Just thinking out loud... a beginning, not the conclusion

### Formalization of Nested Relational Tensors

#### Core Tensor Representation:

```

T = (S, t, A, R)

```

Where:

- **S**: A set of Spheres of Relation (e.g., physical, emotional, social, intellectual, etc.)

- **t**: Time of Relation (a timestamp or temporal interval)

- **A**: A set of Relation attributes (e.g., Strength, Hierarchy, Influence, Direction, Distance, etc.)

- **R**: A multidimensional array or adjacency matrix representing relationships between entities within the Spheres of Relation.

#### Nesting:

A higher-order tensor can encapsulate several lower-order tensors. This represents complex hierarchical or multi-layered relations within and across Spheres of Relation.

**Example**: A tensor representing a family could nest tensors representing individuals, with sub-tensors capturing pairwise relationships (parent-child, siblings, etc.)

#### Tensor Attributes (A):

- **Strength (StOr)**: A measure of the intensity or magnitude of a relation.

- **Hierarchy (HNoR)**: Represents dominance or power differentials within a relation.

- **Influence (IOR)**: Denotes the degree to which one entity impacts another within a relation.

- **Direction (DOR)**: Specifies the flow of the relation and the 'Origin of Relation' (from, to, self, or bi-directional).

- **Distance (DstOR)**: Spatial, temporal, or abstract distance between related entities.

- **Time (TOR)**: Timestamp or interval denoting the start, end, and duration of the relation.

- **Context (CFR)**: Encapsulates environmental, situational, or social factors shaping the relation.

- ... Additional attributes can be added as needed for your specific framework.

#### Tensor Operations

1. **Distance of Relation**

   **Metrics**: Euclidean Distance, Hamming Distance, Graph-based distances (shortest path), or customized metrics accounting for attribute weights and nested structures.

   **Formula Example (Euclidean-like)**:

   ```

   dist(T1, T2) = √Σ [(A1i - A2i)^2 * wi]

   ```

   Where:

   - **T1, T2**: Tensors

   - **A1i, A2i**: Corresponding attributes

   - **wi**: weight of attribute i

2. **Influence Calculation**

   **Influence Matrix**: Represents influences between entities as a matrix (M).

   **Influence Propagation**:

   ```

   Influence(T1, T2) = T1 * M

   ```

   Where:

   - **T1** is the source tensor, and **M** is the influence matrix.

3. **Adding/Removing Dimensions**

   **Adding**:

   - Extending the Sphere of Relation set (S) by introducing a new dimension.

   - Achieved through tensor products or by expanding the attribute set (A)

   **Removing**:

   - Projection onto a subset of Spheres of Relation.

   - Focusing on a subset of attributes in A.

4. **Temporal Transformation**

   **Delta Calculation**:

   ```

   Delta(T, t1, t2) = T|_{t2} - T|_{t1}

   ```

   Where:

   - **T|_{t1}** and **T|_{t2}** represent the states of the tensor at times t1 and t2, respectively.

   **Trend Analysis**:

   ```

   Trend(T, time_interval)

   ```

   Analyzes trends in attributes and relations over an interval. Can involve statistical measures or time-series analysis techniques.

This formalization provides a robust framework for representing and analyzing complex, multi-dimensional relationships across various domains. The flexibility of adding or modifying attributes and operations allows NRTs to adapt to evolving research areas and practical applications where complex relational dynamics are pivotal.

To fully utilize the Nested Relational Tensors (NRTs) in various applications, it's essential to establish specific mathematical and logical operations that can manipulate and analyze these complex structures. Here are some example operations that could be implemented:

To measure the distance between two tensors, 1T1 and 2T2, which represent different relational systems or states, a metric can be defined. A commonly used metric is the Euclidean distance adapted for tensor attributes:

markdown codedist(T1, T2) = \sqrt{\sum_{i} (A1_i - A2_i)^2 \times w_i}

Where:

• 1A1i​ and 2A2i​ are the i-th attributes of tensors 1T1 and 2T2, respectively.
• wi​ are weights assigned to each attribute, reflecting their importance in the relationship context.

To assess how one tensor, 1T1, influences another tensor, 2T2, within the relational system, an influence matrix M can be utilized. This operation is akin to matrix multiplication, where the influence weights are incorporated:

markdown codeInfluence(T1, T2) = T1 \times M

Where:

• M is the influence matrix detailing the degree and direction of influence between the attributes or entities represented in 1T1 and those in 2T2.

