Recently, large language models (LLMs) have gained significant attention in the field of Natural Language Processing (NLP) and have shown promise across various tasks, even when given only a few examples to learn from. However, their ability to understand and reason with natural language remains uncertain. While there have been attempts to evaluate these models using reasoning tests, these evaluations have mostly focused on models' final answers, often overlooking the step-by-step reasoning processes behind their performance. Additionally, these analyses have typically concentrated on just one or a few aspects of reasoning, especially for tasks that do not require much complex thinking to find the answer. This limits our understanding of LLMs' potential and limitations when it comes to more complex and realistic questions. To address this issue, we conduct a comprehensive analysis of LLMs using the existing Fermi reasoning challenge, a task that combines diferent aspects of reasoning into a single question-answering format, requiring deeper levels of reasoning. In this paper, we examine various advanced LLMs in this reasoning challenge and explore how their performance is afected by their size (i.e., the number of parameters). We also investigate how these models behave with diferent levels of supervision, ranging from having all the information to no evidence at all. Furthermore, we compare the two primary methods of teaching these LLMs, fine-tuning, and few-shot learning, using the Chain-of-Thought approach. We provide a detailed case study highlighting the most common limitations of these models. While our results imply that these models may have a long journey ahead to reach human-level reasoning, our work can be considered a robust baseline for the community to strive toward achieving this ambitious goal. Our code is available on GitHub https://github.com/MostafaRahgouy/LLMs_for_FPs for the community.
Throughout history, humans have grappled with the concept of reasoning, seeking to define
and understand it. This pursuit dates back to the early Greek philosophers, who posed profound
questions such as “What can be known?” and “What does that mean someone knows something?”
in their quest to illuminate the nature of reasoning [
The Fermi challenge [17] inspired by Enrico Fermi, the Nobel winner in physics known for his remarkable skill in making accurate estimates of complex numerical problems, is often referred to as “Fermi problems”. These problems typically involve making assumptions and approximations to arrive at a rough estimate, rather than a precise calculation. Owing to the intrinsic complexity of the reasoning questions involved, FPs have been appropriated for use in science Olympiads and interviews. 2.1. Tasks FP encompasses three distinct tasks: perfect-context, distractor-context, and full, which are designated as Task 1, Task 2, and Task 3, respectively. These tasks can be seen as diferent levels of supervision or evidence provided to a model. Figure 1 illustrates an example of the FP for these three tasks.
At this level, alongside the given question, all essential knowledge (defined as a set of facts) required to answer the question is integrated into the input. This task bears resemblance to math word problems, which often feature a concise narrative outlining a scenario and presenting a question related to an unknown quantity [19]. In both of these scenarios, information is explicitly provided, eliminating the need for retrieving knowledge, and instead emphasizing how such information is interconnected and can be used as guidelines to arrive at the final answer.
In realistic scenarios, the input context often comprises non-relevant information, and models must efectively discern which pieces are pertinent to the given question. In line with this If every human that's ever lived was brought back to life. What would be the population density of people per square
kilometer? F1: The total surface area of Earth is 510e+6 km square F2: The total population on earth is 7.2e+9 F3: Around 55e+6 people die each year F4: It has been 120000 years since the start of human-life on earth rFe5q:uTirheed mtoinriempuompunlautmebtheer eonftpireeohpuleman race is 500 s tac F6: There are 192 countries in the world F ittrrsacoD 5FF1720:01:T0Th0eh0et0ot0toatslaqllainncdoamr...eeagoenneeraarttehdislast year in the world is 80e+12 real-world complexity, the FP includes this task that involves the deliberate inclusion of some distractor facts alongside the question and relevant information. 2.1.3. Task 3: Full The ultimate goal is to enable models to answer complex questions without any provided information. In essence, the model should learn to retrieve supporting facts and demonstrate the reasoning behind the answer based on the retrieved supporting facts. Achieving this level of abstraction is akin to human-style problem-solving. 2.1.4. Outputs Regardless of the chosen task, two acceptable outputs can be provided, which may serve as alternatives to each other: Ans and Program P. Ans signifies the direct answer to a given question. Furthermore, each FP question is complemented by an explanation in the form of an executable program. This program delineates the facts, values, and mathematical computations essential for deriving the answer. Program P has the potential to demonstrate how models engage in reasoning across various facets, including question decomposition, fact matching, and the relationship between questions in terms of mathematical operations. Given the paper’s primary focus on LLM behavior and Program P’s role in facilitating the interpretability of
Train 185 Validation 125* 1000 Test 558 1000
models, we will exclusively present experiments based on this output, excluding Ans . This decision allows us to delve deeper into the model’s reasoning processes and provides valuable insights into its decision-making capabilities. 2.2. Datasets [17] presented two distinct datasets for analysis: RealFP, comprising of real-world FP collected from various internet pages, quizzes, and Fermi problem Olympiads; and SynthFP, a larger synthetic dataset created manually from 12 diferent templates. Table 1 shows statistics of the datasets. As the test set of RealFP better represents the Fermi challenges in the real world, throughout this paper, we will use this set to report the performance of the models. Additionally, the RealFP validation set is employed for early stopping mechanisms for the models in our experiments, and we chose not to use the validation and test sets of Synthetic to maintain consistency and fairness across all models. 2.3. Metrics
To evaluate the performance of a predicted direct answer ′ over the true answer , the _ with the following definition will be employed: _ = 0, 1 − ︂{ 1 3 ⃒ ⃒ ⃒⃒ 10 ⃒⃒ ′ ⃒⃒ }︂ (1) This metric considers the imprecision and uncertainty of answers by assigning a full score to a prediction that produces an answer within the same order of magnitude as the reference gold answer. Conversely, for each order of magnitude that the prediction diverges from the reference answer, the score is reduced by 1/3 points.
