Mathematical problem solving is an important skill for Large Language Models (LLMs), both as an important capability and a proxy for a range of reasoning abilities. Existing benchmarks probe a diverse set of skills, but they yield aggregate accuracy metrics, obscuring specific abilities or weaknesses. Furthermore, they are difficult to extend with new problems, risking data contamination over time. To address these challenges, we propose MathCAMPS: a method to synthesize high-quality mathematical problems at scale, grounded on 44 fine-grained "standards" from the Mathematics Common Core (CC) Standard for K-8 grades. We encode each standard in a formal grammar, allowing us to sample diverse symbolic problems and their answers. We then use LLMs to realize the symbolic problems into word problems. We propose a cycle-consistency method for validating problem faithfulness. Finally, we derive follow-up questions from symbolic structures and convert them into follow-up word problems - a novel task of mathematical dialogue that probes for robustness in understanding. Experiments on 23 LLMs show surprising failures even in the strongest models (in particular when asked simple follow-up questions). Moreover, we evaluate training checkpoints of Pythia 12B on MathCAMPS, allowing us to analyze when particular mathematical skills develop during its training. Our framework enables the community to reproduce and extend our pipeline for a fraction of the typical cost of building new high-quality datasets.

We explored the addition bias, a cognitive tendency to prefer adding elements over removing them to alter an initial state or structure, by conducting four preregistered experiments examining the problem-solving behavior of both humans and OpenAl's GPT-4 large language model. The experiments involved 588 participants from the U.S. and 680 iterations of the GPT-4 model. The problem-solving task was either to create symmetry within a grid (Experiments 1 and 3) or to edit a summary (Experiments 2 and 4). As hypothesized, we found that overall, the addition bias was present. Solution efficiency (Experiments 1 and 2) and valence of the instruction (Experiments 3 and 4) played important roles. Human participants were less likely to use additive strategies when subtraction was relatively more efficient than when addition and subtraction were equally efficient. GPT-4 exhibited the opposite behavior, with a strong addition bias when subtraction was more efficient. In terms of instruction valence, GPT-4 was more likely to add words when asked to "improve" compared to "edit", whereas humans did not show this effect. When we looked at the addition bias under different conditions, we found more biased responses for GPT-4 compared to humans. Our findings highlight the importance of considering comparable and sometimes superior subtractive alternatives, as well as reevaluating one's own and particularly the language models' problem-solving behavior.

Recent advancements in pretraining have demonstrated that modern Large Language Models (LLMs) possess the capability to effectively learn arithmetic operations. However, despite acknowledging the significance of digit order in arithmetic computation, current methodologies predominantly rely on sequential, step-by-step approaches for teaching LLMs arithmetic, resulting in a conclusion where obtaining better performance involves fine-grained step-by-step. Diverging from this conventional path, our work introduces a novel strategy that not only reevaluates the digit order by prioritizing output from the least significant digit but also incorporates a step-by-step methodology to substantially reduce complexity. We have developed and applied this method in a comprehensive set of experiments. Compared to the previous state-of-the-art (SOTA) method, our findings reveal an overall improvement of in accuracy while requiring only a third of the tokens typically used during training. For the purpose of facilitating replication and further research, we have made our code and dataset publicly available at \url{https://anonymous.4open.science/r/RAIT-9FB7/}.

Language Models (LMs) are increasingly used for a wide range of prediction tasks, but their training can often neglect rare edge cases, reducing their reliability. Here, we define a stringent standard of trustworthiness whereby the task algorithm and circuit implementation must be verified, accounting for edge cases, with no known failure modes. We show that a transformer model can be trained to meet this standard if built using mathematically and logically specified frameworks. In this paper, we fully verify a model for n-digit integer addition. To exhibit the reusability of verified modules, we insert the trained integer addition model into an untrained model and train the combined model to perform both addition and subtraction. We find extensive reuse of the addition circuits for both tasks, easing verification of the more complex subtractor model. We discuss how inserting verified task modules into LMs can leverage model reuse to improve verifiability and trustworthiness of language models built using them. The reuse of verified circuits reduces the effort to verify more complex composite models which we believe to be a significant step towards safety of language models.

