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.