Constraints: Completeness & Disjointness
◈ 6 cardsThe two constraint axes on every specialization — total vs partial (double vs single line) and disjoint d vs overlap o — their four combinations, and the discriminator each implies.
Two independent questions about every specialization
Once you have a supertype with subtypes, two business rules must be pinned down. They are independent — every specialization answers both — and they are the single most heavily tested notation in Chapter 3.
### 1. Completeness — must a supertype instance be in some subtype?
A completeness constraint addresses whether an instance of the supertype must also be a member of at least one subtype.
- Total specialization rule: every supertype instance must belong to some subtype. There are no "uncategorized" instances. Drawn as a double line from the supertype to the circle. (Insert a new supertype instance → you must simultaneously place it in at least one subtype.)
- Partial specialization rule: a supertype instance is allowed not to belong to any subtype. Drawn as a single line. (The
MOTORCYCLE-under-VEHICLEcase: a valid VEHICLE in no subtype.) Partial is actually the more common case in practice — most real classifications have exceptions.
### 2. Disjointness — may an instance be in two subtypes at once?
A disjointness constraint addresses whether an instance of the supertype may simultaneously be a member of two or more subtypes.
- Disjoint rule: if an instance is in one subtype, it cannot be in any other at the same time — mutually exclusive. Marked by the letter
din the circle. (An instance may change subtype over time — a UNION employee promoted to MANAGEMENT — but at any one moment it is in only one.) - Overlap rule: an instance may be in two or more subtypes simultaneously. Marked by the letter
oin the circle. (Part #4000 has both manufactured and purchased units on hand, so it is both a MANUFACTURED_PART and a PURCHASED_PART at once.)
The subtype discriminator
A subtype discriminator is an attribute of the supertype whose value determines which subtype(s) an instance belongs to. Its form follows directly from the disjointness rule:
- Disjoint → a simple attribute with mutually exclusive values:
Employee_Type='H'(hourly) |'S'(salaried) |'C'(consultant). One value → one subtype. - Overlap → a composite attribute of Boolean components, one per subtype:
Part_TypewithManufactured? = Y/NandPurchased? = Y/N. An instance can be Y on more than one, so it can sit in more than one subtype.
A disjoint discriminator can't represent dual membership (one value, one subtype), which is why overlap demands the composite-Boolean form.
The four combinations (memorize this table)
Because the two axes are independent, every specialization is one of four combinations:
| Combination | Meaning | Textbook example |
|---|---|---|
| total / disjoint | every instance is in exactly one subtype | PATIENT → OUTPATIENT \ |
| total / overlap | every instance is in at least one, possibly several | PART → MANUFACTURED \ |
| partial / disjoint | an instance is in at most one subtype, or none | EMPLOYEE → MANAGEMENT \ |
| partial / overlap | an instance is in zero, one, or many subtypes | PERSON → EMPLOYEE \ |
Worked example — choosing both constraints
An organization has employee types full-time salaried, part-time hourly, and contract. Every employee is at least one of these, and some contract workers are also part-time hourly. Specify the constraints.
- Completeness: every employee must be at least one type → total (double line).
- Disjointness: a contract worker can simultaneously be part-time hourly → overlap (
o). - Discriminator: overlap ⇒ a composite Boolean discriminator on
EMPLOYEE:Full_Time_Salaried? (Y/N),Part_Time_Hourly? (Y/N),Contract? (Y/N). A contract/part-time worker isYon two components.
So this is a total/overlap specialization. Walking the two axes independently — must they be in one? then may they be in several? — and then reading the discriminator off the disjointness answer is the whole exam procedure.