Vector symbolic architectures (VSA) are widely used for reasoning in neuro-symbolic (NeSy) AI, yet high-dimensional codebooks often create severe memory bottlenecks that limit scalability and deployment. In this paper, we propose Gram-Space, a compression framework that applies Gram-Schmidt orthogonalization to represent codebook vectors in a compact orthonormal coordinate system. Gram-Space preserves the dot-product structure required by matrix-based VSA operators, which supports numerically equivalent execution of matrix similarity, probability vectorization, and attention score computations. We provide a correctness analysis showing that inner products are preserved under the orthonormal basis representation. Using modern GPU hardware, we benchmark the Gram-Space framework on standard neuro-symbolic reasoning datasets. Experimental evaluations across state-of-the-art VSA models show that Gram-Space reduces model-level GPU memory usage by up to 15.75x and improves inference latency by up to 3.62x. Profiling results further indicate that Gram-Space reduces allocation-heavy overhead in codebook-associated stages and improves hardware utilization for NeSy workloads.
Tensor Product Representations provide the structural fidelity required for symbolic reasoning in models but suffer from exponential dimensionality growth when encoding deep recursive structures. Conversely, Vector Symbolic Architectures maintain constant dimensionality but sacrifice capacity and fidelity due to noisy compression via superposition. In this work, we propose Orthogonal Subspace Carving (OSC), a memory architecture that binds fillers to roles by projecting onto the null space of the role basis before aggregating into a fixed order-p tensor. OSC uses projections to enforce geometric orthogonality between bound structures within a static memory trace. We show that this mechanism decouples the tensor order from the structural depth, enabling deep recursive binding within a constant memory footprint. By performing retrieval via recognition, this construction allows for component vectors that are orders of magnitude smaller than the memory tensor, giving superior memory efficiency in settings involving high superposition. We also show that TPR is a special case of binding in Clifford algebra, and give a Clifford formulation of OSC.
Sutra is a typed, purely functional programming language whose compiled forward pass is a PyTorch neural network. The compiler beta-reduces the whole program -- primitives, control flow, string I/O -- to one fused tensor-op graph over a frozen embedding substrate. Rotation binding, unbind, bundle, polynomial Kleene three-valued logic, and tail-recursive loops all lower to tensor operations; the Kleene connectives are Lagrange-interpolated polynomials exact on the {-1, 0, +1} truth grid. Validation is one fact tested two ways. (1) The same program runs on four frozen embeddings spanning two modalities -- three text encoders (nomic-embed-text, all-minilm, mxbai-embed-large) and one protein language model (ESM-2) -- and decodes bundles at 100% accuracy through width k=8 on every substrate, where the textbook Hadamard product has already collapsed (2.5% on mxbai-embed-large, 7.5% on all-minilm). (2) PyTorch autograd flows through the actually compiled graph: a fuzzy-rule classifier written in .su trains from random init (18.7 +/- 9.5%; chance = 20%, five classes) to 100.0 +/- 0.0% (three seeds) by backpropagating through the emitted graph, the symbolic source unmodified. A weighted variant additionally trains a scalar cosine gain and writes it back into the .su source as a numeric literal; recompiling reproduces the trained behaviour to ~2e-7 per logit, so the trained model is itself legible, recompilable code. The same artifact is therefore both a logic program and a trainable neural network.