Certifiable Boolean Reasoning Is Universal

TL;DR

The paper addresses the challenge of certifiably universal Boolean reasoning in neural nets by constructing a Stochastic Boolean Circuit (SBC) that samples distributions over $2$-input, $1$-output gates. It proves that SBC samples almost surely yield valid circuits and that, with suitable parameters, any Boolean function $f:\{0,1\\}^B\to\{0,1\}$ can be realized with arbitrarily high probability, with linear parameter scaling for $O(\log B)$-juntas. The approach integrates stochastic bit-lifting, randomized Boolean neurons, and a recursive circuit network to guarantee a circuit-level certificate (structure) and universal expressivity, while enabling differentiable optimization to steer toward target circuits. Empirically, SBC matches exact truth-table completion benchmarks while maintaining a strictly Boolean internal representation, unlike conventional ReLU networks where most hidden units are not two-valued on the Boolean cube. This work thus jointly advances interpretability, safety, and provable reasoning in neural systems by tying learning to explicit, certifiable circuit structures.

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.

Figures (5)

• Figure 2: Gate histograms: SBC-all, SBC-path, and MLP-all (exact recovered gates) across matching regimes.
• Figure 3: SBC sweep: EM vs. $S_{\mathrm{add}}$ for different $L_{\mathrm{add}}$, where $S_{\mathrm{model}}=S_{\mathrm{base}}+S_{\mathrm{add}}$ and $L_{\mathrm{model}}=L_{\mathrm{base}}+L_{\mathrm{add}}$.
• Figure : (a) Stochastic Bit-Lifting Channels: The first layer of our model employs stochastic bit-lifting channels (cf. \ref{['eq:lifted_guy']}), which select $B^{\uparrow}$ among the $B$ input bits and their negations (duplicates allowed).
• Figure : (a) Stochastic Bit-Lifting Channels: The first layer of our model employs stochastic bit-lifting channels (cf. \ref{['eq:lifted_guy']}), which select $B^{\uparrow}$ among the $B$ input bits and their negations (duplicates allowed).
• Figure : (b) Randomized Boolean Neuron: Exactly two edges (fan-in $2$) are selected from the nodes of the previous layer (cf. \ref{['eq:NoCrossing_EdgeWeights']}); then, a Boolean gate is randomly selected (cf. \ref{['eq:EveryPossibleGateAllAtOnce']}) and the output is computed.

Theorems & Definitions (34)

• definition 1: Boolean Neuronal Representability
• definition 2: The Stochastic Boolean Circuit Model
• proposition 1: Stochastic Bit-Lifting Channels
• proposition 2: Only, and Any, Two Edges Can Be Sampled
• proposition 3: Stochastic Boolean Gates: Sample Boolean Gates
• proposition 4: Stochastic Boolean Gates: Universally Sample Boolean Gates
• theorem 1: Certifiable Boolean Reasoning
• theorem 2: Universal Boolean Reasoning
• theorem 3: Efficient Boolean Reasoning for $\mathcal{O}(\log(B))$-Juntas
• lemma 1: Embedding of Unary Gates