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Datasets:
BeeGass
/
Group-Theory-Collection

Tasks:
Text Generation
Languages:
English
Size:
10M<n<100M
ArXiv:
Tags:
mathematics
group-theory
permutations
symbolic-reasoning
algebra
sequence-modeling
DOI:
License:
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Group-Theory-Collection / README.md
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---
license: mit
task_categories:
- text-generation
language:
- en
tags:
- mathematics
- group-theory
- permutations
- symbolic-reasoning
- algebra
- sequence-modeling
- state-space-models
- computational-complexity
pretty_name: Group Theory Collection
size_categories:
- 10M<n<100M
configs:
- config_name: s3
 data_files:
 - split: train
 path: data/s3/train/*.arrow
 - split: test
 path: data/s3/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: s4
 data_files:
 - split: train
 path: data/s4/train/*.arrow
 - split: test
 path: data/s4/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: s5
 data_files:
 - split: train
 path: data/s5/train/*.arrow
 - split: test
 path: data/s5/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: s6
 data_files:
 - split: train
 path: data/s6/train/*.arrow
 - split: test
 path: data/s6/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: s7
 data_files:
 - split: train
 path: data/s7/train/*.arrow
 - split: test
 path: data/s7/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: s8
 data_files:
 - split: train
 path: data/s8/train/*.arrow
 - split: test
 path: data/s8/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: s9
 data_files:
 - split: train
 path: data/s9/train/*.arrow
 - split: test
 path: data/s9/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: a3
 data_files:
 - split: train
 path: data/a3/train/*.arrow
 - split: test
 path: data/a3/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: a4
 data_files:
 - split: train
 path: data/a4/train/*.arrow
 - split: test
 path: data/a4/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: a5
 data_files:
 - split: train
 path: data/a5/train/*.arrow
 - split: test
 path: data/a5/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: a6
 data_files:
 - split: train
 path: data/a6/train/*.arrow
 - split: test
 path: data/a6/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: a7
 data_files:
 - split: train
 path: data/a7/train/*.arrow
 - split: test
 path: data/a7/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: a8
 data_files:
 - split: train
 path: data/a8/train/*.arrow
 - split: test
 path: data/a8/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: a9
 data_files:
 - split: train
 path: data/a9/train/*.arrow
 - split: test
 path: data/a9/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: c2
 data_files:
 - split: train
 path: data/c2/train/*.arrow
 - split: test
 path: data/c2/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: c3
 data_files:
 - split: train
 path: data/c3/train/*.arrow
 - split: test
 path: data/c3/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: c4
 data_files:
 - split: train
 path: data/c4/train/*.arrow
 - split: test
 path: data/c4/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: c5
 data_files:
 - split: train
 path: data/c5/train/*.arrow
 - split: test
 path: data/c5/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: c6
 data_files:
 - split: train
 path: data/c6/train/*.arrow
 - split: test
 path: data/c6/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: c7
 data_files:
 - split: train
 path: data/c7/train/*.arrow
 - split: test
 path: data/c7/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: c8
 data_files:
 - split: train
 path: data/c8/train/*.arrow
 - split: test
 path: data/c8/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: c9
 data_files:
 - split: train
 path: data/c9/train/*.arrow
 - split: test
 path: data/c9/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: c10
 data_files:
 - split: train
 path: data/c10/train/*.arrow
 - split: test
 path: data/c10/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: c11
 data_files:
 - split: train
 path: data/c11/train/*.arrow
 - split: test
 path: data/c11/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: c12
 data_files:
 - split: train
 path: data/c12/train/*.arrow
 - split: test
 path: data/c12/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: c13
 data_files:
 - split: train
 path: data/c13/train/*.arrow
 - split: test
 path: data/c13/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: c14
 data_files:
 - split: train
 path: data/c14/train/*.arrow
 - split: test
 path: data/c14/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: c15
 data_files:
 - split: train
 path: data/c15/train/*.arrow
 - split: test
 path: data/c15/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: c16
 data_files:
 - split: train
 path: data/c16/train/*.arrow
 - split: test
 path: data/c16/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: c17
 data_files:
 - split: train
 path: data/c17/train/*.arrow
 - split: test
 path: data/c17/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: c18
 data_files:
 - split: train
 path: data/c18/train/*.arrow
 - split: test
 path: data/c18/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: c19
 data_files:
 - split: train
 path: data/c19/train/*.arrow
 - split: test
 path: data/c19/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: c20
 data_files:
 - split: train
 path: data/c20/train/*.arrow
 - split: test
 path: data/c20/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: c21
 data_files:
 - split: train
 path: data/c21/train/*.arrow
 - split: test
 path: data/c21/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: c22
 data_files:
 - split: train
 path: data/c22/train/*.arrow
 - split: test
 path: data/c22/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: c23
 data_files:
 - split: train
 path: data/c23/train/*.arrow
 - split: test
 path: data/c23/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: c24
 data_files:
 - split: train
 path: data/c24/train/*.arrow
 - split: test
 path: data/c24/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: c25
 data_files:
 - split: train
 path: data/c25/train/*.arrow
 - split: test