This paper presents RevOrder, a novel technique aimed at improving arithmetic operations in large language models (LLMs) by reversing the output digits in addition, subtraction, and n-digit by 1-digit (nD by 1D) multiplication tasks. Our method significantly reduces the Count of Sequential Intermediate Digits (CSID) to $\mathcal{O}(1)$, a new metric we introduce to assess equation complexity. Through comprehensive testing, RevOrder not only achieves perfect accuracy in basic arithmetic operations but also substantially boosts LLM performance in division tasks, particularly with large numbers where traditional models struggle. Implementation of RevOrder is cost-effective for both training and inference phases. Moreover, applying RevOrder to fine-tune the LLaMA2-7B model on the GSM8K math task results in a considerable improvement, reducing equation calculation errors by 46% and increasing overall scores from 41.6 to 44.4.

We introduce Goat, a fine-tuned LLaMA model that significantly outperforms GPT-4 on a range of arithmetic tasks. Fine-tuned on a synthetically generated dataset, Goat achieves state-of-the-art performance on BIG-bench arithmetic sub-task. In particular, the zero-shot Goat-7B matches or even surpasses the accuracy achieved by the few-shot PaLM-540B. Surprisingly, Goat can achieve near-perfect accuracy on large-number addition and subtraction through supervised fine-tuning only, which is almost impossible with previous pretrained language models, such as Bloom, OPT, GPT-NeoX, etc. We attribute Goat's exceptional performance to LLaMA's consistent tokenization of numbers. To tackle more challenging tasks like large-number multiplication and division, we propose an approach that classifies tasks based on their learnability, and subsequently decomposes unlearnable tasks, such as multi-digit multiplication and division, into a series of learnable tasks by leveraging basic arithmetic principles. We thoroughly examine the performance of our model, offering a comprehensive evaluation of the effectiveness of our proposed decomposition steps. Additionally, Goat-7B can be easily trained using LoRA on a 24GB VRAM GPU, facilitating reproducibility for other researchers. We release our model, dataset, and the Python script for dataset generation.

Answering complex questions that require multi-step multi-type reasoning over raw text is challenging, especially when conducting numerical reasoning. Neural Module Networks(NMNs), follow the programmer-interpreter framework and design trainable modules to learn different reasoning skills. However, NMNs only have limited reasoning abilities, and lack numerical reasoning capability. We up-grade NMNs by: (a) bridging the gap between its interpreter and the complex questions; (b) introducing addition and subtraction modules that perform numerical reasoning over numbers. On a subset of DROP, experimental results show that our proposed methods enhance NMNs' numerical reasoning skills by 17.7% improvement of F1 score and significantly outperform previous state-of-the-art models.

Okay, let's wait a few seconds. Feel free to have a quick stretch. Now, without looking, what was the answer to the first question? You've just performed a series of advanced brain functions. You did math based on prompts designed to appeal to entirely different parts of your brain and you displayed the ability to recall previous information when queried later.

How does google understand how to translate '今日はどうですか？' to'How are you doing today?' or vice versa? How do we get to predict a disease spread such as COVID-19 way into the future beforehand? How do automatic Text generation or Text Summarization mechanisms work? The answer is Recurrent Neural Networks. RNNs have been the solution to deal with most problems in Natural language Processing and not only NLP but in Bio-informatics, Financial Forecasting, Sequence modelling etc.

Looking back at the history of calculators, one can see that they become less functional and more computationally expensive over time. A modern calculator runs on a personal computer and is drawn at 60 fps only to help us click a few digits with a mouse pointer. A search engine is often used as a calculator, which means that nowadays we need the Internet just to add two numbers. In this paper, we propose to go further and train a convolutional neural network that takes an image of a simple mathematical expression and generates an image of an answer. This neural calculator works only with pairs of double-digit numbers and supports only addition and subtraction. Also, sometimes it makes mistakes. We promise that the proposed calculator is a small step for man, but one giant leap for mankind.