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Linear Reasoning vs. Proof by Cases: Obstacles for Large Language Models in FOL Problem Solving

Yuliang Ji, Fuchen Shen, Jian Wu, Qiujie Xie, Yue Zhang

TL;DR

This work introduces a novel first-order logic (FOL) dataset named PC-FOL, annotated by professional mathematicians, focusing on case-based reasoning problems, and provides a theoretical analysis grounded in graphical model, which provides an explanation for the observed disparity between the two types of reasoning problems.

Abstract

To comprehensively evaluate the mathematical reasoning capabilities of Large Language Models (LLMs), researchers have introduced abundant mathematical reasoning datasets. However, most existing datasets primarily focus on linear reasoning, neglecting other parts such as proof by contradiction and proof by cases, which are crucial for investigating LLMs' reasoning abilities. To address this limitation, we first introduce a novel first-order logic (FOL) dataset named PC-FOL, annotated by professional mathematicians, focusing on case-based reasoning problems. All instances in this dataset are equipped with a manually written natural language proof, clearly distinguishing it from conventional linear reasoning datasets. Our experimental results over leading LLMs demonstrate a substantial performance gap between linear reasoning and case-based reasoning problems. To further investigate this phenomenon, we provide a theoretical analysis grounded in graphical model, which provides an explanation for the observed disparity between the two types of reasoning problems. We hope this work can reveal the core challenges in the field of automated natural language mathematical proof generation, paving the way for future research.

Linear Reasoning vs. Proof by Cases: Obstacles for Large Language Models in FOL Problem Solving

TL;DR

This work introduces a novel first-order logic (FOL) dataset named PC-FOL, annotated by professional mathematicians, focusing on case-based reasoning problems, and provides a theoretical analysis grounded in graphical model, which provides an explanation for the observed disparity between the two types of reasoning problems.

Abstract

To comprehensively evaluate the mathematical reasoning capabilities of Large Language Models (LLMs), researchers have introduced abundant mathematical reasoning datasets. However, most existing datasets primarily focus on linear reasoning, neglecting other parts such as proof by contradiction and proof by cases, which are crucial for investigating LLMs' reasoning abilities. To address this limitation, we first introduce a novel first-order logic (FOL) dataset named PC-FOL, annotated by professional mathematicians, focusing on case-based reasoning problems. All instances in this dataset are equipped with a manually written natural language proof, clearly distinguishing it from conventional linear reasoning datasets. Our experimental results over leading LLMs demonstrate a substantial performance gap between linear reasoning and case-based reasoning problems. To further investigate this phenomenon, we provide a theoretical analysis grounded in graphical model, which provides an explanation for the observed disparity between the two types of reasoning problems. We hope this work can reveal the core challenges in the field of automated natural language mathematical proof generation, paving the way for future research.
Paper Structure (48 sections, 3 theorems, 7 equations, 6 figures, 13 tables)

This paper contains 48 sections, 3 theorems, 7 equations, 6 figures, 13 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Dataset for FOL reasoning
  4. LLM Natural Language Reasoning over Mathematical Problems
  5. Preliminaries
  6. FOL Natural-Language Problem
  7. Graphical Model
  8. The PC-FOL Dataset
  9. Experiments
  10. Experiment Setting
  11. Natural language FOL reasoning
  12. Proof Evaluation
  13. Manually Checking Proofs
  14. Main Findings
  15. Theoretical Analysis
  16. ...and 33 more sections

Key Result

Theorem 6.1

Under previous assumptions, for a LLM $F$, a linear-reasoning dataset $D$ with the distribution $P_{D}$ of reasoning steps, the probability that the LLM $F$ can give a correct proof is $\sum\limits_{k=1}^{\infty} p_F^k \cdot P_D(|X|=k)$, or $E_{X \sim P_D}[p_F^{|X|}].$

Figures (6)

  • Figure 1: Abstracted reasoning chain for the left side example of Table \ref{['table_of_difference_between_linear_and_case']}.
  • Figure 2: Upper: The abstracted reasoning chain for right side example of Table \ref{['table_of_difference_between_linear_and_case']}. Lower: The abstracted reasoning chain for a typical example that existing a contradiction in a subcase. The dotted line represents a possible case inferred from a certain premise and the previous steps, and the red double-headed arrow denotes the existence of a contradiction.
  • Figure 3: The distribution of the number of premises in each instance of the PC-FOL dataset. Blue color represents the distribution of the proof-by-case type problems, and the yellow color represents the distribution of the linear reasoning type problems.
  • Figure 4: Comparison between the ground truth of the correct proof ratio and the estimated correctness ration over the PC-FOL Linear-Reasoning dataset.
  • Figure 5: Left: The abstracted reasoning chain for right side example of Table \ref{['table_of_difference_between_linear_and_case']}. Right: The abstracted reasoning chain for a typical example that existing a contradiction in a subcase.
  • ...and 1 more figures

Theorems & Definitions (6)

  • Theorem 6.1
  • Theorem 6.2
  • Theorem 6.3
  • proof
  • proof
  • proof