Differentiable Logic Synthesis: Spectral Coefficient Selection via Sinkhorn-Constrained Composition

TL;DR

Hierarchical Spectral Composition is introduced, a differentiable architecture that selects spectral coefficients from a frozen Boolean Fourier basis and composes them via Sinkhorn-constrained routing with column-sign modulation to enable Boolean negation -- a capability absent in standard doubly stochastic routing.

Abstract

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.

Figures (8)

• Figure 1: Hierarchical Spectral Composition architecture. Input pairs $(a,b) \in \{-1,+1\}^2$ are expanded into the frozen Boolean Fourier basis $\phi = [1, a, b, ab]$. Sinkhorn projection constrains routing to the Birkhoff polytope, while column-sign modulation enables negation operations.
• Figure 2: Phase 1 training dynamics for all four base operations. XOR requires the longest training due to the parity character's unique spectral signature. AND and OR converge in $<500$ steps due to their simpler affine structure.
• Figure 3: XOR parity emergence: the $|c_{ab}|$ coefficient grows from noise to 1.0 while other coefficients ($|c_1|$, $|c_a|$, $|c_b|$) decay to zero. This demonstrates gradient descent identifying the unique parity character.
• Figure 4: Phase 2 routing visualization. Left: Sinkhorn-constrained $P$ learns identity routing. Middle: Column-sign $s$ enables negation (ops 4-7). Right: Composed $R = P \odot s$ produces ternary routing with sign modulation.
• Figure 5: Phase 3 optimal ternary masks for all 10 three-variable operations. Colors indicate coefficient values: blue (+1), white (0), red (-1). Pure 3-var operations (top) vs. cascade compositions (bottom). The sparsity pattern (39% zeros) reflects spectral concentration.

Theorems & Definitions (11)

• Proposition 1: Completeness
• Remark 1: Rectangular Sinkhorn
• Proposition 2: Parity Gradient Accumulation
• Remark 2: "Selection" vs. "Discovery"
• Proposition 3: Negation Inaccessibility in Doubly Stochastic Routing
• proof
• Remark 3: Identity as Optimization Prior, Not Solution Leak
• Theorem 1: Ternary Representability for $n=2$
• proof : Constructive Proof via Exhaustive Enumeration
• proof : LP Certificate (Alternative Verification)