# Conceptual Graphs

Conceptual Graphs (CG) is a logical formalism that includes classes, relations, individuals and quantifiers. This formalism is based on semantic networks, but it has direct translation to the language of first order predicate logic, from which it takes its semantics. The main feature is standardized graphical representation that like in the case of semantic networks allows human to get quick overview of what the graph means. Conceptual graph is a bipartite orientated graph where instances of concepts are displayed as rectangle and conceptual relations are displayed as ellipse. Oriented edges then link these vertices and denote the existence and orientation of relation. A relation can have more than one edges, in which case edges are numbered. An example of a graphical representation, so called Display Form (DF), of a sentence "a cat is on a mat" is shown in the figure below. Simple conceptual graph in the graphical representation DF

Using textual notation Linear Form (LF) this sentence would be written as

`[Cat]-(On)-[Mat]`

DF and LF are intended as representation (and presentation) formats for human. There is also a formal language CG Interchange Form (CGIF) defined. In this language the sentence would be expressed as

`[Cat: *x] [Mat: *y] (On ?x ?y)`

where `*x` is a variable definition and `?x` is a reference to the defined variable. Using syntactical shortcuts, the same sentence could be also written in the same language as

`(On [Cat] [Mat])`

The conversion between the three languages is defined as well as direct conversion between CGIF and KIF (Knowledge Interchange Format, see next section). In the KIF language this example would be expressed as

`(exists ((?x Cat) (?y Mat)) (On ?x ?y))`

All these forms have the same semantics in the predicate logic:

`∃ x,y: Cat(x) ∧ Mat(x) ∧ on(x,y)`

Conceptual graphs have the same expressing power as predicate logic. As we have seen there is again a possibility to define concepts (i.e., ontology) and then use them to express particular state of affairs.  