Goal: Develop reasoning algorithms that operate natively in hyperdimensional space, enabling associative inference beyond traditional symbolic logic.
Key insight: HDC vectors encode semantic relationships in their geometry. This structure enables reasoning operations that discover implicit knowledge.
The foundation of holographic reasoning is the HDC Master Equation:
Where:
This single equation supports retrieval, completion, and inference in constant time.
The relation (B BIND A−1) is extracted and applied to C to find the analogous concept.
// Analogical reasoning in HDC function solveAnalogy(A, B, C, cleanupMemory) { // Step 1: Extract the relation from A→B const relation = bind(B, inverse(A)); // B BIND A⁻¹ // Step 2: Apply relation to C const rawAnswer = bind(C, relation); // C BIND relation // Step 3: Cleanup to nearest known concept return cleanup(rawAnswer, cleanupMemory); } // Example: king:man :: queen:? const answer = solveAnalogy( vectors.king, vectors.man, vectors.queen, conceptMemory ); // answer ≈ vectors.woman (high similarity)
HDC can encode structured knowledge using role-filler bindings:
// Encoding: "John loves Mary" const fact = bundle([ bind(roles.agent, entities.john), bind(roles.action, verbs.love), bind(roles.patient, entities.mary) ]); // Query: "Who does John love?" const query = bind(roles.agent, entities.john); const answer = unbind(fact, query); // Cleanup → entities.mary // Query: "Who loves Mary?" const query2 = bind(roles.patient, entities.mary); const answer2 = unbind(fact, query2); // Cleanup → entities.john
For transitive relations like IS-A, we can chain inferences:
Each application of isA⁻¹ moves one step up the hierarchy.
| Inference Type | HDC Operation | Example |
|---|---|---|
| Direct lookup | cleanup(X BIND relation⁻¹) | isA(Cat) → Mammal |
| Transitive (2 steps) | cleanup(X BIND rel⁻¹ BIND rel⁻¹) | isA(Cat) → Animal |
| Property inheritance | cleanup(X BIND isA⁻¹ BIND has⁻¹) | has(Cat) → warm_blooded |
| Reverse lookup | cleanup(Y BIND relation) | What is a Mammal? → Cat, Dog, ... |
HDC operations accumulate noise. The confidence decreases with inference depth:
Confidence Formula:
Where α ≈ 0.85-0.95 depending on vector dimensionality and noise level.
| Challenge | Current Status | Proposed Solution |
|---|---|---|
| Noise accumulation | Limits depth to ~4-5 steps | Iterative cleanup, higher dimensions |
| Negation handling | No natural representation | Dedicated negation vectors, bipolar encoding |
| Variable binding | Limited to role-filler patterns | Placeholder vectors, substitution algebra |
| Rule application | Requires symbolic fallback | Holographic rule encoding |
| Contradiction detection | Not implemented in HDC | Orthogonality testing |
AGISystem2 combines HDC with symbolic reasoning: