Dual-Architecture Latent Reasoning

• Dual-architecture latent reasoning is a framework that splits reasoning tasks between two interacting neural modules, one for evaluating operations and one for simulating outcomes.
• It leverages graph neural networks to represent mathematical formulas as graphs, supporting precise multi-hop reasoning and effective error diagnosis.
• The approach demonstrates robust generalization across diverse mathematical domains while addressing challenges such as cumulative error and alignment in multi-step deductions.

Dual-architecture latent reasoning refers to computational paradigms in which reasoning processes are formally divided between two distinct, yet interacting, neural architectures or modules. These modules typically assume specialized roles, such as evaluating the success of reasoning actions, simulating the outcome of reasoning steps in a learned latent space, or maintaining parallel representations with different inductive biases. The development of dual-architecture frameworks underpins advances in approximate reasoning, generalization across mathematical domains, and scalable, neural-based theorem proving beyond the reach of traditional symbolic methods.

1. Conceptual Foundation and Architecture

Dual-architecture latent reasoning originates from the need to separate the tasks of reasoning evaluation and reasoning simulation in continuous vector spaces. The canonical structure, as established in "Mathematical Reasoning in Latent Space" (Lee et al., 2019), consists of two principal neural network modules:

• The first module, often labeled as $\sigma$, is designed to predict the success of applying a reasoning operation (e.g., a rewrite or transformation), mapping symbolic entities into latent embeddings and assessing whether an operation is applicable or likely to yield a valid step.
• The second module, referred to as $\omega$, is tasked with directly simulating the result of a successful operation. It predicts the latent representation of what the transformed entity would be, given the initial embedding and the operation in question.
• An alignment component $\alpha$ maps between potentially non-identical latent spaces, ensuring that the outputs of different architectures or training regimes can be semantically synchronized for multi-step reasoning.

This separation permits each architecture to be trained and evaluated for its role, supporting independent optimization and granular error analysis.

2. Latent Space Reasoning Workflow

The dual-architecture approach operationalizes "approximate reasoning in a fixed-dimensional latent space." All entities—formulas, targets, theorems—are embedded as vectors (e.g., in $\mathbb{R}^{1024}$). During reasoning:

• The $\sigma$ model computes a prediction $\sigma(T, P) = p(c(\gamma(T), \pi(P)))$ to determine if applying a particular theorem $P$ to target $T$ should succeed. Here, $\gamma$ and $\pi$ are separate embedding towers for the operands, $c$ is a combiner MLP, and $p$ is a classification layer.
• If a step is likely, $\omega$ computes both an updated success score and, critically, a predicted new latent vector $e'(c'(\gamma'(T), \pi'(P)))$ corresponding to the post-rewrite state.
• Chaining these predictions, the system simulates multiple consecutive reasoning steps, alternating between the main and auxiliary latent spaces via an alignment network $\alpha$.

This architecture enables deduction chains to be performed solely on latent representations, bypassing the slow symbolic reconstruction at each step.

3. Underlying Neural Backbone: Graph Neural Networks

Mathematical formulae are naturally encoded as graphs, with nodes representing atomic terms and edges denoting syntactic relations. In dual-architecture latent reasoning, both $\sigma$ and $\omega$ utilize graph neural networks (GNNs) as their respective embedding towers. The process involves:

• Canonicalizing formulas as graph structures with variable bindings.
• Applying message passing or graph convolution layers, which aggregate features from neighboring nodes over multiple hops.
• Yielding semantic embeddings that encode both local and global properties necessary for high-precision rewrite prediction.

This formalism supports highly structured, domain-agnostic reasoning and is robust to the graph-theoretic complexity inherent in diverse mathematical domains.

4. Multi-step Approximate Deduction and Alignment

For reasoning over multiple steps, the system composes predictions and transformations as follows: $$T_1 \xrightarrow[\gamma']{} l_1' \in L' \xrightarrow[e'(\cdot, P_1)]{} l_2 \in L \xrightarrow[\alpha]{} l_2' \in L' \xrightarrow[e'(\cdot, P_2)]{} \ldots$$ At each interface between latent reasoning networks, the alignment model $\alpha$ projects embeddings from $L$ to $L'$ or vice versa, ensuring consistency across architectures. This chaining permits the system to propagate semantic information for up to four consecutive steps, as demonstrated experimentally. The L$_2$ distance between predicted and ground-truth embeddings, as well as ROC/AUC metrics for rewrite success, are used to quantify the fidelity of the deduction simulation across steps.

5. Empirical Results and Generalization Across Domains

Evidence from HOList, a large and diverse corpus containing topology, calculus, algebra, and other disciplines, underscores the effectiveness and generalization of dual-architecture latent reasoning (Lee et al., 2019). Key empirical findings include:

Number of Steps	ROC/AUC (Rewrite Prediction)	Embedding Error (L$_2$ Dist.)
Step 1	High	Low
Step 2–4	Gradually declines	Slightly increases

The quality of semantic propagation diminishes gradually with depth but remains significantly above random or parameter-only baselines, indicating meaningful information is maintained over multiple operations.

6. Advantages and Potential Limitations

Primary advantages of the dual-architecture approach include:

• Flexible Specialization: Distinct networks optimize for decision-making ($\sigma$) and semantic update ($\omega$), enabling architectural specialization.
• Error Diagnosis: Separation allows identification of bottlenecks in success prediction vs. reasoning simulation.
• Domain Robustness: Performance across distinct mathematical fields suggests strong generalization capacity.

Potential limitations arise from cumulative error over long reasoning chains and the need for robust alignment across latent spaces, particularly when architectures diverge in training or structural design.

7. Significance and Applications

Dual-architecture latent reasoning demonstrates the viability of fully neural, approximate deduction systems without symbolic replay at every step. It supports multi-step proof sketching, rapid preview and planning in theorem proving, and stands as an indicator for future, scalable reasoning approaches in both mathematics and other domains requiring structured, multi-step inference.

The paradigm established in (Lee et al., 2019) by decoupling success evaluation and state simulation in vector spaces prefigures a broad class of neural symbolic algorithms, and has influenced subsequent reasoning architectures in neural theorem proving and machine learning for mathematics.