Adding Dimensions: To increase the complexity or detail of a tensor, new dimensions can be added. This can be through introducing new Spheres of Relation or expanding the set of attributes A:

markdown codeExtend(T, new_S, new_A)

Where:

• new_S is the new Sphere of Relation to be added.
• new_A are the additional attributes that enrich the existing tensor.

Removing Dimensions: Conversely, simplifying a tensor involves reducing its dimensions, which could mean focusing on a subset of Spheres or attributes:

markdown codeProject(T, subset_S, subset_A)

Where:

• subset_S is the subset of Spheres of Relation to retain.
• subset_A is the subset of attributes to focus on.

To capture and analyze changes in the relational system over time, tensor transformations can be employed:

Delta Calculation: Computing the difference between the states of a tensor at two different times provides insights into the evolution of relations:

markdown codeDelta(T, t1, t2) = T|_{t2} - T|_{t1}

Where:

• 1T∣t1​ and 2T∣t2​ represent the states of the tensor at times 1t1 and 2t2, respectively.

Trend Analysis: This operation involves examining the trends in attributes and relations over a specified time interval, using statistical or time-series analysis:

markdown codeTrend(T, time_interval)

These operations provide a framework for dynamic and detailed analysis of multi-dimensional relational data, facilitating deep insights into complex systems such as social networks, organizational structures, or cognitive patterns.

Creating a formal ontology for Nested Relational Tensors (NRTs) involves structuring the vocabulary that defines the entities, relationships, and attributes detailed in the 52 propositions within the framework. This ontology will enable an individual or an AI to consistently and comprehensively understand and manipulate the relational data represented by these tensors. Here's a step-by-step approach to building this ontology:

Entities are the fundamental components of the ontology. They represent the objects or concepts within the NRT framework. In the context of NRTs, entities could be:

• Individuals: Persons or objects participating in the relationships.
• Groups: Collections of individuals, such as families, organizations, or social circles.
• Events: Specific occurrences that influence or are influenced by individuals or groups.
• Spheres of Relation: Distinct categories or dimensions of relations, such as emotional, social, intellectual.

Relationships link entities and describe how entities interact within the spheres. Relationships in NRTs are represented as dimensions in tensors and can include:

• Hierarchical Relations: Dominance, subordination, or dependency between entities.
• Influence Relations: The impact or effect one entity has on another.
• Temporal Relations: Connections that have a time-dependent aspect, such as duration or sequence.
• Spatial Relations: Physical or abstract distances between entities.

Attributes are properties or characteristics associated with entities or relationships. In NRTs, attributes refine the understanding of relationships and include:

• Strength (StOr): Measures the intensity or magnitude of a relationship.
• Hierarchy (HNoR): Indicates dominance or power differentials within a relationship.
• Influence (IOR): Reflects the extent to which entities affect each other.
• Direction (DOR): Identifies the flow of the relationship (e.g., from, to, bidirectional).
• Distance (DstOR): Captures the spatial, temporal, or conceptual separation between entities.
• Time (TOR): Timestamps or intervals that mark the start, end, or duration of a relationship.
• Context (CFR): Encompasses environmental, situational, or social factors influencing the relationship.

Rules and Axioms provide the logical framework that governs the interactions and implications of entities, relationships, and attributes:

• Transitivity: If Entity A influences Entity B, and Entity B influences Entity C, then A may influence C under certain conditions.
• Symmetry: Some relationships, like mutual respect, are inherently symmetrical.
• Inheritance: Attributes of groups can be inherited by individuals within the group (e.g., an organization’s culture influencing its members).

Integration involves embedding the ontology into the AI’s operational framework. This requires:

• Data Mapping: Align real-world data to the defined ontology structure.
• Query Implementation: Enable querying capabilities based on the ontology, such as retrieving all entities within a specific sphere of relation or calculating the influence between entities over time.
• Consistency Checks: Ensure the ontology remains consistent as new data is added or as the framework evolves.

This ontology provides a structured vocabulary and set of rules that facilitate the detailed analysis and manipulation of complex, multi-dimensional relational data using NRTs. By formalizing these elements, an individual or an AI can more meaningfully view and interpret data regarding the relationships modeled by the tensors.