Type Hierarchy Vs Meta Hierarchy

• interlocutor: 4o

do you remember the multi-level modeling session where you provided me an example with "the HR department of Company X"?

Yes, I remember the multi-level modeling session. The example I provided involved the HR Department of Company X to illustrate how different levels in a multi-level model can be represented:

• M0 Level (Instance Level): This is where the specific, real-world instance resides, such as the HR Department of Company X.
• M1 Level (Class/Type Level): At this level, you would define the HR Department class, which generalizes the concept of HR departments that can be instantiated by companies.
• M2 Level (Meta-Class Level): This level could define OrganizationalUnitType, a more abstract class that generalizes types of organizational units, such as HR departments, finance departments, etc.
• M3 Level (Meta-Meta-Class Level): At this top level, you would deal with Types in a very general sense, abstracting further from specific organizational units.

The idea is to distinguish the different levels, where each level progressively abstracts away from specific instances toward more general, reusable concepts.

with the multi-level example, could m2 be a "Department class", m3 a OrganizationUnitType Class, and m4 just Types?

Yes, your proposed multi-level example with M2 as a "Department class", M3 as an "OrganizationUnitType Class", and M4 as "Types" makes sense and aligns with the principles of multi-level modeling. Let’s break it down:

Multi-Level Example:

• M0: HR Department of Company X (a specific instance of an HR Department).

  • This is the real-world instance, where you are modeling a specific HR department in a company.

• M1: HR Department (a specific subclass of "Department" that specializes in HR-related functions).

  • This is a more specific class (like a "concrete" department type) in which "HR Department" is a specialized type of department. This might also include departments like "Finance Department," "Marketing Department," etc.

• M2: Department class (a meta-class that generalizes different kinds of departments).

  • This is where you model the general idea of a department, which can specialize into HR, Finance, etc. This class could define common characteristics that all departments share, like having employees, roles, and organizational responsibilities.

• M3: OrganizationUnitType Class (a meta-meta-class that generalizes different types of organizational units).

  • This layer generalizes departments into broader organizational units. You might use this level to model abstract organizational structures, such as divisions, branches, departments, etc., all of which fall under the broader category of OrganizationUnitType.

• M4: Types (the most abstract, meta-level that classifies even the meta-classes).

  • At this level, you are dealing with abstract, fundamental types that classify other classes (such as "OrganizationUnitType Class"). This layer could be called Types or Meta-types, which define the most fundamental classifications that apply to all entities in the ontology.

How it Works:

• M0: Real-world instances of entities, like "HR Department of Company X."
• M1: Specific classes like "HR Department" that represent concrete specializations.
• M2: More abstract meta-classes like "Department class" that generalize multiple kinds of departments.
• M3: A higher-level meta-meta-class like "OrganizationUnitType Class" that abstracts across different organizational units (not limited to departments).
• M4: The most abstract level, Types, which provides the ultimate generalization for all entities, regardless of their domain.

Conclusion:

Your proposed structure works well for a multi-level model. It provides a clear hierarchical layering of specific instances, specialized classes, general organizational unit types, and abstract types, each existing at a different meta-level. This hierarchy allows for flexibility and scalability, especially in complex domains like organizational structures, where many different levels of abstraction and generalization are needed.

I get confused on superclass vs metaclass. If M2 "Department Class" is a superclass of "HR Department" is it really a meta-class?

You're raising an important distinction between superclass and meta-class, and it's easy to get them confused in multi-level modeling contexts. Let me explain the difference clearly and how it applies to your case.

Superclass vs. Meta-Class:

1. Superclass:

   • A superclass is a generalization of a class in a hierarchy. It defines common properties and behaviors shared by all of its subclasses.
   • Example: If "Department Class" is the superclass, then "HR Department" is a subclass of "Department Class." The Department Class defines shared characteristics like having employees, a budget, and responsibilities, while HR Department adds specifics related to human resources functions.
   • Hierarchy: The superclass/subclass relationship defines a type hierarchy within a single level of modeling (typically within one level, such as M1 in your case).

2. Meta-Class:

   • A meta-class is a class that classifies other classes, not just individuals. Meta-classes exist at a higher meta-level compared to their instances (which are classes themselves).
   • Example: If M2 "Department Class" is a meta-class, then "HR Department" (which is a class at M1) is an instance of the meta-class. In this case, the Department Class (M2) would describe what it means to be a "class of departments," while "HR Department" (M1) would be a specific instance of this meta-class.

Superclass vs. Meta-Class in Your Example:

Scenario 1: "Department Class" as a Superclass

• If M2 "Department Class" is a superclass of "HR Department" (M1), it is not a meta-class.
  • "Department Class" defines common features of all departments, and "HR Department" inherits those features while specializing its own.
  • This is a type hierarchy: "Department Class" -> "HR Department."
  • Hierarchy: M2 and M1 are on the same level of abstraction, dealing with classes and subclasses that inherit properties, not with meta-level distinctions.

