Link Theory in Data Models

Connection Theory: From Relational Algebra to Duplet Networks

Connection theory offers a unified approach to data representation, where the basic element is a connection—an n-tuple of references to other connections. This mapping is defined as \(R \to R^2\), where R is the set of references. Such a foundation allows modeling structures without separate entities for nodes or tables.

Unlike the relational model, which relies on relations as subsets of the Cartesian product of domains \(R \subseteq S_1 \times S_2 \times \dots \times S_n\), connection theory eliminates the need for n-tuples with a fixed order. Data is grouped solely through connections, simplifying recursive definitions.

The graph model uses vertices V and edges \(E \subseteq V \times V\). Here, a vertex is equivalent to a closed connection, and an edge is a duplet of references. Connection theory reduces everything to one type of object, minimizing duplication.

Comparison of Key Models

Consider three approaches:

• Relational algebra: Tables as sets of n-tuples. Limitations: fixed number of columns (often <32), artificial attribute order.
• Directed graphs: Two sets—V (vertices) and E (edges). Issues: duplication when modeling sequences, complex deduplication of chains.
• Connection theory: Only connection-duplets. Network: \(N^2: R \to R \times R\).

| Model | Basic Entities | Sequence Representation | Unification Level |

|--------|------------------|----------------------------------|-------------------|

| Relational | Relations (n-tuples) | Direct via tuples | Low |

| Graph | Vertices + edges | Chains (with duplication) | Medium |

| Connection | Connections (duplets) | Recursive connections | High |

Simon Williams' associative model evolved: from items/links tables to a unified table of triplets or duplets.

Duplets as the Network Foundation

A connection-duplet is an ordered pair of references to connections with its own reference. For a set R of references, possible duplets form the Cartesian product \(R \times R\).

Example for R = {1, 2}:

R × R = {
  (1, 1),
  (1, 2),
  (2, 1),
  (2, 2),
}

This yields 4 possible connections in a network of 2 elements. The Cartesian product matrix visualizes all combinations: rows and columns are references, cells are duplets.

The network of connection-duplets is formalized by the mapping \(N^2: R \to R \times R\), where each reference is mapped to a pair of references. Recursion allows building arbitrary structures without external data types.

Graph vertices are transformed into self-connections (1,1), directed edges into (a,b), undirected edges into pairs (a,b) and (b,a). This eliminates the separate set V.

Advantages for Developers

Connection theory is oriented toward machine data representation, close to the associative processes of the brain. Key aspects:

• Universality: One entity for all structures—from graphs to tables.
• Recursion without references: In set theory, references mimic recursion; in pure theory, they are emergent.
• Storage efficiency: A single duplet table instead of multiple tables or sets.
• Scalability: No limits on arity (n in n-tuples).
• AI compatibility: Direct modeling of associations for neural networks.

For database implementation: store duplets as (id, source_ref, target_ref), index by ref for traversal.

Key Takeaways

• Theory reduces to \(R \to R^2\): all structures are a consequence of duplets of references.
• Simplification of the associative model: from two tables to one.
• Vertices = self-connections; edges = duplets; tables = projections of connection sets.
• Recursive definitions without external references in the base theory.
• Suitable for AI: data as a network of associations without hierarchies.

— Editorial Team