THEORY OF CONCEPTUAL GRAPHS

CHAPTER 1 - CONCEPTS

Basic Ideas

Concepts represent entities in the world. Such entities may be concrete objects, such as bricks or people or anything else represented in normal language by a noun. Similarly concepts may be more abstract “objects” such as instances of running or giving or anything else represented in normal language by a verb. States are also represented as concepts. It is probable that the concept is somewhat overloaded. The term “concept” is possibly a bit pretentious but it is intended to convey the idea that these objects carry content, meaning and definition. In fact they are typed identifiers. However, as will be seen, concepts are defined in terms of other concepts according to their degree of specialisation.

Since concepts represent entities that exist in the world they make the statement that some entity exists. They are complete conceptual graphs in themselves and can stand alone.

There are two parts to any concept. The first part is the type label and the second part is the referent.

Type Label

The type label is a label which specifies some taxonomic group to which the entity represented by the concept belongs. It can take the form of a word, such as cat or throw, or it can take the form of a unique catalogue number. Whatever form it takes the type label captures those aspects of an entity's description or definition that are essential and unchanging.

Additionally the type label can be the generic marker which is a place holder for the actual type label which is not specified or known. Normally the generic marker is unnamed but in situations where it must be referred to elsewhere it can have a label, a coreference marker, attached to it.

Type labels may also be entire graphs, held in contexts, which either define the individual directly or describe some situation. In the case where the context defines the entity that the whole concept represents then the context must contain a concept with the same referent as the main concept. In the case where the main referent identifies a whole context then that context does not contain a copy of the main referent.

Where a concept contains some form of type label then that label will have a type definition held somewhere else. The type label, or its definition, is intended to capture those aspects of an entities description that are necessary for that entity to be of that type. In other words it captures the “necessary and sufficient” conditions that define a type. For example, the type label CAT embodies all those things that all cats must be. The type label, or its definition, is thus an entity's natural type.

Referent

The referent of a concept is a label which names some particular individual entity. It can take the form of a word such as john, or it can be a unique catalogue number, an individual marker, for the entity represented. A concept with such a referent is an individual concept.

Also the referent can be the generic marker, which indicates that the concept represents some entity or other but which one is not specified or known. Normally the generic marker is unnamed but in situations where it must be referred to elsewhere it can have a label, a coreference marker, attached to it. A concept with a generic marker is a generic concept.

Referents may also be entire graphs, held in contexts, which either define the individual directly or describe some situation. In this case the referent is said to be a second order referent and the concept is a second order concept. If a referent is second order then its type label must also be second order.

The referent may also contain a number or literal text which are self referencing. Some authors would even advocate placing pictures or video clips into the referent field. This is satisfactory except that any increase in types of marker increases the computational complexity of a real system. Logically there are only two types of marker: generic and individual. Inclusion of others is merely a convenience.

The referent, and therefore the whole concept, is existentially quantified. Therefore any one concept represents a single entity. Some authors include the universal quantifier as a possible referent. This only serves to confuse the issue and to complicate the processing of concepts in a real system.

Multiple Labels

Two or more type labels or referents may be included in the type or referent field, separated by = symbols in Sowa's notation and, for pragmatic reasons we also permit the use of the “/” symbol. This is to indicate that the different markers refer to the same individual. Logically this is a bit odd but we will see later that it is quite possible to discover that what were thought to be two individuals, with different markers, turn out to be the same one. This simple syntactic device adds equality to the logical primitives of conceptual graphs.

Representation

Concepts are graphically represented as boxes with two fields. The left hand field is the type field and is where the type label goes and the right hand field is the referent field and is where the referent goes. The two fields are separated by a colon. Markers in either field can have one of the following forms:

Table 1.1 – kinds of object placed into type and referent fields

There two types of notation: Display notation (Display form) and Linear notation (Linear form). The display form is normally the preferred notation because all links between concepts can be shown in a very direct way. However, in order to enable conceptual graphs to be typed into a computer there exists and equivalent linear form.

In either form of notation labels are shown in upper case where they are type labels and in lower case where they are individual markers. A type label placed in the referent field would be shown in upper case whereas it is never the case that an individual marker would appear in the type field.

Display Form Of Concepts

In the display form concepts are shown in boxes. Examples of concepts are:

Table 1.2 – a selection of concepts in display form

Other examples can be contrived but these give the idea.

Linear Form Of Concepts

To translate between notations the following equivalences are used:

Table 1.3 – equivalences between display and linear forms

The following table shows the same concepts as in table 1.2 but in linear form:

Table 1.4 – a selection of concepts in linear form

More On Markers

One important principle of conceptual graphs, especially when drawn in display form, is their readablity. What happens inside a computer is another matter. Individual markers are, logically, all of the #nnn... form and any other form is merely a convenience for the human read. Thus where there is an elephant known as Clyde to a real human this may be shown as clyde in the concept. However, a computer system ought to store this as #53246 and keep a table of mappings between the “display name” of the marker and its real version.

Generic markers are free, existentially quantified variables. In principle they could acquire any individual marker as their value in a process known as “instantiation”. Should a generic marker with a coreference marker acquire a value then all coreferent generic marker would immediately acquire the same new value.

Type labels can be arranged into generalisation hierarchies. More will be said about this in the section on lattices but for now the point is that generalisation lattices can be represented as strings of Booleans. These strings can be and have been used as type labels.

Conformity Relations

For any concept [ TYPE : ref ] we say that the entity represented by ref is of type TYPE. We can also say that ref conforms to TYPE, written TYPE :: ref. Every individual referent will have an entry in a conformity relations table, a table of these conformity relations.

The purpose of the conformity relations table is to check that, should a generic marker acquire an individual marker as its value, that that individual marker is appropriate for the type label. If the acquisition were done as part of a series of logical inferences then there would not be a problem but there are certain operations on conceptual graphs that are not truth preserving and which require the conformity to be tested.

More will be said about conformity relations in the section about the joining of graphs.

Updated 14^th December, 2006