Learning to solve arithmetic problems with a virtual abacus

Abstract

Acquiring mathematical skills is considered a key challenge for modern Artificial Intelligence systems. Inspired by the way humans discover numerical knowledge, here we introduce a deep reinforcement learning framework that allows to simulate how cognitive agents could gradually learn to solve arithmetic problems by interacting with a virtual abacus. The proposed model successfully learn to perform multi-digit additions and subtractions, achieving an error rate below 1% even when operands are much longer than those observed during training. We also compare the performance of learning agents receiving a different amount of explicit supervision, and we analyze the most common error patterns to better understand the limitations and biases resulting from our design choices.

Can Neural Networks Do Arithmetic? A Survey on the Elementary Numerical Skills of State-of-the-Art Deep Learning Models

“…Another recent approach proposed to extend the regular transformer architecture to promote problem decomposition into reusable building blocks, applied in sequential steps. This allowed a significant increase in performance in terms of simple arithmetic and list processing tasks [56]. A different modular neurosymbolic architecture was shown to be able to improve on out-of-distribution nested arithmetical expressions when trained only on a subset including the simplest cases, by iteratively applying a learned substitution rule on the input string [57].…”

“…The idea of granting access to an external memory to solve complex problems is reminiscent of the notion of "material representation" introduced in anthropology, which has been recently elaborated on in the context of numerical cognition [63]. According to this view, abstract mathematical concepts would be a relatively recent cultural achievement, which have emerged thanks to the spread of numerical manipulation tools [64]. This perspective has recently been explored in computational models based on deep reinforcement learning, which can simulate the active interaction of a learning agent with external numerical representation devices [65,66].…”

“…This allows this model to significantly outperform the original PaLM model, although its average accuracy remains below the human level. Other recent improvements introduced algorithmic prompting methods [83,84], which provide a detailed description of the algorithm execution on running examples and use explicit natural language instructions to further remove ambiguity. This increases the amount of detail included in the input rationales, whilst at the same time, promoting the composition of basic skills to solve complex reasoning tasks.…”