Do Large Language Models Truly Grasp Addition? A Rule-Focused Diagnostic Using Two-Integer Arithmetic

• URL: http://arxiv.org/abs/2504.05262v3
• Date: Wed, 17 Sep 2025 07:14:56 GMT
• Title: Do Large Language Models Truly Grasp Addition? A Rule-Focused Diagnostic Using Two-Integer Arithmetic
• Authors: Yang Yan, Yu Lu, Renjun Xu, Zhenzhong Lan,
• Abstract summary: Large language models (LLMs) achieve impressive results on advanced mathematics benchmarks but sometimes fail on basic arithmetic tasks.<n>We investigate whether they have truly grasped fundamental arithmetic rules or are merely relying on pattern matching.<n>Our evaluation of 12 leading LLMs reveals a stark disconnect: while models achieve high numeric accuracy, they systematically fail these diagnostics.
• Score: 21.014229380679975
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: Large language models (LLMs) achieve impressive results on advanced mathematics benchmarks but sometimes fail on basic arithmetic tasks, raising the question of whether they have truly grasped fundamental arithmetic rules or are merely relying on pattern matching. To unravel this issue, we systematically probe LLMs' understanding of two-integer addition ($0$ to $2^{64}$) by testing three crucial properties: commutativity ($A+B=B+A$), representation invariance via symbolic remapping (e.g., $7 \mapsto Y$), and consistent accuracy scaling with operand length. Our evaluation of 12 leading LLMs reveals a stark disconnect: while models achieve high numeric accuracy (73.8-99.8%), they systematically fail these diagnostics. Specifically, accuracy plummets to $\le 7.5$% with symbolic inputs, commutativity is violated in up to 20% of cases, and accuracy scaling is non-monotonic. Interventions further expose this pattern-matching reliance: explicitly providing rules degrades performance by 29.49%, while prompting for explanations before answering merely maintains baseline accuracy. These findings demonstrate that current LLMs address elementary addition via pattern matching, not robust rule induction, motivating new diagnostic benchmarks and innovations in model architecture and training to cultivate genuine mathematical reasoning. Our dataset and generating code are available at https://github.com/kuri-leo/llm-arithmetic-diagnostic.

Related papers

• Verifying Large Language Models' Reasoning Paths via Correlation Matrix Rank [71.09032766271493]
  Large language models (LLMs) are prone to errors and hallucinations.<n>How to check their outputs effectively and efficiently has become a critical problem in their applications.
  arXiv  Detail & Related papers  (2025-10-28T11:01:10Z)
• SciML Agents: Write the Solver, Not the Solution [69.5021018644143]
  We introduce two new datasets: a diagnostic dataset of adversarial "misleading" problems; and a large-scale benchmark of 1,000 diverse ODE tasks.<n>We evaluate open- and closed-source LLM models along two axes: (i) unguided versus guided prompting with domain-specific knowledge; and (ii) off-the-shelf versus fine-tuned variants.<n>Preliminary results indicate that careful prompting and fine-tuning can yield a specialized LLM agent capable of reliably solving simple ODE problems.
  arXiv  Detail & Related papers  (2025-09-12T02:53:57Z)
• Probing for Arithmetic Errors in Language Models [86.8227317662622]
  Internal activations in language models can be used to detect arithmetic errors.<n>We show that simple probes can accurately decode both the model's predicted output and the correct answer from hidden states.<n>We train lightweight error detectors that predict model correctness with over 90% accuracy.
  arXiv  Detail & Related papers  (2025-07-16T16:27:50Z)
• Broken Tokens? Your Language Model can Secretly Handle Non-Canonical Tokenizations [83.93566096400723]
  We find that instruction-tuned models retain up to 93.4% of their original performance when given a randomly sampled tokenization.<n>Character-level segmentation improves string manipulation and code understanding tasks by up to +14%.<n>Right-aligned digit grouping enhances large-number arithmetic by +33%.
  arXiv  Detail & Related papers  (2025-06-23T18:02:26Z)
• ASyMOB: Algebraic Symbolic Mathematical Operations Benchmark [0.0]
  Large language models (LLMs) are rapidly approaching the level of proficiency in university-level symbolic mathematics.<n>We introduce ASyMOB, a novel assessment framework focused exclusively on symbolic manipulation.