Program Evaluation: Explanations (programs) are evaluated along three criteria: receives a score of 0.
_. • Validity (valid?): This assesses whether the program is syntactically valid by evaluating it to determine if it results in a numerical output. A Python program executor is employed for this purpose. A score of 1 is assigned if the execution is successful, otherwise, it • Answer Accuracy Evaluation ( ): After the program successfully generates a numeric answer (passed the valid? evaluation with score 1), this metric assesses the accuracy of the generated numeric answer ′ with ground truth based on the previously defined • Fact Identification ( Facts): Determines whether the program includes all and only the specified gold facts F, using an F1 measure for assessment.
Our experiments are structured around diferent LLMs training approaches, including finetuning, few-shot, and zero-shot learning, with and without Chain-of-Thought (CoT) prompting. 3.1. Fine-tuning Setting Fine-tuning without Chain-of-Thought: The introduction of Transformers, along with the attention mechanism [20], marked a new era in transfer learning. This innovation enabled the training of large models in an unsupervised manner on extensive datasets, allowing them to acquire semantic knowledge of the language, thus forming a robust foundation for adapting to downstream NLP tasks. This approach has significantly improved performance across a wide range of NLP tasks with limited training samples, a practice commonly referred to as fine-tuning. Notable examples of these models include BERT, BART, and T5. Among the aforementioned models, T5 stands out as particularly adept at handling reasoning tasks due to its inherent architecture of sequence-to-sequence (seq2seq), which aligns well with the demands of such tasks. Consequently, we have chosen this model as the foundation for our fine-tuning setting. To achieve this, we employ a straightforward approach: we concatenate the available supportive facts with the question (for tasks 1 and 2) to construct the input, which the model processes, ultimately yielding Program P as the output. Furthermore, we assess the performance of T5 using various versions, including T5-small and T5-base. It is worth noting that we also explored a larger variant of T5, namely T5-large, which boasts 770 million parameters. However, our ifndings indicated a degradation in results, which could potentially be attributed to the limited number of samples available within the datasets.
Fine-tuning with Chain-of-Thought: In the realm of instruction-based fine-tuning, [ 21] conducted an investigation into the impact of various factors, including scaling the number of tasks and model size, and the incorporation of CoT data during the fine-tuning process. Specifically, in their paper, they outlined their objective as follows: “The goal of Flan finetuning is to produce an improved checkpoint across a range of evaluations, which includes multi-step reasoning ability in addition to traditional NLP tasks”.
