Fill in the Blank: Exploring and Enhancing LLM Capabilities for Backward Reasoning in Math Word Problems

Thank you for your positive feedback; we appreciate your recognition of the extensive experiments, clear solution, and well-written paper.

Comment 1: The paper explores the LLM capabilities for backward reasoning, it is not clear what the complexity is compared to the traditional math word problem-solving models, such as Graph2Tree, GTS etc.

Response: But based on the reviewer comment, we tried to experiment with some of these models. But unfortunately, Graph2Tree requires data preprocessing to convert the solution equations into a certain format, and scripts for which are not provided by the authors. We have sent an email to the authors in this regard. At the same time, we note that these techniques require fully supervised data, in the form of equations which solve the reasoning problem. Since we do not have such a set of equations for the backward task, we may only be able to evaluate this after training on the forward problem. We do not expect this to do well, but we are working on these experiments (assuming we are able to get the code) and plan to add the results before the end of the discussion phase. GTS assumes specific linguistic data (Chinese) and it is not clear how to extend this to a general purpose setting in other languages.

Comment 2: The implementation details are not clearly described. For example, GPU and memory size, codes and datasets.

Response: For all experiments involving closed-source LLMs, we utilized the respective model APIs (OpenAI, Google Bard). For the experiment (only inference), using the Llama-2-70B model, we used a 4-bit GPTQ quantized version of the model and two 40GB NVIDIA A100 GPUs were used for the inference. Details regarding the dataset and prompts used in the experiments can be found in Appendix A. We will release the code and the dataset upon acceptance of the paper.

Comment 1: The motivation seems to be Ambiguous. The paper's definition of backward reasoning essentially frames it as a fill-in-the-blank task, rather than a genuine backward reasoning or a broader sense of causal reasoning. In reference [1], a similar task is merely a sub-task in backward verification for verifying forward reasoning. The authors have repurposed it as a new backward reasoning task, which seems redundant and lacks research value. The paper's discussion on the practical application scenarios of this task is insufficient, making it challenging to discern its real-world significance. Moreover, the rationale behind comparing the difficulty levels of backward and forward reasoning remains unexplained.

Response: The primary motivation of our work is to study the backward reasoning abilities of LLMs as observed in human problem-solving. With these insights, we can not only differentiate between LLMs and human problem-solving but also leverage this understanding to improve LLMs. Enhancing the backward reasoning capabilities of LLMs can result in the development of more sophisticated and context-aware language models, ultimately advancing their performance and applicability in various domains. For this, we have chosen a fill-in-the-blank task on Math Word Problems (MWPs), a particular type of causal reasoning task. We chose MWPs because LLMs, like GPT-4, demonstrate strong forward reasoning performance, making them natural and adept verifiers for backward reasoning on MWPs. In contrast to reference [1], where the backward reasoning task is used for verifying forward reasoning, our goal is to enhance the performance of LLMs on the backward reasoning task itself.

Comment 2: Lack of Methodological Novelty.

Response: We first presented three methods, Rephrase, PAL-Tools, and Check your Work, for solving the backward reasoning task. However, in references [1], [2], and [3], the goal is to solve the forward reasoning task. Then, we proposed a novel Bayesian ensembling technique to combine these methods. This technique leverages pretrained Language Models (LLMs) as naturally available verifiers without the need for explicit training. Our proposed Bayesian ensembling technique is versatile and can be applied to any backward reasoning task when a strong verifier is readily available.

Comment 3: While the paper presents a range of experimental results, the depth of analysis behind these results is lacking. For instance, the paper doesn't delve deep into the differences between various prompting techniques and why certain techniques are more effective in specific scenarios. The comparative analysis mostly revolves around forward reasoning methods, lacking a direct comparison with potential backward reasoning techniques, making it difficult for readers to gauge the true advantages of the proposed methods.

Response: Thanks for the suggestion. We are working on a comparative analysis of various prompting techniques, we hope to upload some of these findings before the end of the discussion phase. Few preliminary analyses is present in response of comment 7. Apart from that, we observed Check your Work method doing well as compared to self-refine because of having the original question and answer in the context history for solving to find x and also having that solution for forming new question and checking it. This is because of the design of the prompt of doing these in the same iteration.

Comment 4: The authors' modifications to the datasets GSM8k, SVAMP, and MultiArith are minimal, merely replacing numbers in questions with blanks. However, they claim to have created “new datasets”, which seems to exaggerate their contribution.

Response: Thank you for your comment. While we understand your perspective, we would like to clarify that our intention was not to exaggerate the contribution but rather to highlight the modifications made in the existing datasets to make them amenable for backward reasoning. We acknowledge that the term "new datasets" might have caused confusion, and we will adjust the language in the revised version of the paper to better reflect the nature of the modifications made.

Comment 5: The paper contains many grammatical errors. One is the missing period at the end of the paragraph “In order to establish … on this task”, and the second is the incorrect punctuation in the paragraph “A forward or the typical … numeric value of the blank” with the misplaced colon in ’half.’.

Response: Thank you for your feedback. We will carefully review and address the grammatical errors in the revised version of the paper.

We thank the reviewer for recognizing our extensive experiments, showcasing the effectiveness of our proposed approach, and the paper's clarity and well-organized structure.

Comment 1: The motivation for the proposed “backward reasoning” problem is not very clear to me. In practice, if we want to know a condition, it is natural to just rephrase the question and make {given conditions, final answer} as conditions and the missing condition as a question. In other words, I do not really see the essential of defining a so-called "backward reasoning".

