B6. COMPOSITE ENTITIES
1. Definitions
A composite entity is a thing composed of other things. It is one of
the most basic concepts in a knowledge base of commonsense knowledge.
It is hard to think of anything that is not a composite entity, and
much of our everyday vocabulary is for talking about composite
entities.
Under the concept of "composite entity" we mean to include complex
physical objects, such as a door, a cup, a telephone, a chair, and an
automobile; complex events, such as a hike, the process of erosion,
and a concert; and complex information structures, such as an
equation, a sentence, a theory, and a schedule. Some entities, such
as books, have both physical and informational components. In this
theory of composite entities, we do not make any distinctions among
the types of components an entity might have. From the standpoint of
the theory of composite entities, the physical-abstract distinction is
of no interest.
A composite entity is characterized by a set of components, a set of
properties, and a set of relations.
(forall (x) (1)
(iff (compositeEntity x)
(exists (s1 s2 s3)
(and (componentsOf s1 x)(propertiesOf s2 x)
(relationsOf s3 x)))))
The set of components of a composite entity is nonempty.
(forall (s1 x) (2)
(if (componentsOf s1 x)
(and (set s1)(compositeEntity x)(not (null s1)))))
There are no further constraints on the components.
A component of a composite entity is one of the components. That is,
the predicate "componentOf" gives us a quick way to say a single
entity is a component.
(forall (y x) (3)
(iff (componentOf y x)
(and (compositeEntity x)
(exists (s)
(and (componentsOf s x)(member y s))))))
An aggregate of two entities is a composite entity with those two
entities as its components.
(forall (x y z) (4)
(iff (aggregate x y z)
(and (compositeEntity x)
(exists (s)
(and (componentsOf s x)(doubleton s y z))))))
We define a predicate "componentOrWhole" as the disjunction of
"componentOf" and being equal to the whole.
(forall (x1 x) (5)
(iff (componentOrWhole x1 x)
(or (equal x1 x)(componentOf x1 x))))
An entity y is external to a composite entity x if neither it nor any
of its components is equal to x or one of x's components.
(forall (y x) (6)
(iff (externalTo y x)
(not (exists (y1 x1)
(and (componentOrWhole y1 y)
(componentOrWhole x1 x)
(equal y1 x1))))))
The constraints on the properties of a composite entity are that they
are properties, that is, that they have one argument, and that that
argument be the whole or one of the components. For example, a window
x has a pane of glass y as its component, and we may want to say that
the pane is transparent. The property is the e such that
"(transparent' e y)". But we may want embedded properties as well.
For example, suppose we want the property that y is either transparent
or translucent. This would be the property e such that
(and (or' e e1 e2)(transparent' e1 y)(translucent' e2 y))
This is still a property because once we recurse through the
eventuality arguments e1 and e2, we bottom out in a single entity y.
We will capture this property of complex expressions with the
predicate "onlyarg*". Its definition recurses through eventuality
arguments and bottoms out in a single entity y.
(forall (y e) (7)
(iff (onlyarg* y e)
(and (eventuality e)(nequal y e)
(forall (y1)
(if (arg y1 e)
(or (equal y1 y)
(onlyarg* y y1)))))))
Then all the elements in the properties of a composite entity have an
"onlyarg*" that is either a component of the composite entity or the
whole.
(forall (s2 x) (8)
(if (propertiesOf s2 x)
(and (set s2)(compositeEntity x)
(forall (e)
(if (member e s2)
(exists (y)
(and (componentOrWhole y x)
(onlyarg* y e))))))))
The set of relations of a composite entity are relations between a
component or the whole, and something else.
(forall (s3 x) (9)
(if (relationsOf s3 x)
(and (set s3)(compositeEntity x)
(forall (e)
(if (member e s3)
(exists (y z)
(and (componentOrWhole y x)(arg* y e)
(nequal z y)(arg* z e))))))))
Note that z, the thing the component or whole y is related to, can be
another component, the whole if y is not the whole, or something
external. So the set of relations can include relations between two
components, between a component and the whole, or between a component
or the whole and something external. The axiom is also silent about
whether or not entities other than y and z are involved; the relation
could be among three or more entities.
A relation is a relation of a composite entity if it is in the
"relationsOf" set.
(forall (e x)
(iff (relationOf e x)
(exists (s)
(and (relationsOf s x)(member e s)))))
2. Some Simple Examples
Some of the kinds of entities we have previously introduced can be
viewed as composite entities.
Sets are probably the simplest kind of composite entity. As a
composite entity, its components are its members, its only property is
that it is a set, and there are no relations.
(forall (e s s1 s2 s3) (10)
(if (and (set' e s)(not (null s))(singleton s2 e)(null s3)
(Rexist e))
(and (compositeEntity s)(componentsOf s s)
(propertiesOf s2 s)(relationsOf s3 s)))))
Alternatively, we could call the "member" relations between its
elements and the whole the relations.
Ordered pairs can also be viewed as composite entities, where the
components are the first and second elements, there are no properties
other than the whole being a pair, and the relations are the "first"
and "second" relations between the components and the whole.
(forall (p x y e1 e2 e3) (11)
(if (and (pair' e1 p)(first' e2 x p)(second' e3 y p)(Rexist e1))
(exists (s1 s2 s3)
(and (compositeEntity p)
(doubleton s1 x y)(componentsOf s1 p)
(singleton s2 e1)(propertiesOf s2 p)
(doubleton s3 e2 e3)(relationsOf s3 p)))))
A sequence can be viewed as a composite entity, whose components are
the elements of the sequence and whose relations are the ordering
relations in the sequence.