 path: data/c25/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: c26
 data_files:
 - split: train
 path: data/c26/train/*.arrow
 - split: test
 path: data/c26/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: c27
 data_files:
 - split: train
 path: data/c27/train/*.arrow
 - split: test
 path: data/c27/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: c28
 data_files:
 - split: train
 path: data/c28/train/*.arrow
 - split: test
 path: data/c28/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: c29
 data_files:
 - split: train
 path: data/c29/train/*.arrow
 - split: test
 path: data/c29/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: c30
 data_files:
 - split: train
 path: data/c30/train/*.arrow
 - split: test
 path: data/c30/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: d3
 data_files:
 - split: train
 path: data/d3/train/*.arrow
 - split: test
 path: data/d3/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: d4
 data_files:
 - split: train
 path: data/d4/train/*.arrow
 - split: test
 path: data/d4/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: d5
 data_files:
 - split: train
 path: data/d5/train/*.arrow
 - split: test
 path: data/d5/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: d6
 data_files:
 - split: train
 path: data/d6/train/*.arrow
 - split: test
 path: data/d6/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: d7
 data_files:
 - split: train
 path: data/d7/train/*.arrow
 - split: test
 path: data/d7/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: d8
 data_files:
 - split: train
 path: data/d8/train/*.arrow
 - split: test
 path: data/d8/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: d9
 data_files:
 - split: train
 path: data/d9/train/*.arrow
 - split: test
 path: data/d9/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: d10
 data_files:
 - split: train
 path: data/d10/train/*.arrow
 - split: test
 path: data/d10/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: d11
 data_files:
 - split: train
 path: data/d11/train/*.arrow
 - split: test
 path: data/d11/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: d12
 data_files:
 - split: train
 path: data/d12/train/*.arrow
 - split: test
 path: data/d12/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: d13
 data_files:
 - split: train
 path: data/d13/train/*.arrow
 - split: test
 path: data/d13/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: d14
 data_files:
 - split: train
 path: data/d14/train/*.arrow
 - split: test
 path: data/d14/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: d15
 data_files:
 - split: train
 path: data/d15/train/*.arrow
 - split: test
 path: data/d15/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: d16
 data_files:
 - split: train
 path: data/d16/train/*.arrow
 - split: test
 path: data/d16/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: d17
 data_files:
 - split: train
 path: data/d17/train/*.arrow
 - split: test
 path: data/d17/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: d18
 data_files:
 - split: train
 path: data/d18/train/*.arrow
 - split: test
 path: data/d18/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: d19
 data_files:
 - split: train
 path: data/d19/train/*.arrow
 - split: test
 path: data/d19/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: d20
 data_files:
 - split: train
 path: data/d20/train/*.arrow
 - split: test
 path: data/d20/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: q8
 data_files:
 - split: train
 path: data/q8/train/*.arrow
 - split: test
 path: data/q8/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: q16
 data_files:
 - split: train
 path: data/q16/train/*.arrow
 - split: test
 path: data/q16/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: q32
 data_files:
 - split: train
 path: data/q32/train/*.arrow
 - split: test
 path: data/q32/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: f20
 data_files:
 - split: train
 path: data/f20/train/*.arrow
 - split: test
 path: data/f20/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: f21
 data_files:
 - split: train
 path: data/f21/train/*.arrow
 - split: test
 path: data/f21/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: v4
 data_files:
 - split: train
 path: data/v4/train/*.arrow
 - split: test
 path: data/v4/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: z2_1
 data_files:
 - split: train
 path: data/z2_1/train/*.arrow
 - split: test
 path: data/z2_1/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: z2_2
 data_files:
 - split: train
 path: data/z2_2/train/*.arrow
 - split: test
 path: data/z2_2/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: z2_3
 data_files:
 - split: train
 path: data/z2_3/train/*.arrow
 - split: test
 path: data/z2_3/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: z2_4
 data_files:
 - split: train
 path: data/z2_4/train/*.arrow
 - split: test
 path: data/z2_4/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: z2_5
 data_files:
 - split: train
 path: data/z2_5/train/*.arrow
 - split: test
 path: data/z2_5/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: z3_1
 data_files:
 - split: train
 path: data/z3_1/train/*.arrow
 - split: test
 path: data/z3_1/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: z3_2
 data_files:
 - split: train
 path: data/z3_2/train/*.arrow
 - split: test
 path: data/z3_2/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: z3_3
 data_files:
 - split: train
 path: data/z3_3/train/*.arrow
 - split: test
 path: data/z3_3/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: z3_4
 data_files:
 - split: train
 path: data/z3_4/train/*.arrow
 - split: test
 path: data/z3_4/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: z5_1
 data_files:
 - split: train
 path: data/z5_1/train/*.arrow
 - split: test
 path: data/z5_1/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: z5_2
 data_files:
 - split: train
 path: data/z5_2/train/*.arrow
 - split: test
 path: data/z5_2/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: z5_3
 data_files:
 - split: train
 path: data/z5_3/train/*.arrow
 - split: test
 path: data/z5_3/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: z5_4
 data_files:
 - split: train
 path: data/z5_4/train/*.arrow
 - split: test
