How Conceptual Search Works: BFS, SPARQL, TF-IDF and Formal Logic

This page explains the technical mechanics behind each algorithm. Every claim is verifiable — the algorithms are standard computer science, and the data sources are open.

Four Algorithms, One Answer

Each algorithm discovers a different type of connection between two concepts:

• BFS — editorial connections (how Wikipedia editors linked the concepts)
• SPARQL — ontological connections (where the concepts sit in the tree of knowledge)
• TF-IDF — statistical connections (hidden vocabulary overlaps)
• Formal logic — property-based connections (shared attributes with verifiable premises)

Running all four simultaneously gives the most complete picture of how two ideas relate. A connection that appears across multiple methods is stronger than one found by a single method.

BFS Pathfinding: Following the Links

Breadth-First Search is a graph traversal algorithm. It starts from a source node, explores all neighbors at distance 1, then all neighbors at distance 2, and so on. It uses a queue (FIFO) data structure to track which nodes to visit next.

On Wikipedia's graph:

• Nodes = Wikipedia articles (60 million+)
• Edges = internal hyperlinks (hundreds of millions)
• Path = the shortest chain of article-to-article links between two concepts

Time complexity: O(V+E) where V is the number of vertices visited and E is the number of edges traversed. In practice, most Wikipedia paths are found within 3-6 hops, so the search space is manageable.

Why BFS and not other algorithms? Wikipedia links are unweighted — a link is a link, with no inherent "cost." BFS is optimal for shortest-path problems in unweighted graphs. Dijkstra's algorithm is designed for weighted graphs and adds unnecessary complexity. DFS (Depth-First Search) does not guarantee finding the shortest path. A* requires a heuristic function, which is difficult to define for semantic connections.

SPARQL Ontological Reasoning: Climbing the Tree

SPARQL (SPARQL Protocol and RDF Query Language) is a W3C standard for querying structured data in RDF format. Wikidata's public SPARQL endpoint allows anyone to query its 100 million+ entities.

MapOfLogic uses the P279 property ("subclass of") to trace classification hierarchies. Every Wikidata entity has a chain of increasingly general categories. The algorithm traces upward from both concepts until the chains converge at a common ancestor.

This is formal ontological reasoning — not keyword matching, not statistical inference. The connection is derived from the taxonomic structure of human knowledge, encoded by thousands of Wikidata contributors.

TF-IDF Cosine Similarity: Measuring Overlap

TF-IDF (Term Frequency-Inverse Document Frequency) is an information retrieval technique that converts text into numerical vectors:

• Term Frequency (TF) — how often a word appears in a document
• Inverse Document Frequency (IDF) — how rare that word is across all documents
• TF-IDF weight = TF × IDF — words that are frequent in one document but rare overall get the highest weights

Cosine similarity then measures the angle between two TF-IDF vectors. A cosine of 1.0 means identical word distributions. A cosine of 0.0 means completely different vocabulary. Values between 0.0 and 1.0 indicate varying degrees of similarity.

When applied to Wikipedia articles, TF-IDF reveals that two concepts that seem unrelated may share surprisingly similar vocabulary — indicating a structural connection that neither link analysis nor ontological reasoning would detect.

Formal Logic: Verifiable Propositions

When two concepts share properties in Wikidata, the system constructs formal logical propositions:

Each premise is traceable to a specific Wikidata property and entity ID. The conclusion is derived, not generated. This makes formal logic propositions the most auditable form of connection — every step can be independently verified by anyone with access to Wikidata.

The Data Sources

Both sources are open, free, and verifiable. MapOfLogic does not use proprietary datasets. Every result can be independently confirmed using the public APIs.