Exploring the Compositional Deficiency of Large Language Models in Mathematical Reasoning (2405.06680v4)

Abstract: Human cognition exhibits systematic compositionality, the algebraic ability to generate infinite novel combinations from finite learned components, which is the key to understanding and reasoning about complex logic. In this work, we investigate the compositionality of LLMs in mathematical reasoning. Specifically, we construct a new dataset \textsc{MathTrap} by introducing carefully designed logical traps into the problem descriptions of MATH and GSM8K. Since problems with logical flaws are quite rare in the real world, these represent "unseen" cases to LLMs. Solving these requires the models to systematically compose (1) the mathematical knowledge involved in the original problems with (2) knowledge related to the introduced traps. Our experiments show that while LLMs possess both components of requisite knowledge, they do not \textbf{spontaneously} combine them to handle these novel cases. We explore several methods to mitigate this deficiency, such as natural language prompts, few-shot demonstrations, and fine-tuning. Additionally, we test the recently released OpenAI o1 model and find that human-like `slow thinking' helps improve the compositionality of LLMs. Overall, systematic compositionality remains an open challenge for LLMs.

• The paper shows that LLMs struggle to combine mathematical knowledge with trap-specific reasoning when solving complex problems.
• The study introduces the MathTrap dataset with five distinct trap types to rigorously assess compositional reasoning in language models.
• Results indicate significant performance declines on trap problems, highlighting the need for improved training techniques and dynamic reasoning models.

Exploring the Compositional Deficiency of LLMs in Mathematical Reasoning

The paper "Exploring the Compositional Deficiency of LLMs in Mathematical Reasoning" presents an empirical investigation into the capabilities of LLMs to systematically combine learned components of mathematical reasoning when confronted with novel problem cases. The inquiry tackles the intrinsic ability of LLMs to perform compositional reasoning, which is fundamental to human cognition and essential for understanding and resolving complex logical tasks.

Overview

The authors employ a novel dataset called MathTrap to evaluate the compositional ability of LLMs. This dataset is derived by embedding intentional logical traps within problems sourced from the MATH and GSM8k benchmarks. Such traps are rare encounters in typical training data, positioning them as "unseen" scenarios for LLMs. The central hypothesis is that effectively addressing these challenges would necessitate LLMs to synergize (a) the mathematical knowledge used in solving the original problems and (b) the additional reasoning required to navigate the traps.

Methodology

Key aspects of the methodology include the categorization of trap types into five distinct groups: Concept Undefined, Missing Condition, Direct Contradiction, Indirect Contradiction, and Violating Common Sense. Each category embodies unique complexities that challenge the systematic compositional generalization of models. The approach involves assessing LLM performance through several interventions, including prompts, few-shot demonstrations, and fine-tuning, to enhance model capabilities in trap resolution.

Results

The experimentation involves both closed-source and open-source LLMs, measuring their accuracy on conceptual, original, and trap problems. Notably, there was a clear performance decline when models transitioned from original problems to trap problems, elucidating a weakness in the models' ability to harness their mathematical and trap-specific knowledge compositionally. Despite having access to the requisite knowledge base, LLMs frequently failed to utilize it effectively unless explicitly prompted or demonstrated, substantiating the compositional deficiency claim. Human participants, in contrast, excelled significantly, showcasing their innate capacity for compositional reasoning.

Implications and Future Directions

These findings bear critical implications in the context of advancing LLMs towards more robust cognitive models. First, they highlight the clear bottleneck present in current LLM architectures when attempting to simulate the human-like compositional reasoning process. Second, they stress the importance of developing improved training and fine-tuning methodologies that could mitigate these deficiencies, potentially through incorporating more complex, unseen problem scenarios in training data.

Furthermore, the insights drawn from the paper pave the way for enhancements in foundational models by incorporating mechanisms that can better integrate isolated knowledge components on the fly. Future research trajectories could explore innovative architectural changes that endow LLMs with more dynamic reasoning paths and investigate other domains where compositional reasoning is pivotal.

In conclusion, the work underscores a significant challenge within the landscape of LLM development and encourages the ongoing effort to refine these models to better mirror human capabilities in adaptive reasoning. The open challenge of enhancing systematic compositionality remains an area ripe with opportunity for innovative solutions and theoretical advancement in artificial intelligence.