THEORY OF CONCEPTUAL GRAPHS

CHAPTER 2 – CONCEPTUAL RELATIONS

Basic Ideas

Entities exist in relationships with other entities. These relationships are most commonly held between pairs of entities but there is no reason why more might not take part. In addition it is possible for properties to be held by an entity without other entities being involve. The kinds of properties that are captured by conceptual relations are those things that are represented by adjectives and adverbs in natural language.

A conceptual relation is shown by an arrow which points from the relation towards the entity which has the property or attribute expressed by the relation and it points away from all those entities which are themselves some part of the property towards the relation. The property or attribute is a role played by the holder of the property and such properties are known as role types. Any such role will be potentially temporary and is not part of the necessary conditions for defining that nature of the property holder. Examples will be given later. The arrow, or arc, is labelled with the relation label.

Since relations depend on the existence of the entities that the relationship exists between a relation does not stand on its own. Therefore a relation without all attached concepts is not a conceptual graph.

Relation Label

The relation label specifies the relationship between one entity and another. It can take the form of a word, such as father or agent, or it can take the form of a unique catalogue number. Whatever form it takes the relation label captures aspects of an entity's relationship with another entity.

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

Relation labels may also be entire graphs, held in contexts. In the case where the context defines the relationship 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.

Conceptual relations express relationships between entities under the assumption that all aspects of an entity's existence that are not captured by its type label can be captured by relationships with other entities. In other words they capture accidental or transient properties or attributes that a particular entity holds at some point in time.

Types Of Relation

The number of entities involved in any relation is known as its arity. A relation between two entities has an arity of 2 and is a binary relation, also called 2-ary or diadic. A relation between 3 entities is a ternary relation, also called 3-ary or triadic. Similar names exist for other arities.

A monadic relation is one in which some entity has a property that does not depend on the existence of some other entity. Such relations might be temporary states than an entity finds itself in. Also it is possible to take this to the extreme and allow a “relation” to exist without any entities involved at all. Such relations may be thought of as constants, although what the psychological equivalent of this is is not clear.

In general relations can have various arities and within any arity there may be any number of role players, up to the arity.

Representation

The kinds of object that the relation field can contain are the same as those that concepts can contain. As with type labels relation labels are shown in upper case. Relations can be shown in either of two forms:

Display Form Of Relations

The graphical display form for relations consists of an oval, which contains the relation field, and arrows which are attached to the oval and which have arrow heads pointing either towards or away from the oval as appropriate.

Diagram 2.1 – a typical diadic relation in display form

This can be read in either of two ways:

Concept 1 has a Relation which is Concept 2

Concept 2 is a Relation of Concept 1

It is important that a consistent reading of relations is adopted because otherwise it s impossible for a reader to know which concept is the role player and which is not. The following examples illustrate how relations of various arities are represented and read:

Diagram 2.2 – a selection of relations of various arities

Linear Form Of Relations

Since relations are somewhat branched the linear notation for them needs to be quite complex. This is particularly the case where graphs with several relations are written. The basic linear form for relations is to represent the arrows as hyphens (-) with “greater than” (>) signs and “less than” (<) signs as the arrow heads. The relation node is shown between pairs of parentheses ( ( ) ). All well formed relations must include the concepts attached to each arrow.

John Sowa's version of the linear notation is incomplete. It does not take into account arities greater than 2 nor does it take into account the possibility of relations with more than one outward pointing arrow. We will briefly survey Sowa's linear notation for single relations and then go on to discuss extensions that allow any relation to be correctly written.

Sowa's Relations

The general form of a diadic relation is:

[ Concept 1 ]->( Relation )->[ Concept 2]

or, the other way round:

[ Concept 2 ]<-( Relation )<-[ Concept 1]

What Sowa does is always to start with a concept and then follow it with a list of attached relations. Thus the two relations above would become:

[ Concept 1 ]-

( Relation )->[ Concept 2]

or, the other way round:

[ Concept 2 ]-

( Relation )<-[ Concept 1]

In each case the assumption is that the missing arrow points the same way as the included one. However, in real relations this is not necessarily the case and, in particular, a monadic relation such as:

[ MAN ]-

( FATHER )

is somewhat ambiguous. There are other problems which will become apparent when we deal with graphs with several relations.

The following Sowa linear form examples are the equivalents of the display form relations in diagram 2.2:

[ CAT : loki ]-

( LOCATION )->[ KNEE ]

[ MAN : #32546 ]-

( PARENT )<-[ BOY : * ]

[ MAN : #32546 ]-

( FATHER )

[ MAN : #32546 ]-

( * )->[ BOY : * ]

[ SET : *x ]-

( SUBSET )->[ SET : *x ]

In each case the hyphen is assumed to have an arrow head pointing the same way as the arrow. Notice that the third of these is ambiguous. Also notice that the last one linearises what is a cyclic graph by repeating the concept and putting in a coreference link to indicate that the two concepts in the linear form are really the same one.

Extending The Notation Of Relations

We require certain extensions to Sowa's notation for relations to allow n-ary relations and relations with different numbers of inward and outward pointing arrows to be represented.

When dealing with large graphs or large relations it can be convenient for the syntax to allow the placing of the relation node at the start, at the end or anywhere inside the list of attached concepts. Each arc should have its arrow shown. The following are all examples of the BETWEEN relation as applied to three bricks, x, y and z, where x is between y and z:

[ BRICK : *x ]<-( BETWEEN )

<-[ BRICK : *y ]

<-[ BRICK : *z ]

[ BRICK : *y ]->

[ BRICK : *z ]->( BETWEEN )->[ BRICK : *x ]

( BETWEEN )->[ BRICK : *x ]

<-[ BRICK : *y ]

<-[ BRICK : *z ]

[ BRICK : *x ]<-

[ BRICK : *y ]->

[ BRICK : *z ]->( BETWEEN )

Each of these can be written on one line, for example:

[ BRICK : *y ]->[ BRICK : *z ]->( BETWEEN )->[ BRICK : *x ]

More will be said about linear form in the section on graphs.

Numbering Of Arcs

Some authors, including Sowa, add numbers to the arcs of a relation. This, they claim, clarifies the identity of each concept in terms of the correspondence between a mathematical function with an ordered set of formal parameters and the function-like semantics of, at least some, relations. There may be a case for this but we argue that the type label of each attached concept should specify the role of that concept within the relation. Nevertheless the following example shows how arc numbering works and shows the linear form for numbered arcs:

The numbering of arcs in this way requires one to know which of bricks y and z is which, even though they are unspecified in the referents. It also requires the reader somehow to know that the brick attached to arc 1 is always the brick to the left (say) if brick x whilst the brick attached to arc 2 is always the brick to the right of brick x. This is clearly not possible in general. Although any particular group might have decided on this convention it is not universal.

Updated 14^th December, 2006