Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning (2506.21797v2)

Abstract: We develop a theoretical framework that explains how discrete symbolic structures can emerge naturally from continuous neural network training dynamics. By lifting neural parameters to a measure space and modeling training as Wasserstein gradient flow, we show that under geometric constraints, such as group invariance, the parameter measure $\mu_t$ undergoes two concurrent phenomena: (1) a decoupling of the gradient flow into independent optimization trajectories over some potential functions, and (2) a progressive contraction on the degree of freedom. These potentials encode algebraic constraints relevant to the task and act as ring homomorphisms under a commutative semi-ring structure on the measure space. As training progresses, the network transitions from a high-dimensional exploration to compositional representations that comply with algebraic operations and exhibit a lower degree of freedom. We further establish data scaling laws for realizing symbolic tasks, linking representational capacity to the group invariance that facilitates symbolic solutions. This framework charts a principled foundation for understanding and designing neurosymbolic systems that integrate continuous learning with discrete algebraic reasoning.

• The paper introduces a rigorous algebraic and geometric framework that demonstrates how gradient-based training enables neural networks to discover symbolic structures.
• It models training as a gradient flow over probability measures, employing monomial potentials and semi-ring operations to formalize compositional reasoning.
• The study establishes practical implications for sample complexity and architectural design, providing conditions for exponential convergence to symbolic solutions.

Algebraic and Geometric Foundations for Neurosymbolic Reasoning in Neural Networks

This paper develops a rigorous theoretical framework explaining how neural networks, when trained with gradient-based methods, can naturally discover and internalize symbolic structures. The analysis is grounded in measure theory, algebra, and geometry, providing a principled account of how discrete, compositional reasoning emerges from continuous optimization dynamics. The work addresses a central question in neurosymbolic AI: under what conditions and through what mechanisms can neural networks transcend statistical pattern matching to achieve genuine symbolic reasoning capabilities?

Theoretical Framework

The authors model neural network training as a gradient flow in the space of probability measures over the parameter manifold, specifically employing the Wasserstein metric. This "lifting" of parameters to measure space enables the use of tools from optimal transport and functional analysis to paper the evolution of the network's representational structure during training.

A key insight is that, for a broad class of reasoning tasks—including group operations and logical inference—the loss function can be reformulated as a function of monomial potentials (MPs): expectations of specific monomials under the parameter measure. These MPs encode algebraic constraints relevant to the task and act as ring homomorphisms under a commutative semi-ring structure defined on the measure space.

Algebraic Structure and Compositionality

The solution space of the neural network, when viewed in measure space, is shown to possess a rich algebraic structure. The authors define addition and multiplication operations on measures, corresponding to mass fusion and element-wise product, respectively. Under these operations, the space of measures forms a commutative semi-ring, and the MPs preserve this structure as ring homomorphisms.

This algebraic perspective enables the compositional construction of global minimizers: solutions to complex reasoning tasks can be assembled from simpler ones via the semi-ring operations. The framework generalizes previous finite-neuron analyses to the infinite-width (mean-field) regime, providing a unified view of compositionality in neurosymbolic systems.

Emergence of Symbolic Structure via Gradient Flow

A central result is that, under geometric constraints—specifically, when the velocity field of the gradient flow is group-equivariant (e.g., $O(d)$-equivariant)—the dynamics of the MPs decouple. Each MP evolves independently according to a coordinate-wise gradient flow, driving the system toward binary (0/1) assignments that satisfy the logical constraints of the reasoning task. This decoupling reduces the infinite-dimensional optimization problem to a tractable, low-dimensional one over the MP variables.

The analysis further reveals a progressive reduction in the effective degrees of freedom of the parameter measure during training. As the system approaches symbolic solutions, the measure contracts onto a low-entropy, low-dimensional submanifold, reflecting the parsimony and compositionality of symbolic reasoning. This phenomenon is formalized via the eigenspectrum of the Hessian (second variation) of the loss functional, drawing connections to renormalization group theory.

Sample Complexity and Architectural Implications

The framework yields concrete, actionable principles for the design and training of neurosymbolic systems:

• Sample Complexity: The authors derive scaling laws for the number of samples required to learn group-invariant functions, showing that the sample complexity is reduced by the symmetry group’s cardinality or the effective volume of the quotient space. This provides theoretical justification for the empirical efficiency of symmetry-aware architectures.
• Architectural Design: To realize the decoupled gradient dynamics necessary for symbolic reasoning, architectures must preserve the relevant group invariance. This can be achieved through equivariant layers, invariant loss functions, and appropriate initialization schemes.
• Regularization and Dimension Reduction: Entropy regularization, Gaussian initialization, and weight decay are recommended to steer the optimization toward low-entropy, symbolic solutions. Ensuring Lipschitz continuity and smooth activations further supports the contraction of the solution space.

Numerical Results and Claims

The paper provides strong theoretical claims, including:

• Exponential convergence of MPs to their symbolic (binary) assignments under the specified geometric constraints.
• Provable compositionality of global minimizers via the semi-ring structure.
• Explicit sample complexity bounds for learning invariant functions, scaling favorably with group size and function smoothness.

These claims are supported by formal theorems and proofs, with clear connections to empirical observations in neurosymbolic learning.

Implications and Future Directions

The presented framework offers a principled foundation for understanding and designing neurosymbolic systems that integrate continuous learning with discrete algebraic reasoning. It bridges the gap between sub-symbolic and symbolic paradigms, providing conditions under which neural networks can internalize and generalize structured logical patterns.

Potential future developments include:

• Extending the analysis to non-Abelian and continuous groups, which are prevalent in real-world tasks such as robotics and vision.
• Investigating neural scaling laws for parameter count in addition to data, aiming to predict the onset of emergent symbolic capabilities.
• Developing practical algorithms for post-hoc symmetry alignment and low-entropy solution discovery in pre-trained models.
• Empirically validating the theoretical predictions across diverse reasoning tasks and architectures.

Conclusion

This work advances the theoretical understanding of neurosymbolic reasoning by elucidating the algebraic and geometric mechanisms through which neural networks can discover and represent symbolic structures. The measure-theoretic approach, combined with explicit architectural and data-driven prescriptions, provides a robust foundation for the next generation of AI systems capable of both flexible learning and rigorous reasoning.