Discovering a Shared Logical Subspace: Steering LLM Logical Reasoning via Alignment of Natural-Language and Symbolic Views

Large language models (LLMs) have made significant strides in the field of natural language processing, particularly in text generation and understanding tasks. Despite their success in many areas, LLMs still struggle with complex multi-step logical reasoning problems. Traditional approaches often rely on optimizing reasoning chains in natural language form or utilizing external symbolic solvers, but these methods fail to fully leverage the potential capabilities within LLMs. As the demand for integrating natural language processing with symbolic reasoning grows, discovering and utilizing a shared logical subspace within LLMs has become a crucial research direction.

The core problem with LLMs in multi-step logical reasoning is their difficulty in establishing effective alignment between natural-language and symbolic-language expressions. This lack of alignment leads to information loss and incorrect inferences during the reasoning process. Particularly in tasks involving complex rules and multi-step reasoning, existing methods often fail to provide sufficient accuracy and robustness. Therefore, discovering and leveraging a shared logical subspace within LLMs to enhance their reasoning capabilities is a pressing issue that needs to be addressed.

The core innovations of this paper include:

• �� Proposing a method to discover a shared logical subspace within LLMs through the alignment of natural-language and symbolic views.

• �� Utilizing Canonical Correlation Analysis (CCA) to analyze the residual activations of natural-language and symbolic-language reasoning chains, learning a low-dimensional subspace.

• �� Maximizing cross-view correlation to capture shared logical reasoning capabilities within LLMs.

• �� Steering LLMs' reasoning chains along this logical subspace during generation, leveraging complementary reasoning signals without additional training.

The methodology involves several key steps:

• �� Canonical Correlation Analysis (CCA): Analyzing the residual activations of natural-language and symbolic-language reasoning chains to learn a low-dimensional subspace.

• �� Subspace Steering: Guiding LLMs' reasoning chains along this logical subspace during generation, leveraging complementary reasoning signals.

• �� Training-Free Approach: The method requires no additional training, achieving reasoning guidance by linearly amplifying the projection of each token's activation onto the learned subspace.

• �� Experimental Validation: Conducting experiments across four logical reasoning benchmarks to evaluate the method's effectiveness.

The experimental design includes validation across four logical reasoning benchmarks, including FOLIO, PrOntoQA, and ProofWriter datasets. Models used include Meta-Llama-3.1-8B-Instruct, Llama-3.2-3B-Instruct, and Llama-2-13B-Chat. The experiments compare baseline methods such as Greedy-CoT, 3-shot-CoT, and SC-3, evaluating accuracy improvements across different models and datasets. Key hyperparameters include the dimensionality of the subspace and steering strength, with ablation studies conducted to analyze the contribution of each component.

Experimental results show that the proposed method significantly improves accuracy across multiple logical reasoning benchmarks. For instance, on the PrOntoQA dataset, Phi-3-Mini's accuracy increased from 87.2% to 93.2%. On the FOLIO dataset, using the Llama-3.1-8B model, accuracy improved from 51.7% to 61.1%. Additionally, compared to the SC-3 method, LSS-CoT matched or slightly outperformed on PrOntoQA and ProofWriter datasets using the Gemma-2-9B model. These results demonstrate the method's ability to effectively utilize the shared logical subspace within LLMs to enhance reasoning performance.

The method has direct applications in various fields, including mathematics, scientific analysis, planning, and coding tasks that require multi-step decision-making. By enhancing LLMs' logical reasoning capabilities, the method provides higher accuracy and robustness in these areas. Additionally, the method is training-free and applicable to existing LLM architectures, offering low computational overhead.

Despite the method's impressive performance across multiple benchmarks, it may underperform in complex reasoning tasks, especially when alignment between natural-language and symbolic expressions is weak. Additionally, its generalizability might be limited due to reliance on existing model architectures and datasets. The method's effectiveness may also be less pronounced in non-logical reasoning tasks. Future research could explore how to apply this method to a broader range of reasoning tasks and further optimize the subspace learning process.