types is given in Sect. 3.2. Section 3.3 defines SGs. An SG is simply a BG plus a coreference relation. The coreference relation is an equivalence relation over the concept node set with the following meaning: All concept nodes in a given equivalence class represent the same entity.

SG homomorphisms naturally extend BG homomorphisms. In Sect. 3.4, generalization and specialization operations defined on BGs are extended to SGs. Normal SGs are introduced in Sect. 3.5. An SG is normal if its coreference relation is the identity relation, i.e., each node is solely coreferent with itself. A normal SG can be associated with any SG. In fact, normal SGs and normal BGs can be identified, which emphasizes the importance of normal BGs. The notion of coref-homomorphism, which is specific to SGs, is introduced in Sect. 3.6. Instead of mapping concept nodes onto concept nodes as for a homomorphism, a corefhomomorphism maps coreference classes onto coreference classes. Relationships between homomorphisms and coref-homomorphisms are studied, and it is shown that, in the presence of a coreference, the intuitive meaning of generalization or specialization operations is better captured by coref-homomorphism than by homomorphism. The normal form of an SG is in a sense the most compact form of an SG. The antinormal form studied in the last section can be considered as the most scattered form of an SG: Each relation node is separated from any other relation node, and there are no multiple edges.