Conceptual Graphs (CG, www.conceptualgraphs.org) are one of the semantic text
models related to the semantic networks. Conceptual graphs were proposed in J.
Sova’s works and now play an important role as a means of modeling structures in
such areas as mathematical linguistics, bioinformatics, mathematical logic and
artificial intelligence. CG model was considered as a compromise between a
formal language and a graphical language.
Graph-based semantic depiction was extremely
popular in both computational and theoretical linguistics during the 1960s. In
the early 1960s a semantic network including a lattice of concept types was
introduced by Margaret Masterman. A bit later, Silvio Ceccato came up with
correlational nets based on 56 different relations with subtypes, instances and
many kinds of attributes. David Hays followed with his presentation of dependency
graphs. Although all these graph notations represented the relational structure
semantics, none of them could express full first-order logic.
In 1970s many graph notations were introduced to
represent first-order logic or a formally defined subset. In 1976 John F. Sowa
developed his version of conceptual graphs (CGs). He emphasized by CG syntax
the features of natural language and based their logical structure on existential
graphs (EG) of Charles Sanders Peirce (1909).
The difference between the EG and CG notations
Pierce used three operators to
visualize the first-order logic (FOL) using existential graphs (EG). These
an existential quantifier expressed by a bar or
linked structure of bars called a line of identity;
negation expressed by an oval context delimiter;
conjunction expressed by the occurrence of two or
more graphs in the same context.
Sowa in his CGs has used
rectangles, which are called concepts and linked concepts with ovals, which are
called conceptual relations.
Conceptual graphs are usually presented
in the following form, also called a display
Figure 1. CG Display Form
This display form is read as “ConceptB
represents the relation of ConceptA”, where arrows direction points out the
direction of the display form reading. The display form shown on the figure 1
can be translated into text based form, also known as linear form:
The full stop in this linear
form indicates the end of the certain graph. Admittedly, concepts can have
referents referring to an instance or person of the concept. As an example, to
create the type label ‘Student’ and assign referent to it, let’s say, Laura,
the concept would look like:
This concept is read as “The Student known as Laura”, where
Laura is a conformity to the type label Student in the given concept. In other
words, Laura conforms to the label Student.
Figure 2 visualizes the difference
between the EG and CG. Both kinds of graphs describe the sentence ” If a person
owns a dog then he pets it”.
Figure 2. Existential and Conceptual Graphs
In EG an oval enclosure shows
scope and the default operator for oval with no other marking is negation, but
any metalevel relation can be linked to the oval. For CGs, Sowa used rectangles
instead of ovals as rectangles nest better than ovals; and more importantly,
each context box can be interpreted as a concept box that contains a nested CG.
In the given example, there is an arrow or an arc that is directed toward an
oval and identifies the first argument of the relation. An arrow that is
directed away from an oval marks the last argument.
The sentence “if a person owns a
dog then he pets it” can also be represented by the following linear formula:
All the model-theoretic
semantics in the CGs and EGs are specified in the ISO standard for Common Logic
(CL). Each semantic feature of graphs that is represented by a linear notation is
known as the Conceptual Graph
Interchange Format (CGIF). Two
versions of the CGIF exist:
Core CGIF, which represents a type of logic
without a type and expresses the full Common Logic semantics. This mostly corresponds
to Peirce’s EGs: its only primitives are conjunction, negation, and the
existential quantifier. It does permit quantifiers to range over relations, but
Peirce also experimented with that option for EGs.
Extended CGIF, which represents an upward compatible
extension of the core and adds a universal quantifier; type labels for
restricting the range of quantifiers; Boolean contexts with type labels If, Then, Either, Or, Equivalence, and Iff; and the option of importing
external text into any CGIF text.