Differentiable Logic Synthesis: Spectral Coefficient Selection via Sinkhorn-Constrained Composition
This work addresses the problem of hardware-efficient neuro-symbolic logic synthesis for AI/ML systems, representing an incremental advancement by adapting existing manifold-constrained routing techniques to logic synthesis.
The paper tackles the challenge of learning precise Boolean logic via gradient descent by introducing Hierarchical Spectral Composition, which selects spectral coefficients from a Boolean Fourier basis and composes them via Sinkhorn-constrained routing with column-sign modulation. The method achieves 100% accuracy on 2-variable and 4-variable Boolean operations and 76% accuracy on 3-variable operations via gradient descent, with exhaustive enumeration showing optimal ternary masks exist for all tested functions.
Learning precise Boolean logic via gradient descent remains challenging: neural networks typically converge to "fuzzy" approximations that degrade under quantization. We introduce Hierarchical Spectral Composition, a differentiable architecture that selects spectral coefficients from a frozen Boolean Fourier basis and composes them via Sinkhorn-constrained routing with column-sign modulation. Our approach draws on recent insights from Manifold-Constrained Hyper-Connections (mHC), which demonstrated that projecting routing matrices onto the Birkhoff polytope preserves identity mappings and stabilizes large-scale training. We adapt this framework to logic synthesis, adding column-sign modulation to enable Boolean negation -- a capability absent in standard doubly stochastic routing. We validate our approach across four phases of increasing complexity: (1) For n=2 (16 Boolean operations over 4-dim basis), gradient descent achieves 100% accuracy with zero routing drift and zero-loss quantization to ternary masks. (2) For n=3 (10 three-variable operations), gradient descent achieves 76% accuracy, but exhaustive enumeration over 3^8 = 6561 configurations proves that optimal ternary masks exist for all operations (100% accuracy, 39% sparsity). (3) For n=4 (10 four-variable operations over 16-dim basis), spectral synthesis -- combining exact Walsh-Hadamard coefficients, ternary quantization, and MCMC refinement with parallel tempering -- achieves 100% accuracy on all operations. This progression establishes (a) that ternary polynomial threshold representations exist for all tested functions, and (b) that finding them requires methods beyond pure gradient descent as dimensionality grows. All operations enable single-cycle combinational logic inference at 10,959 MOps/s on GPU, demonstrating viability for hardware-efficient neuro-symbolic logic synthesis.