# Polynomial Surrogate Training for Differentiable Ternary Logic Gate Networks

Source: [Polynomial Surrogate Training for Differentiable Ternary Logic Gate Networks](<https://www.sota2.com/research/paper/69a66bc02d603e6de3e3ca8a>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2026-03-04T05:00:05.078Z

Authors: Sai Sandeep Damera, Ryan Matheu, Aniruddh G. Puranic, John S. Baras

Published: 2026-02-27T20:37:17.000Z

## Abstract

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 $K\_3$ 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 $2{,}187\\times$ 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 $2$-$3\\times$ 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.

## Recorded benchmark results

A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage.

- [Image Classification on CIFAR-10 10K (test)](<https://www.sota2.com/research/sota/image-classification-on-cifar-10-10k-test.md>): Image Classification; CIFAR-10 10K (test); Soft Accuracy: 52.1.

- [Image Classification on CIFAR-10 2K samples](<https://www.sota2.com/research/sota/image-classification-on-cifar-10-2k-samples.md>): Image Classification; CIFAR-10 2K samples; Circuit Accuracy: 48.4.

## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2603.00302>)

[PDF](<https://arxiv.org/pdf/2603.00302.pdf>)