 path: data/z5_4/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: psl2_2
 data_files:
 - split: train
 path: data/psl2_2/train/*.arrow
 - split: test
 path: data/psl2_2/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: psl2_3
 data_files:
 - split: train
 path: data/psl2_3/train/*.arrow
 - split: test
 path: data/psl2_3/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: psl2_4
 data_files:
 - split: train
 path: data/psl2_4/train/*.arrow
 - split: test
 path: data/psl2_4/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: psl2_5
 data_files:
 - split: train
 path: data/psl2_5/train/*.arrow
 - split: test
 path: data/psl2_5/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: psl2_7
 data_files:
 - split: train
 path: data/psl2_7/train/*.arrow
 - split: test
 path: data/psl2_7/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: psl2_8
 data_files:
 - split: train
 path: data/psl2_8/train/*.arrow
 - split: test
 path: data/psl2_8/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: psl2_9
 data_files:
 - split: train
 path: data/psl2_9/train/*.arrow
 - split: test
 path: data/psl2_9/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: psl2_11
 data_files:
 - split: train
 path: data/psl2_11/train/*.arrow
 - split: test
 path: data/psl2_11/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: psl3_2
 data_files:
 - split: train
 path: data/psl3_2/train/*.arrow
 - split: test
 path: data/psl3_2/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: psl3_3
 data_files:
 - split: train
 path: data/psl3_3/train/*.arrow
 - split: test
 path: data/psl3_3/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: psl3_4
 data_files:
 - split: train
 path: data/psl3_4/train/*.arrow
 - split: test
 path: data/psl3_4/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: psl3_5
 data_files:
 - split: train
 path: data/psl3_5/train/*.arrow
 - split: test
 path: data/psl3_5/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: m11
 data_files:
 - split: train
 path: data/m11/train/*.arrow
 - split: test
 path: data/m11/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
- config_name: m12
 data_files:
 - split: train
 path: data/m12/train/*.arrow
 - split: test
 path: data/m12/test/*.arrow
 features:
 - name: input_sequence
 dtype: string
 - name: target
 dtype: string
 - name: sequence_length
 dtype: int64
 - name: group_degree
 dtype: int64
 - name: group_order
 dtype: int64
 - name: group_type
 dtype: string
---
# Group Theory Collection
**A benchmark of 11.28 million permutation-composition problems over 94 finite groups, split by computational complexity class (TC⁰ vs NC¹).**
Given a sequence of group elements, the task is to predict their composition. By Barrington's theorem, iterated composition in any fixed **non-solvable** group is **NC¹-complete**, while for solvable groups it lies in **ACC⁰ ⊆ TC⁰**. Architectures whose expressive power is bounded by TC⁰ — fixed-depth transformers and linear state-space models under standard assumptions — are therefore predicted to fail at length on the 22 non-solvable groups while succeeding on the 72 solvable ones. This dataset operationalizes that prediction as a controlled benchmark: same task, same format, same length distribution, with the group's complexity class as the independent variable.
- **Generator, verifier, and docs:** [BeeGass/Group-Dataset-Generator](https://github.com/BeeGass/Group-Dataset-Generator)
- **Motivating theory:** Merrill, Petty & Sabharwal, [*The Illusion of State in State-Space Models*](https://arxiv.org/abs/2404.08819); Barrington, *Bounded-width polynomial-size branching programs recognize exactly those languages in NC¹*
| At a glance | |
|---|---|
| Task | composition (word problem): sequence of group elements → their product |
| Configs | 94 permutation groups across 10 families |
| Rows | 11,280,000 (per config: 100,000 train / 20,000 test) |
| Complexity split | 72 solvable (TC⁰) / 22 non-solvable (NC¹-complete) |
| Sequence lengths | uniform on [3, 1024], all lengths present in every split |
| Format | Apache Arrow (`datasets` save_to_disk layout) + one `metadata.json` per config |
| Size | ≈ 17 GB |
| Version | **v2** (2026-08, deterministic regeneration); v1 frozen at revision `878908af` |
| License | MIT |
## Quick start
```python
import json
import numpy as np
from datasets import load_dataset
from huggingface_hub import hf_hub_download
GROUP = "s5" # any of the 94 config names below
ds = load_dataset("BeeGass/Group-Theory-Collection", name=GROUP, split="test")
# Rows carry element INDICES. The index -> permutation table lives in the
# config's metadata.json (load_dataset does not fetch it for you):
meta = json.load(open(hf_hub_download(
 "BeeGass/Group-Theory-Collection", f"data/{GROUP}/metadata.json", repo_type="dataset"
)))
perms = {int(k): np.array(v) for k, v in meta["permutation_map"].items()}
# Recompute a target to see the convention in action:
row = ds[0]
result = np.arange(meta["group_degree"])
for i in row["input_sequence"].split():
 result = result[perms[int(i)]]
assert np.array_equal(result, perms[int(row["target"])])
# Filter configs by complexity class via metadata:
assert meta["complexity_class"] in ("tc0", "nc1")
```
Streaming, other access routes, and length filtering:
```python
# Stream without downloading the full split
ds = load_dataset("BeeGass/Group-Theory-Collection", name="m12", split="train", streaming=True)
# Equivalent path-based loading
ds = load_dataset("BeeGass/Group-Theory-Collection", data_dir="data/s5")
# Pin a revision for exact reproducibility:
# v2 data release: 2f336519055a1117ddb852a112d815b75a27e61e
# v1 (frozen): 878908af8299019d3fe0a2cc180af576e011f465
ds = load_dataset(
 "BeeGass/Group-Theory-Collection", name="s5",
 revision="2f336519055a1117ddb852a112d815b75a27e61e",
)
# Length curriculum / analysis
short = ds["train"].filter(lambda x: x["sequence_length"] <= 32)
```
## Worked examples
The dataset supports two consumption modes, and the generator repo's API covers both: **train from scratch** on the published index rows, or **benchmark a pretrained LLM** on rendered prompts whose answers score against these same targets. Both examples below were run as-is; the quoted numbers are their real output.