To achieve this goal, they integrated nine CoT datasets into the fine-tuning phase and demonstrated the positive impact of this approach on unseen reasoning tasks. Additionally, they made Flan-T5 checkpoints publicly available, maintaining consistency with prior versions of publicly released T5 checkpoints. As a result of these considerations, we selected the Flan-T5 model as our experimental model, which incorporates the CoT capability. However, prompt engineering can be beneficial in tailoring prompts to suit specific tasks. Nonetheless, we opted to maintain consistency by using the original CoT prompt (“Answer the following question step by step”), as utilized in the original paper, to ensure fairness across diferent tasks and model sizes in our experiments. 3.2. Few-Shot Setting LLMs have popularized the notion of few-shot learning, where these models can acquire new tasks with just a small number of examples [22, 23]. Few-shot learning ofers significant benefits as it reduces the necessity for extensive data collection, which can be costly. Investigating few-shot reasoning for FPs can determine whether such models can address intricate questions in an interpretable manner. To this aim, we explore the capabilities of the GPT-based family with varying numbers of provided examples, as typically encountered in few-shot learning scenarios. 3.3. Zero-Shot Setting Embracing the use of super-large LMs has opened the door to unlocking zero-shot reasoning capabilities. Prominent models in this category include GPT-4, LLama, Flan-PaLM, Flan-T5, and Bloom. Furthermore, the fusion of these models with the CoT prompting has yielded significant improvements in various tasks [ 24]. However, CoT imposes a requirement for these models to generate answers through step-by-step reasoning processes. Nonetheless, these answers may not align with FP’s Program P output. This discrepancy arises because the specified program P does not conform to the standard output generated by such models, and they encounter challenges in producing responses without any prior exposure to relevant samples. Consequently, we explored and evaluated zero-shot reasoning by focusing solely on direct answers (Ans ). While this approach may limit interpretability to some extent, assessing their performance under these conditions can ofer valuable insights. 3.4. Experiments Setup In our fine-tuning process, we used a constant random seed value throughout all experiments. We maintained a batch size of 8 for the entire duration. Additionally, we set the learning rate to 1e-3 and utilized the Adam optimizer [25]. To monitor model performance and ensure reproducibility, we employed the validation set from the real dataset. The best model checkpoints were saved based on this validation set loss. For consistency and fairness in reporting results, we conducted fine-tuning for 50 epochs on real data and 5 epochs each on synthetic and combined (both) data. Moreover, our GPU of choice was the NVIDIA A100 SXM4 40 GB. In few-shot and zero-shot settings, we adjusted the temperature parameter to <= 0.1 to enhance model determinism and minimize variation.
4.1. Quantitative Results Program-Based results: Table 2 provides an overview of our findings obtained through ifne-tuning and few-shot learning experiments. As observed in the table, increasing the model
Valid? Facts
Valid? Facts
Valid? size from small to base resulted in performance improvements for both the T5 and FLAN-T5 models. Notably, FLAN-T5 achieved the highest performance in task-1 with a precision score of 0.49, while in task-2, T5-base obtained the highest score (0.28) among all fine-tuning settings. However, it is important to highlight that task 3 yielded comparatively lower results across all ifne-tuning settings. Of particular interest is the performance in the 5-shot learning scenario, where GPT-3.5-turbo surpassed all fine-tuned models and achieved a new state-of-the-art benchmark on all three tasks. Another noteworthy observation relates to the "Valid?" score in Task 3 when compared to Task 1 and Task 2 in the Fine-tuning setting. In most instances, the model tends to generate more valid programs in Task 3. This can be attributed to the model’s greater freedom in generating its own approach, which results in more preferred ways of generating solutions, often leading to shorter answers and reducing the likelihood of producing invalid responses.
GPT-4 GPT-3.5-Turbo FLAN-T5-XL
Ans 0.66 0.72 0.34
0.55 0.46 0.12
Direct-Answer-based results As previously mentioned, attempting to generate the executable Program P without providing any examples to LLMs is not practically achievable in a zero-shot context. Therefore, in this section, we evaluate the results based on Ans . Table 3 showcases significant findings in this scenario, with scores of 0.72 and 0.55 for tasks 1 and 2, respectively. These scores suggest that LLMs can ofer reasonably accurate answers through implicit reasoning when provided with useful information(either with only pertinent information or in conjunction with distractors.). However, they struggle to clearly explain how they arrived at their answers (see the superior results in table 3 compared to table 2). Regardless of the output format, it becomes evident that the presence of distractor information can pose challenges for models and adversely afect their performance. Importantly, the results indicate that the provision of relevant information can enable models to generate estimations, whereas the absence of such knowledge reduces LLMs to a trivial baseline level. In essence, [17] reported a constant model that predicts a random value (a logarithmic sweep between 1010 and 10− 10), which can achieve a result of 0.22 of Ans . 4.2. Qualitative Results Figure 2 illustrates various examples generated by fine-tuned and few-shot models for tasks 1 and 3. In the leftmost example in task 1, we observe that the few-shot model correctly produced a program, while the fine-tuned model made an error concerning the selection of the appropriate mathematical operation, mistakenly choosing multiplication instead of division to derive the answer. Conversely, on the right side of the figure, the opposite situation occurred, where the few-shot model erroneously included the original question in the decomposition question and incorporated it into the calculation (Program: div(Q1, Q2, Q3)). More interesting, in task 3 where models have the freedom to forge their unique path of reasoning, leveraging their supportive facts and calculations, the left example showcases the performance of the few-shot model. In this task, the few-shot model interprets the question and deduces a suitable estimated number based on its own knowledge. Notably, in this particular example, the few-shot learner employs “liters” as a unit of measurement to provide the answer. Conversely, the fine-tuned model, while correctly identifying the appropriate mathematical operation (division) for the two facts, falls slightly short in its generation of numbers and decomposition questions. Finally, in the rightmost example, where the question can be considered more challenging than the other examples, the fine-tuned model encountered a significant issue. It duplicated its generation, resulting in an answer that lacked reasonability, essentially attempting to mimic the structure it learned during the fine-tuning phase. Conversely, the few-shot model demonstrated a more Task 1 Task 3
INPUT: F1: The volume of the swimming pool is
2500 cubic meters F2: The volume of a single water drop is
18e-10 cubic meter Q: How many drops of water are needed to fil an Olympic swimming pool?
