A logical re-conception of neural networks: Hamiltonian bitwise part-whole architecture

We introduce a simple initial working system in which relations (such as part-whole) are directly represented via an architecture with operating and learning rules fundamentally distinct from standard artificial neural network methods. Arbitrary data are straightforwardly encoded as graphs whose edges correspond to codes from a small fixed primitive set of elemental pairwise relations, such that simple relational encoding is not an add-on, but occurs intrinsically within the most basic components of the system. A novel graph-Hamiltonian operator calculates energies among these encodings, with ground states denoting simultaneous satisfaction of all relation constraints among graph vertices. The method solely uses radically low-precision arithmetic; computational cost is correspondingly low, and scales linearly with the number of edges in the data. The resulting unconventional architecture can process standard ANN examples, but also produces representations that exhibit characteristics of symbolic computation. Specifically, the method identifies simple logical relational structures in these data (part-of; next-to), building hierarchical representations that enable abductive inferential steps generating relational position-based encodings, rather than solely statistical representations. Notably, an equivalent set of ANN operations are derived, identifying a special case of embedded vector encodings that may constitute a useful approach to current work in higher-level semantic representation. The very simple current state of the implemented system invites additional tools and improvements.

💡 Research Summary

The paper introduces a novel neural architecture called Hamiltonian Bitwise Part‑Whole Network (HNet) that departs fundamentally from conventional artificial neural networks (ANNs). Instead of treating inputs as high‑precision vectors processed by weighted sums, HNet first converts any data set into a graph where each node is a binary variable and each edge encodes one of four elementary binary relations (00, 01, 10, 11). These four relations correspond to logical operations (NOR, negative converse, negative implication, AND). For every edge a 2 × 2 Hamiltonian matrix is defined; its entries are limited to the set {‑1, ‑½, 0, ½, 1}. This extreme quantisation enables the entire system to run on low‑precision bitwise arithmetic.

Composite Hamiltonians for whole graphs are built by projecting each edge‑level Hamiltonian into the sub‑space of its two incident nodes and then simply summing all projected matrices. The resulting n‑dimensional Hermitian matrix H defines an energy function E(ψ)=ψᵀ H ψ for any state ψ (the binary vector of node values). By construction, E is non‑negative and equals zero exactly when ψ matches the graph that generated H. Thus recognition reduces to finding a zero‑energy configuration.

Learning proceeds through four high‑level operations: MEMORIZE stores each training example as a graph together with two edge‑type views (leading NIMPL edges and trailing NCONV edges). EXTRACT partitions the graph into connected components called “parts”; for non‑image data parts are defined by statistical regularities. COMPOSE transforms each part’s edge states into the corresponding unit Hamiltonians and aggregates them into a part‑specific composite Hamiltonian. For image‑like data, CONVOLVE creates shifted copies of each part within a small spatial window (‑2 to +2 pixels) to achieve translation invariance, implemented via permutation matrices.

During inference a test graph is processed by all stored part Hamiltonians; each yields an energy value. The parts with the lowest energies are selected, and their class labels are combined by a logical OR (voting) to produce the final prediction. All operations are embarrassingly parallel and scale linearly with the number of edges, because the Hamiltonians are extremely sparse and the arithmetic is limited to a handful of bitwise values.

The authors evaluate HNet on two benchmarks. First, a simplified MNIST experiment uses binarized 28 × 28 images; 32 prototypical samples per digit are selected, edges are derived from adjacent pixel pairs, and parts are extracted as connected edge clusters. Second, a credit‑card‑application dataset demonstrates the method on non‑topographic data, where parts are defined by frequent co‑occurring feature pairs. In both cases, a simple downstream classifier (e.g., SVM) applied to HNet‑derived representations outperforms the same classifier on raw data, and the overall accuracy exceeds that of comparable shallow ANN baselines. Moreover, the learned representations explicitly encode relational concepts such as “part‑of” and “next‑to,” offering a level of interpretability rarely achieved by standard deep models.

The paper positions HNet as an alternative to the current wave of neuro‑symbolic hybrids, which typically overlay symbolic modules on top of high‑precision neural nets, incurring substantial computational overhead. By embedding relational logic at the most primitive computational level, HNet unifies symbolic reasoning and statistical learning within a single, low‑cost framework. Limitations include reliance on hand‑crafted part extraction rules, modest scale of experiments, and lack of a demonstrated training algorithm that can discover parts automatically. Future work is suggested in automatic part discovery, extension to richer relational vocabularies (e.g., “above,” “before”), and hardware implementations that exploit the bitwise nature of the computations.

In summary, HNet offers a proof‑of‑concept that relational knowledge can be represented, learned, and inferred using only low‑precision Hamiltonian operators, achieving linear‑time inference, sparse memory usage, and interpretable hierarchical representations—potentially opening a new direction for efficient, logic‑aware neural computation.