Scenario 2: "Department Class" as a Meta-Class

• If M2 "Department Class" is a meta-class, then "HR Department" (at M1) is not a subclass but an instance of the meta-class.
  • In this case, "Department Class" (M2) represents a class of classes, and "HR Department" (M1) would be one example of a department class.
  • Hierarchy: M2 -> M1 is a meta-level hierarchy, meaning that M2 classifies classes, while M1 classifies instances.

Key Differences:

• Superclass relationship exists within the same meta-level (e.g., "Department Class" and "HR Department" are both classes, with one being more general).
• Meta-class relationship crosses meta-levels, where a meta-class (M2) classifies another class (M1) instead of simply generalizing it.

In Summary:

• "Department Class" as a superclass: It stays within the same level as its subclass (HR Department), and they exist at the M1 level together.
• "Department Class" as a meta-class: It moves up a level of abstraction, existing at M2, where it classifies the class "HR Department" at M1.

If you’re looking at multi-level modeling, then "Department Class" at M2 should be treated as a meta-class, meaning that "HR Department" (M1) is an instance of that meta-class. But if you’re only interested in inheritance within a single level, you’d treat "Department Class" as a superclass.

The difference boils down to whether you're crossing meta-levels (meta-classification) or staying within the same level (superclass/subclass inheritance).

which makes more sense?

In the context of multi-level modeling, it typically makes more sense to treat "Department Class" (M2) as a meta-class, and "HR Department" (M1) as an instance of that meta-class. Here’s why:

Why Meta-Class (M2) Makes More Sense:

1. Multi-Level Modeling Framework:

   • In multi-level modeling, each level represents a different degree of abstraction, and meta-classes exist at a higher level to describe classes at a lower level.
   • If you put "Department Class" at M2, it naturally fits the idea that it classifies classes like "HR Department" at M1. The HR Department is just one specific type of department, but it’s classified at a lower level of abstraction (M1), and "Department Class" defines the concept of what a department class is (M2).

2. Meta-Level Distinction:

   • If M2 represents a meta-class, then the next level down, M1, is a class level where HR Department is an instance of a department class, not a subclass. This fits well with multi-level modeling's goal to separate types of types across different meta-levels.
   • It aligns with your structure of progressively abstracting from instances (M0) to types of classes (M3), and ultimately to types in general (M4).

3. Clear Separation of Levels:

   • Multi-level modeling's strength lies in separating each level clearly. Treating M2 as a meta-class helps maintain this separation:
     • M2 (meta-class): "Department Class" describes the properties of department classes (at M1).
     • M1 (class): "HR Department" is an instance of a "Department Class," defining what a specific type of department (HR) looks like.
     • M0 (instance): "HR Department of Company X" is a specific instance of the "HR Department" at the lowest level.

4. Avoiding Overlap Between Levels:

   • If you treat "Department Class" (M2) as a superclass rather than a meta-class, it would imply that M2 and M1 are at the same level of abstraction, which can make the structure more confusing and collapse the separation between levels. This would be more of a typical type hierarchy, not a multi-level hierarchy.

Conclusion:

In a multi-level model, treating M2 "Department Class" as a meta-class and M1 "HR Department" as an instance of it makes more sense because it preserves the intended abstraction hierarchy across different levels. This approach fits naturally with the goals of multi-level modeling, where each level represents a progressively more abstract concept.

In which case, it would be wrong to say that "Department class" generalizes multiple kinds of departments

Correct, if "Department class" at M2 is treated as a meta-class, then it would indeed be wrong to say that it "generalizes multiple kinds of departments" in the traditional superclass-subclass sense. Here's why:

Meta-Class vs. Superclass Generalization:

• Meta-Class (M2): A meta-class classifies classes, not instances or subclasses. In your case, "Department class" at M2 does not generalize or define a hierarchy of subclasses (like HR Department or Finance Department); instead, it defines what it means to be a department class. The HR Department (M1) is not a subclass but an instance of the meta-class "Department class" at M2.

  • Example: "Department class" would define characteristics or constraints of department classes (like HR Department or Finance Department) at M1, but it would not generalize these departments as a superclass would.

• Superclass (Same level): A superclass exists at the same level of abstraction as its subclasses and generalizes them. If "Department class" were a superclass, it would generalize various subclasses like HR Department, Finance Department, etc., within the same level (e.g., M1).

  • Example: If "Department class" were a superclass at M1, it could generalize specific types of departments such as HR and Finance departments, which would be its subclasses.

Summary:

• If "Department class" is a meta-class at M2, it does not "generalize" multiple kinds of departments in the usual sense. Instead, it defines the characteristics that classes like HR Department (M1) must have.
• A superclass, on the other hand, generalizes its subclasses at the same level. If you want to express that "Department class" generalizes HR and other department types, you would need to consider Department class as a superclass at the same level (M1), not a meta-class at M2.

Thus, in a multi-level modeling context, saying "Department class" generalizes multiple types of departments would be incorrect if you're treating it as a meta-class. The correct phrasing would be that it classifies department classes at M1.

Maybe it would be better to call M2 a "Department type" class. So then, how would you generalize a department. i..e, at the M1 level, could you just have a "generic department" superclass, to harness the power of a type hierarchy within a "meta hierarchy"

Yes, that approach makes sense! By calling M2 a "Department type" class, you’re clearly distinguishing it as a meta-class that classifies different department types at M1. Then, within M1, you can introduce a "Generic Department" superclass to generalize different kinds of departments like HR, Finance, etc., while still leveraging the power of both the type hierarchy and the meta-hierarchy.

How This Would Work:

1. M2 - "Department type" Class (Meta-Class):

   • At M2, you would have the meta-class "Department type", which defines what it means to be a class of departments. This would classify any department class at M1, such as HR Department or Finance Department.
   • Example:
     • Department type (M2): Class that defines department types.

2. M1 - "Generic Department" Superclass (Type Hierarchy):

   • At M1, you can introduce a "Generic Department" superclass, which generalizes specific kinds of departments like HR Department, Finance Department, etc. This is part of a type hierarchy within the M1 level.
   • This allows you to organize specific department classes within a hierarchy for shared properties (such as departments having employees, budgets, etc.).
   • Example:
     • Generic Department (Superclass at M1): Generalizes common properties for all departments.
     • HR Department, Finance Department (Subclasses at M1): Inherit from Generic Department and add specific characteristics.

3. M0 - Real Instances:

   • At M0, you have specific instances of departments like "HR Department of Company X", which are instances of HR Department (M1).
   • Example:
     • HR Department of Company X (M0): An instance of HR Department at M1.

Power of Combining Type Hierarchy and Meta-Hierarchy:

• By introducing a type hierarchy at M1, you can still benefit from inheritance and shared properties between different department types (through the Generic Department superclass). This allows you to avoid redundancy and manage common behaviors or attributes across specific department types.
• Meanwhile, the meta-hierarchy ensures that you can manage the classification of different department classes (like HR Department and Finance Department) through the Department type (M2). This provides flexibility and abstraction across different meta-levels.

Example Breakdown:

• M2 - Department type class:

  • Defines what it means to be a department type.

• M1 - Generic Department superclass:

  • Generalizes properties of all department types, such as having employees, a budget, and objectives.
  • Specific subclasses, like HR Department and Finance Department, inherit from Generic Department and specialize those properties.

• M0 - Real-World Instances:

  • Instances like HR Department of Company X or Finance Department of Company Y exist at the real-world level.

Conclusion:

This structure works well for combining the strengths of a type hierarchy at M1 with a meta-hierarchy at M2. You get the benefit of generalization and inheritance at the class level while keeping clear distinctions between meta-levels. This makes your model both flexible and scalable, allowing you to handle different types of departments efficiently across different levels of abstraction.

It'd be nice to be able to visualize this clearly, but it feels like you'd need a 3D visualization... one axis for meta-hierarchy, and then two axes for any contained type hierarchies. I was also thinking likeC4 could be a good tool, with it's multi-level capabilities.

You’re absolutely right that visualizing multi-level models with both meta-hierarchy and type hierarchies can be complex, and a 3D approach could indeed help clarify the relationships.

Visualization Idea:

• Meta-Hierarchy Axis: This could be the vertical axis, representing different levels of abstraction (e.g., M0 for instances, M1 for classes, M2 for meta-classes, and so on).
• Type Hierarchy Axes: The other two axes could represent relationships within each level, showing inheritance and specialization among entities (like a type hierarchy for specific departments within the M1 level).

C4 as a Tool:

The C4 model (Context, Container, Component, Code) could indeed be useful for visualizing your multi-level model because it’s designed to handle multi-level systems and relationships. C4 is especially good at representing architectural relationships across different levels of abstraction. Although it’s often used in software architecture, you could potentially adapt its multi-level capabilities for your modeling needs.

• Context Level (C4): Could represent the broader system or domain context.
• Container/Component Levels (C4): Could represent specific departments or organizational units and how they relate within the meta- and type hierarchies.
• Code Level (C4): Could represent specific real-world instances (like an individual HR department).

3D Visualization Tools:

If you’re thinking about creating a truly 3D visualization, you might consider tools that allow for graph-based modeling with 3D space:

• Neo4j (with 3D rendering): Great for relationships and hierarchies, though typically used for graph databases.
• Gephi: A network visualization tool that could be adapted to multi-dimensional models.
• Blender or Three.js: For custom 3D modeling if you want complete control over visualization.

Conclusion:

Your idea of using a 3D visualization (with axes for meta-hierarchy and type hierarchies) makes a lot of sense, especially for complex, layered systems like multi-level models. Adapting tools like C4 for its multi-level capabilities or even exploring 3D visualization options could give you the clarity and insight needed to represent both the abstraction and type relationships effectively.