# Polynomial Surrogate Training for Differentiable Ternary Logic Gate Networks

arXiv: https://arxiv.org/abs/2603.00302 · HTML: https://mlsanity.com/paper/2603.00302/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 18 references that are in the map.
- First posted on arXiv 27 Feb 2026.

## Builds on (most cited first, 9 of 9)

- [The Lottery Ticket Hypothesis: Finding Sparse, Trainable Neural Networks](https://mlsanity.com/paper/1803.03635/) (2018)
- [Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation](https://mlsanity.com/paper/1308.3432/) (2013)
- [Trained Ternary Quantization](https://mlsanity.com/paper/1612.01064/) (2016)
- [Equinox: neural networks in JAX via callable PyTrees and filtered transformations](https://mlsanity.com/paper/2111.00254/) (2021)
- [Deep Differentiable Logic Gate Networks](https://mlsanity.com/paper/2210.08277/) (2022)
- [Convolutional Differentiable Logic Gate Networks](https://mlsanity.com/paper/2411.04732/) (2024)
- [Weightless Neural Networks for Efficient Edge Inference](https://mlsanity.com/paper/2203.01479/) (2022)
- [Recurrent Deep Differentiable Logic Gate Networks](https://mlsanity.com/paper/2508.06097/) (2025)
- [A Method for Optimizing Connections in Differentiable Logic Gate Networks](https://mlsanity.com/paper/2507.06173/) (2025)

## Built on by (most cited first, 2 of 2)

- [THEIA: Learning Complete Kleene Three-Valued Logic in a Pure-Neural Modular Architecture](https://mlsanity.com/paper/2604.11284/) (2026)
- [On the Stability and Realizability of Recurrent Polynomial Surrogate Ternary Logic Gate Networks](https://mlsanity.com/paper/2605.24649/) (2026)

## Abstract (arXiv)

Differentiable logic gate networks (DLGNs) learn compact, interpretable Boolean circuits via gradient-based training, but all existing variants are restricted to the 16 two-input binary gates. Extending DLGNs to Ternary Kleene K3 logic and training DTLGNs where the UNKNOWN state enables principled abstention under uncertainty is desirable. However, the support set of potential gates per neuron explodes to 19,683, making the established softmax-over-gates training approach intractable. We introduce Polynomial Surrogate Training (PST), which represents each ternary neuron as a degree-(2,2) polynomial with 9 learnable coefficients (a 2,187× parameter reduction) and prove that the gap between the trained network and its discretized logic circuit is bounded by a data-independent commitment loss that vanishes at convergence. Scaling experiments from 48K to 512K neurons on CIFAR-10 demonstrate that this hardening gap contracts with overparameterization. Ternary networks train 2-3× faster than binary DLGNs and discover true ternary gates that are functionally diverse. On synthetic and tabular tasks we find that the UNKNOWN output acts as a Bayes-optimal uncertainty proxy, enabling selective prediction in which ternary circuits surpass binary accuracy once low-confidence predictions are filtered. More broadly, PST establishes a general polynomial-surrogate methodology whose parameterization cost grows only quadratically with logic valence, opening the door to many-valued differentiable logic.