### Train from scratch: state tracking with dense supervision
The composition task is a running-state problem, so supervise the *running product at every prefix*, not just the final target — `permutation_map` lets you compute every prefix label locally. (With final-target supervision alone, small models tend to memorize rows instead of learning the group: an identical setup trained this way sat at chance on held-out rows.)
<details>
<summary><b>Full training script</b> (PyTorch; runs on CPU in ~10 minutes)</summary>
```python
import json
import numpy as np
import torch
import torch.nn as nn
from datasets import load_dataset
from huggingface_hub import hf_hub_download
GROUP, MAX_LEN = "c10", 64 # train short, evaluate longer
ds = load_dataset("BeeGass/Group-Theory-Collection", name=GROUP)
meta = json.load(open(hf_hub_download(
 "BeeGass/Group-Theory-Collection", f"data/{GROUP}/metadata.json", repo_type="dataset"
)))
order, degree = meta["group_order"], meta["group_degree"]
perms = {int(k): np.array(v) for k, v in meta["permutation_map"].items()}
index_of = {tuple(perms[i]): i for i in range(order)}
PAD, IGNORE = order, -100
def encode(row):
 """labels[t] = index of p_1 . ... . p_(t+1); labels[-1] == int(row["target"])."""
 ids = [int(t) for t in row["input_sequence"].split()]
 state, labels = np.arange(degree), []
 for i in ids:
 state = state[perms[i]] # same fold as the decode recipe above
 labels.append(index_of[tuple(state)])
 return {"ids": ids, "labels": labels}
train = ds["train"].filter(lambda r: r["sequence_length"] <= MAX_LEN).map(encode)
def collate(batch):
 width = max(len(r["ids"]) for r in batch)
 x = torch.full((len(batch), width), PAD, dtype=torch.long)
 y = torch.full((len(batch), width), IGNORE, dtype=torch.long)
 for i, r in enumerate(batch):
 x[i, : len(r["ids"])] = torch.tensor(r["ids"])
 y[i, : len(r["labels"])] = torch.tensor(r["labels"])
 return x, y
loader = torch.utils.data.DataLoader(train, batch_size=128, shuffle=True, collate_fn=collate)
class Composer(nn.Module):
 """Recurrent baseline: an RNN can carry the running product in its state.
 Swap this module for your transformer or SSM to probe the TC0/NC1
 boundary — fixed-depth parallel architectures are the ones predicted to
 fail at length where this recurrent baseline succeeds.
 """
 def __init__(self, order, d=128):
 super().__init__()
 self.emb = nn.Embedding(order + 1, d) # + PAD
 self.rnn = nn.LSTM(d, d, batch_first=True)
 self.head = nn.Linear(d, order)
 def forward(self, x):
 hidden, _ = self.rnn(self.emb(x))
 return self.head(hidden)
model = Composer(order)
opt = torch.optim.AdamW(model.parameters(), lr=1e-3)
for epoch in range(30):
 for x, y in loader:
 logits = model(x)
 loss = nn.functional.cross_entropy(logits.flatten(0, 1), y.flatten(), ignore_index=IGNORE)
 opt.zero_grad(); loss.backward(); opt.step()
@torch.no_grad()
def final_accuracy(rows):
 hits = total = 0
 for x, y in torch.utils.data.DataLoader(rows, batch_size=256, collate_fn=collate):
 last = (x != PAD).sum(1) - 1
 idx = torch.arange(len(x))
 pred = model(x).argmax(-1)[idx, last]
 hits += (pred == y[idx, last]).sum().item(); total += len(x)
 return hits / total
test = ds["test"].map(encode)
in_dist = test.filter(lambda r: r["sequence_length"] <= MAX_LEN)
longer = test.filter(lambda r: MAX_LEN < r["sequence_length"] <= 4 * MAX_LEN)
print(f"len<={MAX_LEN}: {final_accuracy(in_dist):.3f} | len {MAX_LEN+1}-{4*MAX_LEN}: {final_accuracy(longer):.3f}")
```
</details>
Measured output of that script, run verbatim (CPU, ~10 min; training loss falls from 2.30, chance level, to below 0.01):
```
len<=64: 1.000 | len 65-256: 0.999
```
The LSTM baseline learns the group and generalizes to 4× its training length, because a recurrent state can implement the group's multiplication directly. The experiment this benchmark exists for: replace `Composer` with a fixed-depth transformer or SSM, sweep `GROUP` across the TC⁰/NC¹ boundary (say `c10`, `d10`, `s5`, `a5`, `m11`), and compare accuracy-versus-length curves per class.
### Benchmark a pretrained LLM: rendered prompts via the generator API
The generator package renders these exact composition tasks as natural-language prompts (cycle or one-line notation), parses free-form completions back to element indices, and scores them against the same targets. Task files are self-describing JSONL (each carries its own element table), and `gdg verify` recomputes every target and answer before you spend model calls on a file.