(forall (e s) (12)
(if (and (sequence' e s)(Rexists e)
(exists (x)(inSeq x s)))
(exists (s1 s2 s3)
(and (compositeEntity s)
(forall (x)(iff (member x s1)(inSeq x s)))
(componentsOf s1 s)
(singleton s2 e)(propertiesOf s2 s)
(forall (e1)
(iff (member e1 s3)
(exists (x y)
(beforeInSeq' e1 x y s))))
(relationsOf s3 s)))))
Line 3 of this axiom says that the sequence is not empty. Lines 6 and
7 say that the components are the elements of the sequence. Line 8
says that the only property is "(sequence s)". Lines 9 through 13 say
that the relations are all the "beforeInSeq" relations.
3. The Figure-Ground Relation
The figure-ground relationship is of fundamental importance in
language and cognition. We encode this with the predication
"(at x y s)", saying that a figure x is at a point y in the ground s.
To prime our intuitions about the figure-ground relation, consider the
following examples of the uses of the preposition "at".
Spatial Location:
John is at the back of the store.
Ground: Physical space
Location on a Scale:
Nuance closed at 57 3/8.
Ground: Scale of numbers/prices
Membership in an Organization:
John is now at a competing company.
Ground: Set of organizations
Location in a Text:
At this point, we should provide some background.
A table is attached at the end of the article.
Ground: The text
(One might argue that this is simply spatial, the location
in the physical object that is the text, or temporal, the
time in the reading of the text, but it is true of every
copy of the text and every reading of it.)
Time of an Event:
At that moment John stood up.
Ground: Time scale
Event at an Event:
Let's have the discussion at lunch.
Ground: Set of events
At a Predication:
His intonation was at variance with his words.
John is at work.
Emily was at ease in his company.
Ground: Set of predications
We cannot simply locate a figure at some isolated point, not viewed as
part of a larger system or composite entity, because that would tell
us nothing. In any given instance, what counts as a ground and what
counts as a possible location are context-dependent. The background
structure s of which y is a component may differ in different
circumstances. Sometimes s will be a loose organization of vague
locations; sometimes it will be a coordinate system.
If x is at y in s, then y must be a component of s and x must be
external to s.
(forall (x y s) (13)
(if (at x y s)(and (componentOf y s)(externalTo x s))))
To be a ground, s must at least be a composite entity. (This follows
from "(componentOf y s)".) But the elements of s must be sufficiently
similar to support the same interpretation of the "at" relation in
every case. Thus, we would not have a single space consisting of
organizations and times, where "at an organization" means membership
and "at a time" means the time of occurrence of an event.
Or consider a book as a composite entity consisting of pages, a cover,
a binding, and content. The first three are sufficiently similar that
they can support some interpretations of "at", e.g., a bookworm at the
cover, the binding, or one of the pages. The pages themselves are
similar enough that they can support more complex relationships; for
example, "John is at page 235" may mean that John has read from the
beginning to page 235. But there is probably no property common to
all four components, including the content, that would support a
single interpretation of the "at" relation.
The parts of a bicycle are all similar in that they are parts of a
bicycle or in that they are physical objects. This will support a
spatial "at" relation. But the "at" relation cannot be something that
is conditional on further properties of the parts, such as "holding
the handlebars if the part is the handlebars, turning the chain if the
part is the chain, and leaning over the front wheel if the part is the
front wheel.
Thus, a ground is a composite entity whose parts are all uniform in
that they all share some property. We will take this to be
a sufficient condition as well and define a potential ground as
follows:
(forall (s) (14)
(iff (ground s)
(and (compositeEntity s)
(exists (e y)
(and (arg* y e)
(forall (y1)
(if (componentOf y1 s)
(exists (e1)
(and (subst y e y1 e1)
(if (Rexist s)
(Rexist e1)))))))))))
Something is a potential ground if it is a composite entity and there
is some property type e that holds when its parameter y is
instantiated with any component y1 of s. That is, there is some
property e that all the components share.
This could also be written as
(forall (s) (15)
(iff (ground s)
(and (compositeEntity s)
(exists (e y s1)
(and (componentsOf s1 s)(typelt y s1)(arg* y e)
(if (Rexist s)(Rexist e)))))))
The real existence of e ensures the real existence of all of its
instantiations.
Then we can say that the third argument of an "at" relation must be a
ground.
(forall (x y s)(if (at x y s)(ground s)))
The predicate "ground" means that its argument is a potential ground
for some figure and some "at" relation.
This figure-ground relation is central in the development of scales
and in the characterizations of many verbs of change of state.
Predicates Introduced in this Chapter
(compositeEntity x): x is a composite entity.
(componentsOf s1 x): s1 is the set of components of composite
entity x.
(componentOf y x): y is a component of x.
(aggregate x y z): x is a composite entity whose components are y
and z.
(componentOrWhole y x): y is either a component of x or x itself.
(externalTo y x): Neither y nor any of its components is equal
to x or one of its components.
(onlyarg* y e): y is the only arg* of e, after recursing
through all the eventuality arguments.
(propertiesOf s2 x): s2 is the set of properties of composite
entity x.
(relationsOf s3 x): s3 is the set of relations of composite
entity x.
(relationOf e x): e is one of the relations of x.
(at x y s): x is at component y in composite entity s.
(ground s): The components of s are sufficiently similar
for it to be a potential ground in an "at"
relation.
We used the following predicate from Chapter B9:
(geq n1 n2): n1 is greater than or equal to n2.