Classes

Hierarchies, taxonomies, and knowledge subjects

In programming, classes appear as source code in languages such as C++, Java, or Python. In TAoKE, classes are represented as knowledge subjects (sets of tuples in the graph). Classes collect similar knowledge subjects—animals, buildings, persons, plays, and so on. Similarity is defined by properties: data properties (e.g. .Gender, .Firstname) and, later, object properties. An instance relationship can be written as an atomic knowledge tuple such as (Linnaeus, ◊iof, IndividualScientists) or (^C14, ◊subClassOf, ^C) when a class is both instance and subclass of another class (e.g. a carbon isotope).

Sound taxonomies are discussed in [BaAl2022]; multi-level conceptual modeling in [AlCa2018] and [FoAl2021]. In deriver.app, class-like structure appears as typed nodes and predicates; use modules to keep large taxonomies maintainable.

Class hierarchies are built with object properties such as ◊is, ◊subClassOf, and ◊hasSubClass. These relations are anti-symmetric, so cycles in class hierarchies are not allowed.

Inheritance is central to object orientation: a derived (child) class specializes a base (parent) class; the link is often described as is-a or subClassOf. Generalization is primarily a model-level term. Inheritance underlies instantiation of objects.

Inherence means instantiating a single data-property definition into a knowledge subject (depending on the three data-property types in the treatise). Inheritance, inherence, and instantiation are closely related operational notions; the literature often neglects inherence even where instance and instantiation are used frequently.

If a class has more than one direct superclass, one speaks of multiple inheritance (known from C++, Python, Eiffel, etc.). Single-inheritance languages avoid the “diamond” problem but cannot express some taxonomies without interfaces or mix-ins.

More specific entities are grouped in subclasses: e.g. ^Species with subclasses ^Polar_Bear, ^Homo_Sapiens, and ^Scientist under ^Homo_Sapiens. Every subclass inherits definitions from all superclasses along ◊subClassOf paths. See the example knowledge graph and, on the canonical site, nuclide or BoW examples linked from the original Classes page.

For classes ^c, ^d, … the treatise defines sets of subclasses and superclasses using transitive closure ◊op+ of a property ◊op. The set of subclasses SBC^c, subclass hierarchy SBH^c, set of superclasses SPC^c, superclass hierarchy SPH^c, and full class hierarchy CH^c are given as formal definitions on taoke.de; OQL-style queries can enumerate subclass and superclass identifiers.

Root classes have no superclass; leaf classes have no subclasses. Class hierarchy depth measures the longest ◊subClassOf chain from a chosen root to a leaf.

For a data property instantiation on a class Σ^C, the particular that carries the value must sit under Σ^C in the superclass hierarchy: a data property defined on ^D may apply to particulars of subclasses of ^D. The canonical page lists illustrative sets such as SBC^Species and SBH^Species_by_Gender.

Following [ScSe2012] (p. 15), the subClassOf relation is characterised by: transitivity (if ^A is under ^B, it is under all superclasses of ^B), reflexivity (^A ◊subClassOf ^A), and antisymmetry for distinct classes (if ^A ◊subClassOf ^B then not ^B ◊subClassOf ^A).