```python
# pip install git+https://github.com/BeeGass/Group-Dataset-Generator
from gdg.bench import registry
from gdg.bench.generate import generate_tasks
from gdg.bench.render import get_renderer
spec = registry.get("s5")
manifest, tasks = generate_tasks(spec, lengths=(4, 16, 64), n=50, seed=0, renderer_name="cycle")
renderer = get_renderer(manifest.renderer)
correct = 0
for task in tasks:
 completion = my_llm(task.prompt) # your model call goes here
 predicted = renderer.parse_answer(completion, spec, manifest.elements)
 correct += predicted == task.target # None (unparseable) scores wrong
print(f"accuracy: {correct / len(tasks):.3f}")
```
A rendered `cycle` prompt looks like:
> Compute the product (0 4 3 2) . (0 1)(2 3) . (0 2)(1 3) . (0 3)(2 4) in the group S5, a permutation group on 5 points labelled 0 to 4. Compose so that the rightmost factor acts on a point first. Answer in cycle notation.
with reference answer `(0 4 1)(2 3)`. `parse_answer` accepts any valid spelling of the same permutation (cycle notation is not unique), returns an element index rather than a string, and returns `None` on junk instead of raising, so one garbled completion cannot abort a sweep. Sanity-checked as-is: an oracle that replays each task's reference answer scores 1.000; a model that answers "banana" scores 0.000.
The same harness works offline from files: `gdg generate --group s5 --lengths 4,16,64 --n 50 --renderer cycle --out s5.jsonl` writes the tasks (prompts, answers, and the element table) as JSONL, `gdg verify s5.jsonl` proves the file self-consistent, and three renderers ship (`cycle`, `inline` one-line arrays, `index` raw indices — the published corpus format).
## Dataset structure
### Data instances
One real row from `s5` (test split):
```json
{
 "input_sequence": "43 13 115",
 "target": "7",
 "sequence_length": 3,
 "group_degree": 5,
 "group_order": 120,
 "group_type": "symmetric"
}
```
Read: elements 43, 13, 115 of S₅ compose to element 7 (indices into `permutation_map`). Typical rows are much longer — lengths are uniform on [3, 1024].
### Data fields
| field | type | description |
|---|---|---|
| `input_sequence` | `string` | Space-separated element indices, in composition order (see convention below) |
| `target` | `string` | Index of the composed element. A string, not an int, because it keys `permutation_map`, whose JSON keys are strings |
| `sequence_length` | `int64` | Number of indices in `input_sequence`; uniform on [3, 1024] inclusive |
| `group_degree` | `int64` | Number of points the group acts on (constant per config) |
| `group_order` | `int64` | Number of elements in the group (constant per config) |
| `group_type` | `string` | Family name, e.g. `"symmetric"`, `"psl"`, `"mathieu"` (constant per config) |
Each config directory additionally ships a **`metadata.json`** with: `permutation_map` (index → permutation in one-line notation — the decoder ring for the row fields), `group_name`, `group_type`, `group_parameters`, `group_order`, `group_degree`, `solvable`, `complexity_class` (`"tc0"` or `"nc1"`), `num_train_samples`, `num_test_samples`, `min_seq_length`, `max_seq_length`, the composition convention in prose, the master `seed`, and the producing `gdg_version`.
### Data splits
Every one of the 94 configs has the same split sizes:
| split | rows per config | total rows |
|---|---:|---:|
| train | 100,000 | 9,400,000 |
| test | 20,000 | 1,880,000 |
Train and test are drawn from disjoint deterministic RNG streams (no leakage by construction). Rows are i.i.d. samples; there is no length stratification between splits, and both splits cover every length in [3, 1024].
### Repository layout
```
data/<config>/
 metadata.json # permutation_map + group facts + provenance
 dataset_dict.json
 train/ data-00000-of-00001.arrow # + dataset_info.json, state.json
 test/ data-00000-of-00001.arrow # + dataset_info.json, state.json
```
All 94 configs are declared in this card's YAML, so `load_dataset(..., name="<config>")` works out of the box. Class membership is machine-readable from each config's `complexity_class` metadata field and enumerated in the inventory tables below.
## Composition convention
For an input sequence [p₁, p₂, p₃] the target is:
- **Mathematical notation:** p₁ ∘ p₂ ∘ p₃
- **Operational reading:** result(x) = p₁(p₂(p₃(x))) — **the last element listed acts on the point first**
Permutations are stored in one-line notation, so `p[i]` is the image of `i`:
```python
result = np.arange(degree)
for i in input_sequence.split():
 result = result[permutation_map[i]]
```
**Worked example** in S₃ with a = [1,2,0] = (0 1 2) and b = [1,0,2] = (0 1):
| sequence | result | cycle notation |
|---|---|---|
| `[a, b]` | `[2,1,0]` | (0 2) |
| `[b, a]` | `[0,2,1]` | (1 2) |
These differ, so the example distinguishes this convention from its reverse. Checking the first: b(0) = 1, then a(1) = 2, giving result(0) = 2.
> **Correction.** Revisions of this card before 2026-08-07 described the opposite order, in both the composition formula and its operational gloss ("First apply p₁"). That was wrong. The data was always correct and is unchanged; only the description was wrong. Recomputing 300 test rows per group confirms the convention above matches 300/300 for s5, a5 and m11, while the old wording matched 3, 6 and 0 respectively.
## Group inventory
> **Correction.** Earlier revisions of this card headlined a 58 / 36 split. The correct split is 72 / 22, which is what the enumerated tables below have always shown.
