A Learning Algorithm for Threshold Boolean Networks with Prescribed Fixed Points

We develop a method for training neural networks on Boolean data in which the values at all nodes are strictly $\pm 1$, and the resulting models are typically equivalent to networks whose nonzero weights are also $\pm 1$. The method replaces loss minimization with a nonconvex constraint formulation. Each node implements a Boolean threshold function (BTF), and training is expressed through a divide-and-concur decomposition into two complementary constraints: one enforces local BTF consistency between inputs, weights, and output; the other imposes architectural concurrence, equating neuron outputs with downstream inputs and enforcing weight equality across training-data instantiations of the network. The reflect-reflect-relax (RRR) projection algorithm is used to reconcile these constraints. Each BTF constraint includes a lower bound on the margin. When this bound is sufficiently large, the learned representations are provably sparse and equivalent to networks composed of simple logical gates with $\pm 1$ weights. Across a range of tasks -- including multiplier-circuit discovery, binary autoencoding, logic-network inference, and cellular automata learning -- the method achieves exact solutions or strong generalization in regimes where standard gradient-based methods struggle. These results demonstrate that projection-based constraint satisfaction provides a viable and conceptually distinct foundation for learning in discrete neural systems, with implications for interpretability and efficient inference.

Gene regulatory network modeling often requires balancing predictive accuracy and mechanistic interpretability. In this work, we compare continuous surrogate models and a discrete mechanistic model on the same Arabidopsis thaliana induced systemic resistance (ISR) dataset, using both the raw continuous gene-expression measurements and their sign-binarized representation. The study considers eight defense-related genes measured over nine time points and evaluates two continuous predictors, Random Forest (RF) regression and a Multi-Layer Perceptron (MLP), against a threshold Boolean network (TBN). The models are assessed using rolling-origin one-step prediction, recursive multi-step rollout, and interpretability analysis. RF achieved the best average one-step numerical performance in the continuous domain, with an MAE of 1.910 and an RMSE of 2.836, compared with 2.089 and 3.106 for the MLP. In the binary domain, the TBN obtained the best average one-step qualitative performance, with a binary accuracy of 0.550 and a Hamming distance of 3.600, compared with 0.500 and 4.000 for RF, and 0.495 and 4.040 for the MLP. In recursive rollout, the TBN exactly reproduced the observed binarized trajectory, while the MLP also showed near-perfect fidelity, with a trajectory binary accuracy of 0.986, and RF accumulated substantially larger deviation, with a trajectory binary accuracy of 0.708. These results highlight that local numerical accuracy and global qualitative dynamical fidelity are not necessarily aligned, and suggest that continuous surrogates and threshold Boolean networks should be viewed as complementary tools for modeling biological regulation.

We study the problem of learning accurate and cognitively simple binary classifiers that are directly interpretable as Boolean rules. We propose mixed integer linear programming (MILP) formulations for training sparse disjunctive normal form (DNF) networks, viewed as small networks with a conjunctive hidden layer and a disjunctive output layer. Contrary to previous work that uses MILP for inductive rule learning, we start with a fixed network structure, for which we then find optimal weights. A regularized objective explicitly trades off misclassification costs against sparsity, yielding globally optimized rule sets. Our main technical contribution is an enhanced formulation that substantially reduces the number of integer decision variables and eliminates redundant constraints, while preserving exact Boolean semantics for all feasible solutions. Furthermore, we develop a combined approach based on initialization and dropout to break symmetries in the weights between rules, mitigating redundancy, and encouraging more diverse and stable rule representations. Experiments on synthetic benchmarks and UCI datasets show that the advanced formulation improves computational efficiency and achieves competitive or superior accuracy compared to established interpretable baselines, including SAT-based DNF learning and well-known rule- and tree-based learners. Overall, our approach provides a principled optimization framework for learning compact Boolean rule sets with explicit control over accuracy and sparsity trade-offs.