Certifiable Boolean Reasoning Is Universal

2026-02-04

AI Summary

Although existing deep learning models theoretically possess the capacity to represent any Boolean function, they often fail to guarantee verifiable Boolean reasoning during training. This work proposes a novel neural architecture that parameterizes a distribution over Boolean circuits, ensuring that any sampled circuit is almost surely valid under arbitrary parameter configurations. This approach achieves certified correctness, universality, and parameter efficiency—offering linear parameter complexity for low-complexity Boolean functions—thereby unifying these three desirable properties for the first time. Evaluated on truth table completion tasks, the method demonstrates stable training dynamics, high accuracy, and strictly Boolean-valued internal units throughout. In contrast, conventional multilayer perceptrons exhibit Boolean representations in only approximately 10% of their hidden units.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Large pretrained models with web data

Abstract

The proliferation of agentic systems has thrust the reasoning capabilities of AI into the forefront of contemporary machine learning. While it is known that there \emph{exist} neural networks which can reason through any Boolean task $f:\{0,1\}^B\to\{0,1\}$, in the sense that they emulate Boolean circuits with fan-in $2$ and fan-out $1$ gates, trained models have been repeatedly demonstrated to fall short of these theoretical ideals. This raises the question: \textit{Can one exhibit a deep learning model which \textbf{certifiably} always reasons and can \textbf{universally} reason through any Boolean task?} Moreover, such a model should ideally require few parameters to solve simple Boolean tasks. We answer this question affirmatively by exhibiting a deep learning architecture which parameterizes distributions over Boolean circuits with the guarantee that, for every parameter configuration, a sample is almost surely a valid Boolean circuit (and hence admits an intrinsic circuit-level certificate). We then prove a universality theorem: for any Boolean $f:\{0,1\}^B\to\{0,1\}$, there exists a parameter configuration under which the sampled circuit computes $f$ with arbitrarily high probability. When $f$ is an $\mathcal{O}(\log B)$-junta, the required number of parameters scales linearly with the input dimension $B$. Empirically, on a controlled truth-table completion benchmark aligned with our setting, the proposed architecture trains reliably and achieves high exact-match accuracy while preserving the predicted structure: every internal unit is Boolean-valued on $\{0,1\}^B$. Matched MLP baselines reach comparable accuracy, but only about $10\%$ of hidden units admit a Boolean representation; i.e.\ are two-valued over the Boolean cube.

Problem

• Certifiable reasoning
• Universal Boolean reasoning
• Deep learning
• Boolean circuits
• Neural network expressivity

Innovation

• Certifiable Reasoning
• Boolean Circuits
• Universal Approximation
• Neural Boolean Representation
• Parameter-Efficient Learning