### TC⁰ configs (solvable) — 72
| family | configs | orders | notes |
|---|---|---|---|
| Symmetric | S3, S4 | 6, 24 | solvable for n ≤ 4 |
| Alternating | A3, A4 | 3, 12 | solvable for n ≤ 4 |
| Cyclic | C2–C30 (all 29) | 2–30 | abelian |
| Dihedral | D3–D20 (all 18) | 6–40 | symmetries of regular polygons |
| Klein | V4 | 4 | ≅ Z₂², smallest non-cyclic abelian group |
| Quaternion | Q8, Q16, Q32 | 8, 16, 32 | non-abelian 2-groups; correct in v2 — in v1 all three were dihedral, see Erratum below |
| Elementary abelian | Z2^k (k≤5), Z3^k (k≤4), Z5^k (k≤4) | 2–625 | regular representations |
| Frobenius | F20, F21 | 20, 21 | C5⋊C4 and C7⋊C3, natural Frobenius actions |
| PSL | PSL(2,2), PSL(2,3) | 6, 12 | the two solvable PSLs (≅ S3, A4) |
### NC¹ configs (non-solvable) — 22
| family | configs | orders | notes |
|---|---|---|---|
| Symmetric | S5–S9 | 120–362,880 | non-solvable for n ≥ 5 |
| Alternating | A5–A9 | 60–181,440 | simple for n ≥ 5 |
| PSL | PSL(2,q) for q ∈ {4,5,7,8,9,11}; PSL(3,q) for q ∈ {2,3,4,5} | 60–372,000 | simple (PSL(2,4) ≅ A5, PSL(3,2) ≅ PSL(2,7)) |
| Mathieu | M11, M12 | 7,920, 95,040 | sporadic simple; sharply 4-/5-transitive |
Every solvability label was verified computationally (derived series for all 94 groups), and every group's isomorphism type was pinned by discriminating invariants during the 2026-08 audit — including the sharp transitivity of the Mathieu groups and the element-order spectra separating PSL(3,4) from A₈ and the quaternion groups from dihedral ones.
## Benchmarking guidance
- **The independent variable is `complexity_class`.** Hold the renderer, length distribution and training protocol fixed; compare accuracy-vs-length curves between TC⁰ and NC¹ configs.
- **Degree-matched comparisons.** Rendered prompt width scales with `group_degree`. Degrees 5–13 and 21 have both solvable and non-solvable configs (e.g. degree 5: C5, D5, F20, Z5 vs A5, PSL(2,4), S5), letting you hold width fixed across the class boundary.
- **Dedupe before averaging per class.** Several configs realise the same group (next section). In particular `psl3_2` ≡ `psl2_7` (identical tables) and `a5` ≡ `psl2_4` (same subgroup of Sym(5)) each appear twice within a single degree cohort.
- **Duplicates within tiny configs are expected.** A group of order 2 has only 2^L length-L words, so exact-duplicate rows in configs like `c2` are a birthday certainty at short lengths, not a data defect.
- **Length generalization.** All lengths 3–1024 are present in both splits, so you can train short / test long entirely within the published data.
## Isomorphic duplicates
Some configs name the same group twice. `psl3_2` and `psl2_7` are the same permutation group with the same element indexing (PSL(3,2) is constructed by delegation to PSL(2,7)); `a5` and `psl2_4` are the same subgroup of Sym(5) with different element indexings; `a3`/`c3`/`z3_1`, `a4`/`psl2_3`, `d3`/`s3`/`psl2_2`, `v4`/`z2_2`, `c2`/`z2_1` and `c5`/`z5_1` likewise coincide. Abstractly (with different actions), PSL(2,4) = PSL(2,5) = A5 and PSL(2,9) = A6, so the 22 non-solvable configs realise 18 distinct abstract groups. Distinct actions are distinct datapoints by design — prompt width and index space differ — but any per-class average or degree-matched comparison should dedupe these clusters rather than weight a group by how many names it carries. The full machine-readable map ships in the generator repo as `gdg.bench.registry.SAME_PERMUTATION_GROUP` and `ISOMORPHISM_CLASSES`.
## Dataset creation
Rows are generated, not collected. For each row, an RNG is derived from `(schema, split, config, row_index, master_seed)` via BLAKE2b → PCG64, the sequence length is drawn uniformly from [3, 1024], element indices are drawn uniformly from the group, and the target is computed by one shared composition routine. Every row is therefore independently regenerable, and the corpus is bit-reproducible from the generator at `seed = 0` (gdg 0.2.0).
Verification before release: all 94 element tables were regenerated and checked against v1 element-for-element (87 identical; exactly the 7 repaired configs changed); group axioms, solvability (derived series), and isomorphism-type invariants were verified computationally for every group; targets were recomputed from the shipped `permutation_map` for sampled rows of every family under the documented convention; sequence-length marginals were confirmed uniform on [3, 1024] in all 188 splits; and the generator's full test suite (924 tests, including Hub-backed validation of this artifact) passes.
## Leaderboard
Open-weight models evaluated on freshly generated composition tasks from this
dataset's groups (spanning both complexity classes), scored end to end by the
generator repository's runner: local llama.cpp inference, temperature 0, a
4,096-token reasoning budget, and identical tasks for every model (seed 0).