Response: We appreciate your feedback on the motivation behind the "backward reasoning" problem. The essential motivation lies in capturing a distinct cognitive process where the task involves deducing missing conditions from a given answer, resembling human problem-solving strategies. Human problem-solving and decision-making often involve backward reasoning, a powerful technique in mathematics for problem-solving and theorem proving. Presently, Large Language Models (LLMs) excel in forward reasoning, especially in solving Math Word Problems by identifying answers to given questions. However, the exploration of LLM capabilities in backward reasoning, involving deducing missing conditions from a known answer, is surprisingly limited. In our work, we formally defined the task of backward reasoning to understand the limitation of current LLMs that can allow us to enhance the capabilities of language models in a manner more aligned with human-like deductive reasoning. For more motivation, please see the Second paragraph of the Section 1 Introduction.

Comment 2: I have non-trivial concerns about the novelty and contribution of this paper. The "rephrase" techniques of the three proposed solutions that use x to replace the missing condition are not novel, which is already proposed in the advanced chain-of-thoughts methods [1]. Also, I do not see the uniqueness of the proposed "PAL-Tools", it is basically a special case of PAL [2]. Similar novelty concerns the proposed ensemble stage as well, since the ensemble is definitely not a new technique and in most cases, it can improve the performance. - The utility of the ensemble lies in the fact that when verification is easier than backward reasoning, we can boost performance significantly.

Response: While we agree that rephrasing is already proposed in [1], we note that they do not evaluate the technique specifically on the backward reasoning problem. We will clarify in the updated version of the paper. We argue that PAL-Tools is a significant enhancement from PAL, and this is evident from the results in Table 2, where PAL does significantly worse on all the datasets. Though our key idea in PAL-Tools is simple, that we can write the code which will then call a math solver, it is critical for good performance on the backward reasoning problem. Lastly, regarding ensemble, as the reviewer has already pointed out, the utility of ensemble is in the fact that we naturally get a verifier of high accuracy due to the forward reasoning problem being easier to solve. Further, we believe our Bayesian formulation and the use of small labeled dataset, to estimate the accuracy of the verifier are also novel contributions of our work.

Comment 3: The ``backward reasoning'' problem and proposed approach are only evaluated on math benchmarks. However, if we target for uncovering a missing condition, it is essential to do evaluations on other types of reasoning tasks, such as commonsense and symbolic

Response: This is a fair comment. At the same time, we do believe this is the first study of its kind, where we have shown that LLMs significantly suffer on performance in the backward reasoning problem compared to forward reasoning, in this simple setting. We have formally defined the task, and also given a few methods to enhance the performance of LLMs, showing significant improvement in accuracy. We are currently experimenting on a more challenging version of the task, where we mask out an entire phrase instead of a single number. The preliminary experiments show similar trends in numbers, where we see that there is a significant drop in performance compared to forward reasoning, and some of our methods seem to help. We plan to upload a fuller set of experimental results before the end of the discussion phase.

We believe there are interesting connections of our backward reasoning with problems such as code in-filling [1], or purely common sense abductive reasoning problems, and exploring this connection is a direction for future work.

References:

1. Fried, Daniel, et al. "InCoder: A Generative Model for Code Infilling and Synthesis." ICLR 2022.

We thank the reviewer for the review, for recognizing our contributions, and providing comments to further improve our paper.

Comment 1: Only the verifier with Bayesian ensembling gives some novelty and some insights to the community. But they did not expand more analysis except Table 5 (it should be Table 5 rather than Figure 5 I guess)

Comment 2: The other content seems more engineering efforts than scientific efforts. I think the dataset is also really important, which should be described (with more details) in the main paper rather than in the appendix.

Response to above comments: We thank the reviewer for acknowledging the significance and relevance of the dataset proposed in our work. We will move the details of the dataset from the appendix to the main paper.

Sorry for the typo, it is Table 5 instead of Figure 5.

We respectfully beg to differ from the reviewer regarding novelty and insights for the community. Current Large Language Models (LLMs) have achieved remarkable results in forward reasoning, particularly in Math Word Problems, where the task involves finding the answer to a given question. In contrast, human problem-solving and decision-making often involve backward reasoning, a powerful technique in mathematics for problem-solving and theorem proving. Surprisingly, the capabilities of LLMs in backward reasoning remain relatively unexplored. So, other than proposing datasets, our other major contributions include:

1. Our experiment on Math word problems demonstrates the challenges LLMs face in replicating the intuitive backward reasoning abilities inherent in human problem-solving. Our findings shed light on the divergence between LLMs and human cognitive processes. This understanding opens avenues for further research to bridge this gap and advance the capabilities of language models in mimicking human-like reasoning strategies.
2. We believe we are the first to formally define the backward reasoning task in math word problems and demonstrate a significant decrease in the accuracy of state-of-the-art Language Models (LLMs) on backward reasoning compared to forward reasoning.
3. We have introduced novel techniques that enhance the performance of LLMs on the backward reasoning task.

Comment 3: Not enough experiments to compare, for example, to show that the ensembling is better, we probably need to compare with other ensembling techniques. Comparison with majority voting.

Response: We conducted experiments using majority voting and observed that our proposed ensembling method outperforms majority voting. Below table shows the comparison of ensembling method with other methods including majority voting.

Table: Comparison of Ensembling with other methods

Comment 4: I think the paper should focus more on the task itself, rather than propose something and just put some short results.

Response: In this work, we have defined a new task of backward reasoning, proposed a new dataset to test the performance of LLMs on backward reasoning, and suggested novel techniques to improve the performance of LLMs on backward reasoning. Please let us know if you want us to include anything specific regarding the task.

Comment 5: What's the difference between the reprompt and verification and check your work, are they the same thing? Why we have two names for the same method?

Response: Thank you for your observation, and we apologize for the confusion. Both reprompt and verification and check your work refer to the same method. We will update the paper and ensure consistency by using a unified term for clarity.