AGGREGATION

1. AGGREGATION Reading Assignment Supplementary module 5 Object Aggregation

2. Introduction to Aggregation A complete, self contained and consistent structure of knowledge; a mathematical topos, or stock theme • The aggregation relationship creates the aggregate • Examples of aggregate objects • Anything with parts • Eg: Cars ( a structured collection of parts), object classes (a class of instances), a family, a household, a pattern, a perspective, a set, a list, a domain, a value set, etc • An aggregate object has its own identity, distinct from its parts (a distinct instance identifier) • The identity of the aggregate may or may not depend on some of its parts • Eg: Is a car without its chassis a car? A car without its radio? A car without its wheels? • This relationship between the aggregate and its part is called “existence dependency” • A more constrained form of aggregation: a polymorphism • An aggregate may have no members • Eg: the empty set, an organizational unit with no members • Aggregates may be structured or not • A composition is an aggregate for which we have at least some nominal information about associations between contents • A composition is therefore a subtype of the generic aggregate object • The state of the aggregate is the combined states of its constituents (Eg: the inventory example we discussed under state charts) • An aggregate may contain events • May change state spontaneously • In response to unknown events hidden inside the aggregate A list distinguishes between repeating members, a set or class does not

3. Aggregates have emergent properties Count of its members Eg: total insurance claim amount in a class of individual insurance claims The object class is a special kind of aggregate object Emergent properties emerge from mutual interactions between contents of the aggregate and also interactions between the contents and the aggregate itself The aggregate normalizes emergent properties Eg: Total insurance claim amount, number of claims etc. When we do not have information about the structures within an aggregate, it seems like emergent properties just “appear” Eg: behavior of the engine of your car is more than just the behavior of its parts collected together Emergent properties are sometimes distinguished from “Resultant Properties”, which may be inferred from the known structure of the aggregate. These Resultant properties are therefore polymorphisms of emergent properties. The meaning of “Empirical” and “Inferred” emerge thus; “Inferred” is a polymorphism of “Empirical” Aggregates give birth to emergent attributes as well as emergent object instances and classes. Eg: Emergent attribute: Count of instances Emergent object instance: The largest insurance claim Emergent object class: The top ten customers An aggregate may be an aggregate of aggregate objects; will have an identity distinct from its contents Eg: A city is an aggregation of its roads, lots and buildings; buildings are aggregations of rooms etc. Introduction to Aggregation (2) Eg: The horse power of the engine (normalized by the engine, not any part that constitutes the engine) Eg: Weight of an engine is the sum of the weights of its parts

4. Information content of Aggregation Information on existence of parts • “Aggregate of” and “consists of” are synonyms • “Part of” is the inverse of “Consists of” No information on parts AGGREGATE OBJECT Aggregate of 0 or more [part of 0 or more] Included in SUBTYPE OF OBJECT SUBTYPE OF Composed of 1 or more [component in 0 or more] OBJECT COMPOSITION Information on parts and their interactions

5. The genesis of “Part of” • If one object locates another, the located object also returns the favor • The proximity metric is symmetrical • Location is a symmetrical relationship • The genesis of the concept of “Place” • Any object may be a Place • Need not be a physical object • Containment is an asymmetrical polymorphism of “locate” • The located pattern is constrained to be inside the state space of another; the containing pattern encapsulates the contained pattern without necessarily incorporating it into its identity • Eg: Furniture may be inside a house without being a part of the house • The concept of size emerges from the cardinality constraints on this relationship • Size is a polymorphism of Capacity. Capacity emerges from upper bounds on cardinality of relationships in general. • When infinite cardinalities are involved, this relationship can become symmetrical • Other mutually exclusive polymorphisms of Locate: • Over, Under.. • Containment becomes more constrained and takes the form of incorporation • Although the incorporated object retains its identity and may exist independently • Although it has a distinct identity, an object may depend on a part to lend it its identity. • Eg: the concept of a disk depends on the concept of a delimiting circle to exist. The circle is a part of the disk; a car needs a chassis to be called a car • Existence dependency is a stricter form of incorporation and hence a polymorphism of “Part of” • For patterns of infinite extent, a part may equal the whole! • The Part of relationship can become symmetrical CAN AN OBJECT CONTAIN ITSELF? • Incorporation subsumes the very identity of the incorporated object • Incorporation becomes subtyping • The subtyped object is subsumed into the supertype object, and cannot exist independently.

6. Locate is transitive with its polymorphisms LOCATE contain Object 3 OBJECT 1 locate OBJECT 2 Object 4 contain The twisted surface is not only a part of the volume, but also implies the existence of the volume With patterns of finite extent, an object locating or containing itself itself conveys no information When we deal with infinite cardinalities, a part may contain the whole

7. TRANSITIVITY OF AGGREGATION HOUSE PERSON WALL • The aggregate and its components are related via the “consists of” relationship. • “part of” is the inverse relationship • Any object that relates to the aggregate will relate to its components via the “consists of” relationship • Which specific objects in the aggregate it connects with will crystallize in step with information content (starting with “Do not know” • Ownership is transitive with at “consists of” • Implies ownership of all polymorphisms of “consists of” • If ownership is specified higher up the ontology, it will imply ownership of all subtypes at lower levels • Eg: if ownership of objects Near another is a part of a rule, it will automatically imply ownership of all contained objects, parts and subtypes Owns 0 or more [owned by 0 or more] TRANSITIVE Consists of 1 or more [part of 0 or 1] Transitive relationship Owns 0 or more [owned by 0 or more] (automatically implied)

8. HOUSE TERMITE WALL • Location and its subtypes are mutually transitive • If A locates B, which in turn locates C, then A locates C • If A contains B, which in turn contains C, then A contains C • Similar relationships hold for the other polymorphisms of locate like Part of, Subtype of, List, Live in etc. • Containment and Part of are mutually transitive • If polymorphisms of location are joined in tandem, the composition will be transitive with respect to the least constrained member of the pair Live in 0 or 1 [lived in by by 0 or more] NOT TRANSITIVE Consists of 1 or more [part of 0 or 1] [part of] (inverse of “Consists of”) Non-transitive relationship TRANSITIVE Live in 0 or 1 [lived in by 0 or more] (not automatically implied) (also applies to location, containment etc)