PROGRAM: Q1: What is the volume of an Olympic
swimming pool? Q2: What is the volume of a single drop of
water? A1: 2500 m**3 A2: 18e-10 m**3 Q2 -> A2 | F2 Q1 -> A1 | F1
P: Div(Q1, Q2) Q1: What is the volume of the swimming pool? Q2: What is the volume of a single water drop?
A1: 2500 m**3 A2: 18e-10 m**3 Q2 -> A2 | F2 Q1 -> A1 | F1
P: Mul (Q1, Q2) Q1: What is the volume of the swimming pool? Q2: What is the volume of a single water drop?
A1: 2500 m**3 A2: 18e-10 m**3 Q2 -> A2 | F2 Q1 -> A1 | F1
P: Div (Q1, Q2) x
INPUT: F1: The total heat generated by a heater
is 9600 W F2: The average heat given off by a single
person is 80 W Q: How many children would a couple have to have in order to negate the need of a heating system in a modern average-sized
home?
PROGRAM: Q1: What is the total heat needed to generate to compensate for the need of a heating
system? Q2: What is the average heat produced by the
presence of a single child? A1: 9600 W A2: 80 W Q2 -> A2 | F2 Q1 -> A1 | F1
P: Div (Q1, Q2) Q1: What is the total heat generated by a heater? Q2: What is the average heat given off by a single
person? A1: 9600 W Q2 -> A2 | F2
A2: 80 W
Q1 -> A1 | F1
P: Div (Q1, Q2) Q1: What is the total heat generated
by a heater? Q2: What is the average heat given off by a
single person? Q3: How many children does a couple have to have in order to negate the need of a heating system in a modern average-sized home? A1: 9600 W A2: 80 W Q2 -> A2 | F2 Q1 -> A1 | F1
P: Div (Q1, Q2, Q3) x
INPUT: Q: How many jel y beans fil a bucket?
PROGRAM: F1: The average volume of a bucket is 0.67
cubic feet F2: The average volume of a jel y bean is
0.00012 cubic feet Q1: What is the average volume of a bucket? Q2: What is the average volume of a jel y
bean? A1: 0.67 ft**3 Q2 -> A2 | F2
A2: 0.00012 ft**3
Q1 -> A1 | F1
P: Div(Q1, Q2) F1: The volume of a bucket of jel y beans
is 62000 cc F2: The volume of a single jel y bean is
62000 cm**3 Q1: What is the volume of a single bucket of
jel y beans? Q2: What is the volume of a single jel y bean? A1: 62000 cm**3 Q2 -> A2 | F2
A2: 62000 cm**3
Q1 -> A1 | F1 P: Div(Q1, Q2)
x F1: The volume of a bucket is 5 liters F2: The volume of a jel y bean is 0.001 liters Q1: What is the volume of a bucket? Q2: What is the volume of a single jel y bean?