Full protocol, per-task receipts and submission instructions live in the
[generator repository](https://github.com/BeeGass/Group-Dataset-Generator)
under `docs/LEADERBOARD.md`.
| # | Model | Quant. | Score | TC0 | NC1 | L50 TC0 | L50 NC1 | Unparsed | Truncated |
|---|-------|--------|-------|-----|-----|---------|---------|----------|-----------|
| 1 | gpt-oss-20b | MXFP4 | 0.390 | 0.355 | 0.413 | 4 | 8 | 101 | 258 |
| 2 | Qwen3-4B-Instruct-2507 | Q8_0 | 0.318 | 0.340 | 0.303 | 4 | 4 | 88 | 324 |
| 3 | Qwen3.6-35B-A3B | UD-Q4_K_XL | 0.132 | 0.185 | 0.097 | 0 | 2 | 101 | 487 |
| 4 | Qwen3.6-27B | UD-Q4_K_XL | 0.080 | 0.120 | 0.053 | 0 | 0 | 102 | 500 |
| 5 | LFM2.5-2.6B | Q8_0 | 0.078 | 0.115 | 0.053 | 0 | 2 | 210 | 391 |
| 6 | Llama-3.1-8B-Instruct | Q6_K | 0.020 | 0.030 | 0.013 | 0 | 0 | 274 | 253 |
Score is the macro-average over 25 (group, sequence-length) cells; L50 is the
largest tested length still reaching 0.5 accuracy for that class; Truncated
counts completions that exhausted the token budget.
### Exact scores per model, group and sequence length
Accuracy over 20 tasks per cell.
**gpt-oss-20b** — score 0.390 (TC0 0.355, NC1 0.413); 258/500 truncated at the 4,096-token budget
| group (class) | len 2 | len 4 | len 8 | len 16 | len 32 |
|---|---|---|---|---|---|
| c10 (TC0) | 0.75 | 0.75 | 0.30 | 0.10 | 0.05 |
| d10 (TC0) | 0.80 | 0.50 | 0.25 | 0.00 | 0.05 |
| psl2_9 (NC1) | 0.90 | 0.75 | 0.30 | 0.00 | 0.00 |
| s5 (NC1) | 1.00 | 0.75 | 0.70 | 0.20 | 0.00 |
| m11 (NC1) | 0.85 | 0.45 | 0.30 | 0.00 | 0.00 |
**Qwen3-4B-Instruct-2507** — score 0.318 (TC0 0.340, NC1 0.303); 324/500 truncated at the 4,096-token budget
| group (class) | len 2 | len 4 | len 8 | len 16 | len 32 |
|---|---|---|---|---|---|
| c10 (TC0) | 0.90 | 0.60 | 0.10 | 0.10 | 0.00 |
| d10 (TC0) | 0.90 | 0.70 | 0.00 | 0.00 | 0.10 |
| psl2_9 (NC1) | 0.70 | 0.70 | 0.00 | 0.00 | 0.00 |
| s5 (NC1) | 1.00 | 0.95 | 0.15 | 0.10 | 0.00 |
| m11 (NC1) | 0.75 | 0.20 | 0.00 | 0.00 | 0.00 |
**Qwen3.6-35B-A3B** — score 0.132 (TC0 0.185, NC1 0.097); 487/500 truncated at the 4,096-token budget
| group (class) | len 2 | len 4 | len 8 | len 16 | len 32 |
|---|---|---|---|---|---|
| c10 (TC0) | 0.30 | 0.35 | 0.10 | 0.10 | 0.10 |
| d10 (TC0) | 0.45 | 0.25 | 0.05 | 0.00 | 0.15 |
| psl2_9 (NC1) | 0.15 | 0.00 | 0.00 | 0.00 | 0.00 |
| s5 (NC1) | 0.75 | 0.20 | 0.10 | 0.00 | 0.00 |
| m11 (NC1) | 0.25 | 0.00 | 0.00 | 0.00 | 0.00 |
**Qwen3.6-27B** — score 0.080 (TC0 0.120, NC1 0.053); 500/500 truncated at the 4,096-token budget
| group (class) | len 2 | len 4 | len 8 | len 16 | len 32 |
|---|---|---|---|---|---|
| c10 (TC0) | 0.30 | 0.20 | 0.05 | 0.10 | 0.15 |
| d10 (TC0) | 0.25 | 0.05 | 0.05 | 0.05 | 0.00 |
| psl2_9 (NC1) | 0.00 | 0.00 | 0.10 | 0.00 | 0.00 |
| s5 (NC1) | 0.25 | 0.10 | 0.10 | 0.05 | 0.00 |
| m11 (NC1) | 0.05 | 0.15 | 0.00 | 0.00 | 0.00 |
**LFM2.5-2.6B** — score 0.078 (TC0 0.115, NC1 0.053); 391/500 truncated at the 4,096-token budget
| group (class) | len 2 | len 4 | len 8 | len 16 | len 32 |
|---|---|---|---|---|---|
| c10 (TC0) | 0.25 | 0.15 | 0.10 | 0.10 | 0.10 |
| d10 (TC0) | 0.30 | 0.05 | 0.05 | 0.05 | 0.00 |
| psl2_9 (NC1) | 0.05 | 0.00 | 0.05 | 0.00 | 0.00 |
| s5 (NC1) | 0.50 | 0.10 | 0.00 | 0.00 | 0.00 |
| m11 (NC1) | 0.10 | 0.00 | 0.00 | 0.00 | 0.00 |
**Llama-3.1-8B-Instruct** — score 0.020 (TC0 0.030, NC1 0.013); 253/500 truncated at the 4,096-token budget
| group (class) | len 2 | len 4 | len 8 | len 16 | len 32 |
|---|---|---|---|---|---|
| c10 (TC0) | 0.10 | 0.05 | 0.00 | 0.05 | 0.10 |
| d10 (TC0) | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| psl2_9 (NC1) | 0.05 | 0.00 | 0.00 | 0.05 | 0.00 |
| s5 (NC1) | 0.00 | 0.00 | 0.00 | 0.10 | 0.00 |
| m11 (NC1) | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
## Versions
- **v2 (current).** All 94 configs regenerated with per-item deterministic sampling (`gdg export-hf`, master seed 0). Corrects the seven defective v1 configs (`psl2_4`, `psl2_8`, `psl2_9`, `psl3_4`, `q8`, `q16`, `q32`); the other 87 configs keep their v1 element indexing exactly, with freshly sampled rows. Adds `complexity_class` and provenance fields to `metadata.json`, and removes the duplicated `TC0/` and `NC1/` trees.