  arXiv  Detail & Related papers  (2025-05-28T23:11:14Z)
• Out-of-Context Relational Reasoning in Large Language Models [14.326344469446438]
  We study how well can Large Language Models (LLM) reason out-of-context on binary relations by only learning the representations of newly introduced tokens.<n>Our experiments focus on equality ($=$), inequality ($$), and inclusion ($subset$) and the properties they satisfy.<n>We show that LLMs achieve better than random accuracy, but are still far from perfect, even on relatively simple reasoning tasks involving binary relations.
  arXiv  Detail & Related papers  (2025-03-13T14:32:30Z)
• Token-by-Token Regeneration and Domain Biases: A Benchmark of LLMs on Advanced Mathematical Problem-Solving [0.0]
  This study evalu-ates 10 large language models (LLMs) with 7 to 8 billion parameters using the MATH dataset.<n>The focus is on their ability to generate executable Python code as a step in their reasoning process, involving over 9,450 code executions.
  arXiv  Detail & Related papers  (2025-01-28T17:11:36Z)
• UGMathBench: A Diverse and Dynamic Benchmark for Undergraduate-Level Mathematical Reasoning with Large Language Models [11.964085209696051]
  UGMathBench comprises 5,062 problems across 16 subjects and 111 topics, featuring 10 distinct answer types.<n>Each problem includes three randomized versions, with additional versions planned for release as leading open-source LLMs become saturated in UGMathBench.<n>Our evaluation of 23 leading LLMs reveals that the highest EAcc robustness achieved is 56.3% by OpenAI-o1-mini, with large $Delta$ values observed across different models.
  arXiv  Detail & Related papers  (2025-01-23T15:46:43Z)
• BoostStep: Boosting mathematical capability of Large Language Models via improved single-step reasoning [83.03531832811386]
  BoostStep is a method that enhances reasoning accuracy through step-aligned ICL examples.<n>It integrates seamlessly with chain-of-thought (CoT) and tree search algorithms.<n>It improves DeepSeek-R1-671B's performance on AIME by 2.2%, leveraging simple examples only from the MATH dataset.
  arXiv  Detail & Related papers  (2025-01-06T18:59:13Z)
• IGC: Integrating a Gated Calculator into an LLM to Solve Arithmetic Tasks Reliably and Efficiently [17.525220958618988]
  We introduce the Integrated Gated Calculator (IGC), a module that enables Large Language Models to perform arithmetic by emulating a calculator on the GPU. We finetune a Llama model with our module and test it on the BigBench Arithmetic benchmark, where it beats the State of the Art. Our approach takes only a single iteration to run and requires no external tools.
  arXiv  Detail & Related papers  (2025-01-01T00:01:27Z)
• RuleArena: A Benchmark for Rule-Guided Reasoning with LLMs in Real-World Scenarios [58.90106984375913]
  RuleArena is a novel and challenging benchmark designed to evaluate the ability of large language models (LLMs) to follow complex, real-world rules in reasoning.<n> Covering three practical domains -- airline baggage fees, NBA transactions, and tax regulations -- RuleArena assesses LLMs' proficiency in handling intricate natural language instructions.
  arXiv  Detail & Related papers  (2024-12-12T06:08:46Z)
• Reasoning Robustness of LLMs to Adversarial Typographical Errors [49.99118660264703]
  Large Language Models (LLMs) have demonstrated impressive capabilities in reasoning using Chain-of-Thought (CoT) prompting. We study the reasoning robustness of LLMs to typographical errors, which can naturally occur in users' queries. We design an Adversarial Typo Attack ($texttATA$) algorithm that iteratively samples typos for words that are important to the query and selects the edit that is most likely to succeed in attacking.
  arXiv  Detail & Related papers  (2024-11-08T05:54:05Z)
• Language Models are Symbolic Learners in Arithmetic [8.34588487873447]
  Large Language Models (LLMs) are thought to struggle with arithmetic learning due to inherent differences between language modeling and numerical computation. We first investigate whether LLMs leverage partial products during arithmetic learning. We find that although LLMs can identify some partial products after learning, they fail to leverage them for arithmetic tasks, conversely.
