Constraints: Completeness & Disjointness

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-VEHICLE case: 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 d in 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 o in 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_Type with Manufactured? = Y/N and Purchased? = 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 \| RESIDENT (every patient is exactly one) | | total / overlap | every instance is in at least one, possibly several | PART → MANUFACTURED \| PURCHASED (every part is at least one; some are both) | | partial / disjoint | an instance is in at most one subtype, or none | EMPLOYEE → MANAGEMENT \| UNION (an employee may be neither; never both) | | partial / overlap | an instance is in zero, one, or many subtypes | PERSON → EMPLOYEE \| STUDENT \| ALUMNUS (any combination, including none) |

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 is Y on 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.