GSM-Symbolic: Understanding the Limitations of Mathematical Reasoning in Large Language Models

Abstract: Recent advancements in LLMs have sparked interest in their formal reasoning capabilities, particularly in mathematics. The GSM8K benchmark is widely used to assess the mathematical reasoning of models on grade-school-level questions. While the performance of LLMs on GSM8K has significantly improved in recent years, it remains unclear whether their mathematical reasoning capabilities have genuinely advanced, raising questions about the reliability of the reported metrics. To address these concerns, we conduct a large-scale study on several SOTA open and closed models. To overcome the limitations of existing evaluations, we introduce GSM-Symbolic, an improved benchmark created from symbolic templates that allow for the generation of a diverse set of questions. GSM-Symbolic enables more controllable evaluations, providing key insights and more reliable metrics for measuring the reasoning capabilities of models.Our findings reveal that LLMs exhibit noticeable variance when responding to different instantiations of the same question. Specifically, the performance of all models declines when only the numerical values in the question are altered in the GSM-Symbolic benchmark. Furthermore, we investigate the fragility of mathematical reasoning in these models and show that their performance significantly deteriorates as the number of clauses in a question increases. We hypothesize that this decline is because current LLMs cannot perform genuine logical reasoning; they replicate reasoning steps from their training data. Adding a single clause that seems relevant to the question causes significant performance drops (up to 65%) across all state-of-the-art models, even though the clause doesn't contribute to the reasoning chain needed for the final answer. Overall, our work offers a more nuanced understanding of LLMs' capabilities and limitations in mathematical reasoning.

• The paper demonstrates that GSM-Symbolic, a benchmark using symbolic templates, significantly challenges LLMs’ reasoning capabilities.
• It employs an 8-shot Chain-of-Thought approach and diverse problem instantiations to reveal model fragility and sensitivity to minor numeric variations.
• Findings indicate that LLMs rely on pattern matching rather than genuine logical reasoning, with performance dropping as question complexity increases.

GSM-Symbolic: Understanding the Limitations of Mathematical Reasoning in LLMs

Introduction

The paper "GSM-Symbolic: Understanding the Limitations of Mathematical Reasoning in LLMs" explores the mathematical reasoning capabilities of LLMs, specifically addressing their performance limitations with the GSM8K benchmark. The authors propose GSM-Symbolic, an enhanced benchmark designed to provide more reliable metrics for evaluating mathematical reasoning capabilities of these models. By introducing a diversity of problem variants through symbolic templates, GSM-Symbolic extends beyond static question sets to enable controlled evaluations of LLMs over numerous instantiations.

Figure 1: 8-shot CoT performance across 50 sets generated from GSM-Symbolic templates. All state-of-the-art models exhibit notable variance in accuracy.

Methodology

GSM-Symbolic Benchmark

The GSM-Symbolic benchmark was developed to capture a wider array of mathematical reasoning scenarios than are possible with the GSM8K dataset. By utilizing symbolic templates, GSM-Symbolic can generate an extensive variety of questions while preserving consistent logical steps required for solutions. The templates facilitate changes in variables such as names and numerical values, allowing the study of how these alterations affect LLM performance.

Figure 2: The performance of all state-of-the-art models on GSM-Symbolic drops compared to GSM8K.

Evaluation Setup

The study employs an 8-shot Chain-of-Thought prompting approach across multiple sub-benchmarks generated from GSM-Symbolic templates. Evaluations are structured around different variants by manipulating characteristics such as the insertion of extraneous clauses or altering variable values to assess the impact on model reasoning.

Results

Performance Variation and Fragility

The models demonstrate considerable variance in performance when tested across different GSM-Symbolic generated instances. Statistical analysis suggests that the original benchmark scores, represented by dashed lines, often lie at the distribution tails of the GSM-Symbolic performances, hinting at potential data contamination and indicating a model's potential memorization rather than reasoning capability.

Figure 3: How sensitive are LLMs when we change \textcolor{teal60}{only names}, \textcolor{purple70}{only numbers}, or \textcolor{cyan60}{both names and numbers}?

Sensitivity to Superficial Changes

The study reveals a heightened sensitivity of LLMs to minor numeric adjustments compared to nominal alterations. This is evidenced by a noticeable performance drop and increased variance when changes extend beyond superficial names to numeric elements, reinforcing the hypothesis that LLMs frequently rely on pattern matching rather than genuine reasoning.

Impact of Question Complexity

The authors investigate the influence of question complexity by evaluating LLMs' performance on GSM-Symbolic variants with different levels of difficulty. Variations in performance distributions consistently show a decrease in accuracy and increase in variance as complexity escalates by adding additional clauses to the questions.

Figure 4: The impact of increasing the number of clauses on performance: As the difficulty increases from $\rightarrow$ $\rightarrow$ $\rightarrow$ , the distribution of performance shifts to the left (i.e., accuracy decreases), and the variance increases.

Analysis of Logical Reasoning

GSM-NoOp Dataset

The introduction of the GSM-NoOp dataset further tests LLMs' reasoning capabilities by including irrelevant but seemingly significant clauses within problems. The paper highlights substantial performance declines (up to 65%) when the models, constrained by pattern-matching strategies, fail to recognize the irrelevance of the extraneous information.

Figure 5: Additional results on performance variation on GSM-Symbolic.

Conclusion

This comprehensive study underscores significant LLM limitations in mathematical reasoning, suggesting a predominant reliance on training data patterns over true logical derivation. While GSM-Symbolic advances robust evaluation methodologies, the findings reveal inherent fragility in LLM reasoning, marked by susceptibility to minor input changes and amplified difficulty effects. These insights provide a roadmap for future exploration into developing models with authentic reasoning capabilities, highlighting a critical challenge in progressing towards AI systems capable of human-like cognitive operations and general intelligence.