A1: 5 liters A2: 0.001 liters Q2 -> A2 | F2 Q1 -> A1 | F1
P: Div(Q1, Q2) blue components represent answers obtained from a 5-shot GPT-3.5-turbo, and the orange components show the answers generated by the fine-tuned FLAN-T5-base model. The ground truth for each example is depicted in gray. meaningful expansion of its decomposition, even though it failed to identify a correct path to the original question, yielding an incorrect result. Based on these examples, it becomes evident that models, especially larger ones like GPT-based models, have the capacity to retrieve relevant information, but the relationship between such information requires further investigation. 4.3. Fine-tuning vs Few-shot Learning compared to the few-shot model across all tasks (denoted as Fp’s tasks). In Task 1, depicted as the leftmost image, where all relevant facts are provided, the fine-tuned model generated programs with hop counts that closely matched the two most frequent counts (specifically, 2-hop and 4-hop). This suggests that fine-tuning may be influenced by an unbalanced distribution of examples and may tend to produce the most common strategy learned during the fine-tuning phase. Furthermore, this tendency can explain the high value of the “valid?” score (0.86) for this model. In contrast, the few-shot model appears to mimic the ground-truth behavior more closely. For instance, in Task 1, the few-shot model generates answers with hop counts of 1 and 5, which the fine-tuned model did not produce. Interestingly, in Task 2, where the fine-tuned model displays a similar behavior, the few-shot learning approach generates answers with higher hop counts (e.g., 7, 8, 9) due to the increased number of facts in this task. Furthermore, in Task 3, where models are given the freedom to explore their own reasoning methods, both models exhibit a tendency to generate programs with a lower number of hops. This behavior is noteworthy and may provide insight into why these models face challenges with this task. The inclination towards simpler answers in response to complex questions, such as in the case of Task 3, can potentially lead to suboptimal results.
Facts’ Units Identifiers: One common issue found in LLMs is the mishandling of FP identiifers. This often results in incorrect responses, where LLMs choose mathematical operations for facts that lack compatibility or fail to yield a correct answer. For example, in Figure 2 leftmost hand side, a fine-tuned model selected multiplication despite the query explicitly requesting a unitless numerical answer. A potential solution involves integrating fact’s unit checking into the generation process, using a multi-task approach or a non-sequential procedure, as shown in [19] for addressing equation-based questions with a tree-structured decoder. Reasoning Deadlock (LLMs’ Predicament): A significant challenge in LLMs is their performance in Task 3 especially in the Few-shot setting, where they must provide autonomous reasoning. In this task, LLMs often struggle with their reasoning processes, leading to invalid or incorrect answers. Common issues, including but not limited to, raising sub-questions without answers, assigning numeric values to undecomposed questions, failing to link supportive facts to questions, and condensing multiple mathematical relations into a single equation, such as “Div(Q1, Mul(Q2, Q3), Q4)” which increases the likelihood of encountering impossible equations. An efective approach to address this issue involves the deployment of LLMs within iterative loops, as opposed to requiring them to perform reasoning in a single comprehensive round. Utilizing LLMs in multiple rounds ensures the consistency and reliability of the generated answers. This perspective can yield significant benefits, as the model becomes progressively informed about the components it has expanded upon and those that remain unexplored, resulting in a more coherent and accurate reasoning process.
Recently, LLMs have emerged as a promising avenue for improving reasoning tasks, particularly in the context of retrieving relevant and supportive information. For instance, [26] demonstrates that LLMs possess the ability to discern implicit relations. They achieve this by decoupling the process of inferring reasoning steps from their execution. This partially aligns with our findings for task 3 (implicit reasoning) of FPs, where LLMs appear to be more successful at retrieving information than conducting reasoning over the retrieved information. Additionally, there have been attempts to enhance existing reasoning and QA tasks by generating intermediate knowledge to facilitate multi-hop reasoning. In their work, [27, 28] employ LLMs to create benchmarks, and they find that this technique can improve both the performance of models and their interpretability. In line with these eforts, other studies [ 29, 30] have demonstrated that incorporating intermediate supervisory information into the input can enhance the performance of such models. This aligns with our findings, wherein the fine-tuning of task 1, incorporating supportive facts concatenated with the input yielded superior results than the absence of such knowledge. Furthermore, [31] introduced an agent communication mechanism for addressing complex reasoning questions. In this approach, a model engages in a series of QA interactions with agents, such as TextQA and TableQA, to arrive at the final answer. However, this approach has the potential to harness the capabilities of LLMs as agents for solving FPs. Nevertheless, accomplishing this without auxiliary supervision remains a challenging endeavor, necessitating significant dataset modifications to adapt it for FPs. In addition, exploring other directions in the context of complexity is also noteworthy. In terms of complexity, some studies, such as [32], propose that introducing a progressive task complexity framework can yield advantages for LLMs. [33] proposed TELeR, a general guideline for LLMs in prompt designing to perform complex tasks. However, Fermi Problems regard this complexity as comprising three distinct and isolated tasks. The possibility of merging these complex paradigms represents a promising avenue for future research, which we defer to further exploration. Finally, similar to our work, [34] also assessed the capabilities of LLMs in the domain of logical reasoning tasks. They carried out their experiments in a few-shot learning scenario, utilizing methodologies such as prompt engineering and CoT. However, it’s important to note that their primary focus was solely on logical reasoning, whereas our study diverges by primarily delving into mathematical reasoning, particularly within the context of Fermi Problems.
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