README.md

metadata

license: mit
task_categories:
  - text-generation
language:
  - en
tags:
  - mathematics
  - group-theory
  - permutations
  - symbolic-reasoning
pretty_name: Permutation Groups Dataset
size_categories:
  - 100K<n<1M
configs:
  - config_name: s3_data
    data_files:
      - split: train
        path: data/s3_data/train/*
      - split: test
        path: data/s3_data/test/*
  - config_name: s4_data
    data_files:
      - split: train
        path: data/s4_data/train/*
      - split: test
        path: data/s4_data/test/*
  - config_name: s5_data
    data_files:
      - split: train
        path: data/s5_data/train/*
      - split: test
        path: data/s5_data/test/*
  - config_name: s6_data
    data_files:
      - split: train
        path: data/s6_data/train/*
      - split: test
        path: data/s6_data/test/*
  - config_name: s7_data
    data_files:
      - split: train
        path: data/s7_data/train/*
      - split: test
        path: data/s7_data/test/*
  - config_name: a3_data
    data_files:
      - split: train
        path: data/a3_data/train/*
      - split: test
        path: data/a3_data/test/*
  - config_name: a4_data
    data_files:
      - split: train
        path: data/a4_data/train/*
      - split: test
        path: data/a4_data/test/*
  - config_name: a5_data
    data_files:
      - split: train
        path: data/a5_data/train/*
      - split: test
        path: data/a5_data/test/*
  - config_name: a6_data
    data_files:
      - split: train
        path: data/a6_data/train/*
      - split: test
        path: data/a6_data/test/*
  - config_name: a7_data
    data_files:
      - split: train
        path: data/a7_data/train/*
      - split: test
        path: data/a7_data/test/*

Permutation Groups Dataset

A comprehensive collection of permutation composition datasets for symmetric and alternating groups, designed for training and evaluating models on group theory operations.

Dataset Description

This dataset contains permutation composition problems for various mathematical groups:

• Symmetric Groups: S3, S4, S5, S6, S7
• Alternating Groups: A3, A4, A5, A6, A7

Each dataset consists of sequences of permutations that need to be composed to produce a target permutation. This is useful for:

• Training models on symbolic reasoning
• Evaluating mathematical understanding
• Testing compositional generalization
• Studying group theory properties in neural networks

Usage

from datasets import load_dataset

# Load a specific group dataset
s5_dataset = load_dataset("BeeGass/permutation-groups", name="s5_data", trust_remote_code=True)

# Load alternating group A5
a5_dataset = load_dataset("BeeGass/permutation-groups", name="a5_data", trust_remote_code=True)

# Load all datasets combined
all_datasets = load_dataset("BeeGass/permutation-groups", name="all", trust_remote_code=True)

# Access the data
train_data = s5_dataset["train"]
test_data = s5_dataset["test"]

# Example data point
print(train_data[0])
# {'input_sequence': '23 45 12', 'target': '67'}

Dataset Structure

Each example contains:

• input_sequence: A space-separated sequence of permutation IDs to be composed
• target: The ID of the resulting permutation after composition

The composition follows standard mathematical convention: for input [p1, p2, p3], the result is p3 ∘ p2 ∘ p1.

Available Configurations

Configuration	Group Type	Group Order	Elements	Train Samples	Test Samples
s3_data	Symmetric	S3	6	8,000	2,000
s4_data	Symmetric	S4	24	16,000	4,000
s5_data	Symmetric	S5	120	40,000	10,000
s6_data	Symmetric	S6	720	80,000	20,000
s7_data	Symmetric	S7	5,040	160,000	40,000
a3_data	Alternating	A3	3	4,000	1,000
a4_data	Alternating	A4	12	12,000	3,000
a5_data	Alternating	A5	60	24,000	6,000
a6_data	Alternating	A6	360	64,000	16,000
a7_data	Alternating	A7	2,520	120,000	30,000
all	Combined	-	-	528,000	132,000

Dataset Features

• Variable sequence length: Input sequences range from 3 to 512 permutations
• Consistent formatting: All permutations use space-separated integer IDs
• Metadata included: Each dataset includes a metadata.json file mapping IDs to permutation array forms
• Train/test split: 80/20 split for all configurations

Understanding the Data

Each permutation is represented by a unique integer ID. The metadata.json file in each dataset folder provides the mapping from IDs to permutation array forms.

For example, in S3:

• ID 0 might map to [0, 1, 2] (identity)
• ID 1 might map to [0, 2, 1] (transpose elements 1 and 2)
• etc.

Citation

If you use this dataset in your research, please cite:

@software{permutation_groups_dataset,
  author = {Bryan Gass},
  title = {Permutation Groups Dataset},
  year = {2024},
  publisher = {Hugging Face},
  url = {https://huggingface.co/datasets/BeeGass/permutation-groups}
}

Acknowledgments

This dataset was inspired by the work of William Merrill and his paper "The Illusion of State in State-Space Models", which explores the computational properties of state-space models through group theory.

License

This dataset is released under the MIT License.

Contact

For questions or issues, please open an issue on the GitHub repository.