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Immagine del redattoreAndrea Viliotti

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

In recent years, Large Language Models (LLMs) have attracted considerable interest for their logical reasoning abilities, particularly in mathematics. Despite significant progress in performance, doubts remain about whether these models are truly capable of developing genuine logical reasoning. To address this issue, Mirzadeh, Alizadeh, Shahrokhi, Tuzel, and Bengio (2024) conducted an in-depth study on the GSM8K benchmark, used to evaluate the mathematical reasoning abilities of these models, highlighting various limitations in terms of reliability. In response to these limitations, they developed a new benchmark called GSM-Symbolic, designed to provide a more rigorous and detailed evaluation of the mathematical reasoning capabilities of LLMs.

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

Limitations of the GSM8K Benchmark

The GSM8K benchmark consists of over 8,000 elementary-level math questions, making it a popular tool for evaluating the mathematical reasoning capabilities of models. However, as a static and well-known dataset, GSM8K presents some fundamental issues: the risk of data contamination and the inability to dynamically vary the complexity level of the questions, thus limiting the depth of model evaluation.


Data contamination is a particularly relevant issue. Since GSM8K is one of the most widely used benchmarks, there is a significant probability that examples from this dataset have been included in the training data of the models. This introduces a bias that makes it difficult to accurately assess the true generalization capabilities of LLMs, leading to an overestimation of the models' abilities, which may show seemingly good results but fail to tackle new or varied questions effectively.


Moreover, GSM8K offers only a single level of difficulty, focusing on elementary-level school math problems. This static nature represents a significant limitation, as it does not allow evaluation of how well models can handle increasingly complex problems. Without the ability to adjust difficulty, it is impossible to gain a full understanding of the models' abilities to adapt to more complex situations or manage higher levels of abstraction.


Another issue concerns the structure of the questions within GSM8K, which often follow a repetitive pattern. This makes the benchmark less effective in evaluating the models' ability to generalize to new types of problems or understand structural variations in questions. Language models tend to learn repetitive patterns and may therefore perform well on questions similar to those seen previously, without actually gaining a true understanding of the underlying concepts. Consequently, GSM8K can lead to misleading evaluations of the models' capabilities, overestimating their reasoning abilities.


Furthermore, GSM8K mainly uses questions requiring only simple arithmetic operations. While this is useful for evaluating some basic skills, it fails to provide an adequate measure of the abilities needed to tackle more advanced mathematical problems, such as those involving algebra, geometry, or formal logic concepts. The lack of diversity in the types of problems limits GSM8K's ability to explore and understand the depth of the models' reasoning, which extends beyond basic arithmetic to include understanding complex relationships, managing variables, and formulating solution strategies.


GSM-Symbolic Benchmark: Diversity and Control

GSM-Symbolic was developed as a more adaptable and versatile framework to address the limitations of GSM8K. It uses symbolic templates to generate numerous variants of the original math questions, allowing a deeper analysis of LLM capabilities and ensuring more precise control over difficulty.


For example, in tests conducted with GSM-Symbolic, it was observed that the average performance of models on questions generated with symbolic templates varied significantly, with performance dropping by up to 15% compared to the results obtained on the standard GSM8K benchmark. In particular, models such as Gemma2-9B showed performance variations ranging from 70% to 82%, with an average of 79.1% on GSM-Symbolic, while their performance on GSM8K was 87%. This variability indicates the sensitivity of the models to small changes in question parameters, suggesting that reasoning abilities are heavily influenced by the specificity of input data.


Analyses conducted on fifty sets generated from GSM-Symbolic templates showed that all tested models exhibited significant standard deviation, with an average of ±3.0%. For some models, such as Phi-3.5-mini, the difference between the worst and best recorded performance exceeded 12%, indicating a structural fragility in mathematical reasoning. This fragility becomes even more evident when numerical values are altered: changing simple numerical parameters led to an average performance drop of over 5% in many cases, highlighting how the apparent robustness of the models is only superficial.


The Fragility of Mathematical Reasoning in Language Models

One of the main findings from using GSM-Symbolic is that language models experience significant performance degradation when small modifications are made to the questions, such as changing numerical values or adding seemingly relevant but actually unnecessary information to solve the problem. This phenomenon, called GSM-NoOp, shows how models tend to treat any new information as operational, leading to significant errors. In specific experiments, adding irrelevant clauses led to a performance drop of up to 65% in models like Phi-3-mini and Gemma2-9B, demonstrating the inability of the models to distinguish between crucial and superfluous information.


Furthermore, increasing the number of clauses in a question has been shown to have a negative impact on model performance, in proportion to the complexity of the added clauses. For example, a linear increase in the number of clauses caused the performance of the GPT-4o model to drop from 94.9% on standard questions to 72.4% on questions with two additional clauses, with a standard deviation of ±4.6%. The Phi-3.5-mini model saw an even more drastic decline, dropping from 82.1% to 44.8%, with a standard deviation of ±6.3%, indicating that performance is inversely proportional to the complexity level of the questions.


