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CategoricalDomain — Semantic Specification

1. Definition

A CategoricalDomain is a ValueDomain whose admissible values are drawn from a finite, explicitly enumerated set of symbols.

It defines admissibility by exact membership, without ordering, arithmetic meaning, or implicit hierarchy.

2. Declaration

A CategoricalDomain exists as a semantic specialization that:

  • Declares a finite set of admissible values
  • Performs exact membership validation
  • Preserves value identity without transformation

Minimal declaration requirements:

  • A non-empty finite set of admissible values
  • A deterministic equality predicate

No ordering, distance, or numeric interpretation is implied.

3. Semantic Invariants

These invariants extend all ValueDomain invariants.

C1: Finite Membership

Only values explicitly declared in the domain are admissible.

C2: Exact Equality

Membership MUST be determined by exact equality, not similarity or coercion.

C3: Stable Enumeration

The admissible set MUST NOT change after domain declaration.

C4: Non-Ordinality

No ordering relation is defined between values.

4. Negative Definition

What CategoricalDomain is NOT.

  • Not ordered
  • Not numeric
  • Not probabilistic
  • Not hierarchical
  • Not fuzzy or similarity-based
  • Not extensible at runtime

5. Impossibilities

States or operations that cannot occur.

  • A value not in the declared set cannot be admissible
  • Partial matches cannot be accepted
  • New categories cannot be introduced dynamically
  • Categories cannot be merged or split implicitly

6. Boundary Conditions

Defined behavior at semantic limits.

  • Empty set: A CategoricalDomain with no admissible values is invalid.

  • Single-element set: Valid but degenerate.

  • Duplicate values: Duplicates collapse to a single admissible symbol.

  • Mixed types: Mixed-type categories are allowed but discouraged; equality semantics must be explicit.

7. Composition Rules

Allowed

  • CategoricalDomain ∘ Variable
  • CompositeDomain(CategoricalDomain, …)

Forbidden

  • CategoricalDomain ∘ NumericDomain
  • CategoricalDomain ∘ BooleanDomain

Categorical meaning must remain discrete and uninterpreted.

8. Example (Non-Normative)

Domain = {"red", "green", "blue"}

validate("red")   → admissible
validate("yellow") → inadmissible

9. Validation Rules

  • Static: Domain must declare at least one admissible value.

  • Runtime: Values not exactly matching an admissible symbol are invalid.

  • Structural: Any transformation of values violates semantics.

10. Semantic Notes (Non-Binding)

  • Designed for symbolic states, labels, and modes
  • Avoids hidden hierarchies or ordering assumptions
  • Enables strict categorical reasoning