A Learning Algorithm for Threshold Boolean Networks with Prescribed Fixed Points

Gonzalo A. Ruz

cs.LGAug 18, 2026

A Learning Algorithm for Threshold Boolean Networks with Prescribed Fixed Points

Authors: Gonzalo A. Ruz

Organizations: Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Santiago, Chile · Millennium Nucleus for Social Data Science (SODAS), Santiago, Chile · Millennium Nucleus in Data Science for Plant Resilience (PhytoLearning), Santiago, Chile · Center of Applied Ecology and Sustainability (CAPES), Santiago, Chile

Abstract

We present a learning algorithm for inferring threshold Boolean networks (TBNs) with a prescribed set of fixed points. The proposed method employs a custom differentiable loss function that jointly enforces fixed point preservation, penalizes spurious attractors, encourages binary outputs, and promotes sparsity through L1 regularization. Applied to the FOS-GRN model of Arabidopsis thaliana, the approach achieved perfect reconstruction (i.e., all 10 desired fixed points and no spurious ones) in 5 out of 30 independent runs, recovering on average 8.53 $\pm$ 0.90 correct fixed points with no spurious attractors. In contrast, standard methods such as the Perceptron and Logistic Regression recovered up to 10 fixed points but introduced between 8 and 31 spurious ones. An additional analysis varying the sparsity coefficient ( $λ$ ) confirmed that the method's performance and the structural properties of the inferred networks remain robust within a practical range (up to 0.01) of regularization strengths. Overall, the results demonstrate the effectiveness and stability of the proposed algorithm in capturing meaningful network dynamics under prescribed dynamical constraints.

Figures & tables

	INF1	INF2	INF3	INF4	SEP	PET1	PET2	STM1	STM2	CAR
AG	0	0	0	0	0	0	0	1	1	1
AP1	0	0	0	0	1	1	1	0	0	0
AP2	0	0	0	0	1	1	1	1	1	1
AP3	0	0	0	0	0	1	1	1	1	0
EMF1	1	1	1	1	0	0	0	0	0	0
FT	0	0	0	0	1	1	1	1	1	1

Table 1: FOS-GRN fixed points . The first row shows the names given to the attractors.

Method	Success / Runs	Fixed Points ( $\bar{x}\pm$ sd)	Matched ( $\bar{x}\pm$ sd)	Spurious ( $\bar{x}\pm$ sd)	Time (s)
Custom Loss	5/30	8.53 $\pm$ 0.90	8.53 $\pm$ 0.90	0.00 $\pm$ 0.00	19.68 $\pm$ 0.84
Perceptron	0/30	26.50 $\pm$ 5.72	9.83 $\pm$ 0.38	16.67 $\pm$ 5.65	0.0045 $\pm$ 0.0026
Logistic Regression (L2)	0/30	18.00 $\pm$ 0.00	10.00 $\pm$ 0.00	8.00 $\pm$ 0.00	0.0118 $\pm$ 0.0069
Logistic Regression (L1)	0/30	38.47 $\pm$ 20.39	7.47 $\pm$ 1.04	31.00 $\pm$ 20.37	0.0047 $\pm$ 0.0042

Table 2: Experimental results . Comparison of TBN learning methods over 30 independent runs. A run is considered successful when it recovers exactly the 10 desired fixed points and no spurious fixed points.

Figure 1: A TBN successfully inferred by the proposed approach. Edge labels indicate weights; solid edges denote activation ( $w>0$ ) and dashed edges denote inhibition ( $w<0$ ); edge thickness is proportional to $|w|$ .

Figure 2: The state transition graph of the TBN of Fig. 1 . The fixed points are represented by nodes with a self-loop.

Figure 3: Effect of L1 regularization on model performance. Number of successful networks (out of 30 runs) as a function of the L1 regularization coefficient ( $\lambda_{\ell_{1}}$ ).

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Learning with Boolean threshold functions

Veit Elser, Manish Krishan Lal

Department of Physics, Cornell · Mathematics, Independent Researcher, India

Boolean Function Learning Binary Neural Networks Neural Network Optimization

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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.

Abstract

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Learning with Boolean threshold functions

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