A particularly relevant aspect that emerged from the experiments is that models tend to fail when faced with questions containing distractors that have no impact on the correct answer. In these cases, the models often interpret the additional clauses as relevant to the resolution process, leading to unnecessary or even incorrect operations. This phenomenon was particularly evident in less sophisticated models like Gemma2-2B, which saw a performance drop from 77% to 29.6% when distractors were added, demonstrating that these models are still far from being able to handle complex contexts requiring a clear distinction between relevant and superfluous information.


Implications for Companies

The implications of these results are significant for companies looking to implement LLM-based solutions for analysis or complex problem-solving tasks. The results from GSM-Symbolic demonstrate that, despite progress, current language models still have substantial limitations in formal reasoning capabilities. Their tendency to respond variably to questions with small modifications and their sensitivity to irrelevant information suggest that they are not yet reliable for tasks requiring logical rigor and consistency.


For companies, it is crucial to understand that current LLMs, while powerful, require a cautious and targeted approach to avoid critical errors in practical applications. Advanced evaluation techniques, such as those offered by GSM-Symbolic, can help companies identify gaps in existing models and better understand their reasoning limitations. Using GSM-Symbolic can be crucial for testing a model's robustness in greater detail before implementing it in contexts that require rigor and reliability, thereby reducing the risk of errors related to logical fragility.


For companies wishing to leverage LLMs for process automation or advanced analysis, it is essential to integrate these technologies with human supervision systems, especially for tasks that require the interpretation of complex information or critical evaluations. GSM-Symbolic can highlight those cases where models tend to fail, such as with distractors or irrelevant information. This allows companies to design hybrid systems, where the language model is used for its efficiency in pattern recognition, but the final validation is performed by a human expert.


Another important implication concerns the customization and adaptation of models to specific business contexts. GSM-Symbolic provides the ability to adjust the difficulty and complexity of questions, making it possible to adapt models to contexts with specific precision and robustness needs. Companies can use this approach to train models that are better suited to their operational contexts, thus reducing the risk of errors resulting from standardized applications not adapted to the company's actual needs.


Moreover, the ability of GSM-Symbolic to generate variants of the original questions makes it possible to continuously evaluate models over time, allowing companies to progressively monitor and improve model capabilities. This iterative approach is essential to ensure that LLM-based systems remain reliable and robust even as business needs and operating conditions evolve. Companies can therefore adopt a cyclical approach of continuous evaluation and improvement, using GSM-Symbolic to test new versions of models and verify that any changes made actually improve logical reasoning and the handling of irrelevant information.


Conclusions

The research surrounding GSM-Symbolic reveals an important and novel picture of the limitations of current language models in mathematical and logical reasoning capabilities, a topic with crucial implications for companies looking to artificial intelligence (AI) to improve processes and strategic decisions. The study results highlight that, although large language models (LLMs) have shown remarkable potential in terms of linguistic processing, they present significant shortcomings in distinguishing between relevant and superfluous information and in handling increases in logical and numerical complexity. This limitation results in high variability in performance even in the face of small changes in questions, a vulnerability that highlights an intrinsic structural fragility in their approach.


For companies, these findings are essential because they raise an important warning: current LLMs cannot yet be considered reliable for tasks requiring logical rigor and the ability to generalize in complex contexts. In adopting such models, companies must therefore act with extreme caution, especially for applications involving critical decisions or in-depth analyses. This means that adopting LLMs requires hybrid solutions, where models are integrated with human supervision systems to bridge the gaps in reasoning capabilities. Advanced evaluation techniques, such as GSM-Symbolic, offer companies an opportunity to thoroughly verify these gaps, allowing them to identify weaknesses in models before they are implemented in sensitive operational contexts.


Another strategic implication for companies concerns the importance of customizing LLMs to meet specific business needs. The adoption of GSM-Symbolic, which allows for modulating question difficulty and generating controlled variants, enables companies to configure models according to their operational needs, avoiding the risk of erroneous results stemming from the application of standard models that are not adapted. This approach also makes it possible to obtain a cyclical evaluation of performance, which is essential for monitoring model improvements and ensuring that their reliability levels are maintained over time, even as business needs and data evolve.


The fragility of LLMs highlighted by the GSM-Symbolic framework also leads to reflection on a broader perspective: to develop truly effective models for mathematical reasoning tasks, a profound revision of the LLM architecture will be required, shifting the paradigm from simple probabilistic matching to a model that integrates structured memory elements and formal symbolic reasoning. For companies, this opens the door to strategic collaborations with the research community: by supporting experiments and sharing complex use cases, companies can contribute to developing more robust and sophisticated AI models. Such collaboration can not only accelerate the improvement of models' reasoning capabilities but also ensure that future versions of LLMs better meet companies' operational and strategic requirements.


Ultimately, the work on GSM-Symbolic highlights how moving from simple pattern-based models to models with more formalized reasoning capabilities is essential for reliable use of LLMs in business contexts. In the meantime, companies wishing to take advantage of AI must adopt careful implementation approaches, integrating verification and supervision measures to mitigate the risks arising from the current logical limitations of these systems.


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