  arXiv  Detail & Related papers  (2024-10-21T01:57:16Z)
• MathHay: An Automated Benchmark for Long-Context Mathematical Reasoning in LLMs [61.74749961334557]
  MathHay is an automated benchmark designed to assess the long-context mathematical reasoning capabilities of LLMs. We conduct extensive experiments on MathHay to assess the long-context mathematical reasoning abilities of eight top-performing models.
  arXiv  Detail & Related papers  (2024-10-07T02:30:07Z)
• Symbolic Working Memory Enhances Language Models for Complex Rule Application [87.34281749422756]
  Large Language Models (LLMs) have shown remarkable reasoning performance but struggle with multi-step deductive reasoning. We propose augmenting LLMs with external working memory and introduce a neurosymbolic framework for rule application. Our framework iteratively performs symbolic rule grounding and LLM-based rule implementation.
  arXiv  Detail & Related papers  (2024-08-24T19:11:54Z)
• OccamLLM: Fast and Exact Language Model Arithmetic in a Single Step [7.7168728919692855]
  We propose a framework that enables exact arithmetic in a single autoregressive step. We use the hidden states of a LLM to control a symbolic architecture that performs arithmetic. Our implementation using Llama 3 with OccamNet as a symbolic model (OccamLlama) achieves 100% accuracy on single arithmetic operations.
  arXiv  Detail & Related papers  (2024-06-04T04:17:40Z)
• Common 7B Language Models Already Possess Strong Math Capabilities [61.61442513067561]
  This paper shows that the LLaMA-2 7B model with common pre-training already exhibits strong mathematical abilities. The potential for extensive scaling is constrained by the scarcity of publicly available math questions.
  arXiv  Detail & Related papers  (2024-03-07T18:00:40Z)
• GSM-Plus: A Comprehensive Benchmark for Evaluating the Robustness of LLMs as Mathematical Problem Solvers [68.77382332826167]
  Large language models (LLMs) have achieved impressive performance across various mathematical reasoning benchmarks. One essential and frequently occurring evidence is that when the math questions are slightly changed, LLMs can behave incorrectly. This motivates us to evaluate the robustness of LLMs' math reasoning capability by testing a wide range of question variations.
  arXiv  Detail & Related papers  (2024-02-29T15:26:14Z)
• SatLM: Satisfiability-Aided Language Models Using Declarative Prompting [68.40726892904286]
  We propose a new satisfiability-aided language modeling (SatLM) approach for improving the reasoning capabilities of large language models (LLMs) We use an LLM to generate a declarative task specification rather than an imperative program and leverage an off-the-shelf automated theorem prover to derive the final answer. We evaluate SATLM on 8 different datasets and show that it consistently outperforms program-aided LMs in the imperative paradigm.
  arXiv  Detail & Related papers  (2023-05-16T17:55:51Z)
• MathPrompter: Mathematical Reasoning using Large Language Models [7.953723258038284]
  Large Language Models (LLMs) have limited performance when solving arithmetic reasoning tasks. MathPrompter uses the Zero-shot chain-of-thought prompting technique to generate multiple Algebraic expressions or Python functions to solve the same math problem in different ways.
  arXiv  Detail & Related papers  (2023-03-04T04:43:49Z)
• PAL: Program-aided Language Models [112.94785609781503]
  We present Program-Aided Language models (PaL) to understand natural language problems. PaL offloads the solution step to a programmatic runtime such as a Python interpreter. We set new state-of-the-art results in all 12 benchmarks.
  arXiv  Detail & Related papers  (2022-11-18T18:56:13Z)
• Large Language Models are Zero-Shot Reasoners [28.6899375595088]
  Chain of thought (CoT) prompting is a technique for eliciting complex multi-step reasoning through step-by-step answer examples. We show that LLMs are decent zero-shot reasoners by simply adding Let's think step by step'' before each answer. Experimental results demonstrate that our Zero-shot-CoT, using the same single prompt template, significantly outperforms zero-shot LLM performances.
  arXiv  Detail & Related papers  (2022-05-24T09:22:26Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.