- **v1 (frozen).** The July 2025 upload, pinned at Hub revision `878908af`. Cite it by that revision; it is not bit-reproducible from source (its generator used a global RNG seed), and its seven defective configs are documented in the errata below.
> **Layout change in v2.** v1 shipped `TC0/` and `NC1/` directory trees that duplicated `data/` byte-for-byte (16.99 GB); v1's card marked them as slated for removal, and v2 removed them. Since their removal, `load_dataset(..., data_dir="TC0/...")` and `data_dir="NC1/..."` stop working; `load_dataset(name=...)` and `data_dir="data/..."` are unaffected. Use the `complexity_class` metadata field instead.
## Known issues and errata
### Erratum: four PSL configs were defective in v1
In v1 (Hub revision `878908af`), `psl2_4`, `psl2_8`, `psl2_9` and `psl3_4` did not contain the group their name claims; v2 rebuilds all four correctly. The v1 generator built these groups from elementary matrices with prime-field coefficients, which over GF(p^n) with n > 1 generate only a proper subgroup.
| config | should be | v1 actually contained |
|---|---|---|
| `psl2_4` | PSL(2,4), order 60 | a **solvable** group of order 10 (D5) |
| `psl2_8` | PSL(2,8), order 504 | a **solvable** group of order 18 (D9) |
| `psl2_9` | PSL(2,9), order 360 | A5, order 60 |
| `psl3_4` | PSL(3,4), order 20160 | PSL(2,7), order 168, acting intransitively |
**If you ran a TC⁰-versus-NC¹ comparison on v1, exclude these four and re-run on v2.** Two of them were solvable groups labelled non-solvable, and `psl2_4` acts on 5 points, so it falls in the degree-matched cohort against S5 and A5. The other two were non-solvable but duplicated `a5` and `psl2_7`, adding no new information.
All PSL configs with prime q — `psl2_2`, `psl2_3`, `psl2_5`, `psl2_7`, `psl2_11`, `psl3_2`, `psl3_3`, `psl3_5` — were never affected, and their element indexing is unchanged between v1 and v2.
### Erratum: three quaternion configs were dihedral in v1
In v1 (Hub revision `878908af`), `q8`, `q16` and `q32` did not contain generalized quaternion groups; v2 rebuilds all three correctly. The v1 generator dropped the b² = a^(2^(k-2)) relation — every a^i b element it built squared to the identity — so the enumerated group satisfied the dihedral presentation instead:
| config | should be | v1 actually contained |
|---|---|---|
| `q8` | Q8, order 8 — exactly one involution | D4, order 8 — five involutions, regular representation |
| `q16` | Q16, order 16 — exactly one involution | D8, order 16 — nine involutions, regular representation |
| `q32` | Q32, order 32 — exactly one involution | D16, order 32 — seventeen involutions, regular representation |
v1's orders, degrees and targets were internally consistent, and dihedral 2-groups are solvable, so the TC⁰ placement still held — but every "quaternion" conclusion drawn from v1 is really about dihedral groups, which in v1 duplicated `d4`, `d8` and `d16` abstractly. You can verify from the v1 data alone: count the elements p of `q8`'s v1 `permutation_map` with p[p] = identity — five involutions, where a generalized quaternion group has exactly one. The v2 tables do.
In total **87 of the 94** v1 configs were correct as labelled; v2 corrects all seven, so every v1 target for those seven is void, and the other 87 configs keep their v1 element indexing exactly.
## Limitations
- **Synthetic and exhaustive by construction.** Rows are i.i.d. uniform draws; there is no natural-language noise, distribution shift, or annotation ambiguity. This is by design (the benchmark isolates a single computational property) but means results do not speak to natural-data robustness.
- **Index-based format.** The published rows use element indices, suited to training from scratch. Prompting a pretrained LLM requires rendering elements concretely (e.g. cycle notation); the generator repo ships three reference renderers and a verifier so rendered prompts stay consistent with these targets.
- **Duplicate abstract groups.** See *Isomorphic duplicates*: naive per-class averages overweight A5, A6 and PSL(2,7).
- **Class labels are per group, not per row.** Short sequences over a non-solvable group are still easy; the NC¹-hardness prediction concerns length scaling, not individual rows.
## Citation
```bibtex
@dataset{gass2026grouptheorycollection,
 author = {Gass, Bryan},
 title = {Group Theory Collection: permutation composition over 94 finite
 groups, split by {TC}$^0$/{NC}$^1$ complexity},
 year = {2026},
 version = {2.0},
 publisher = {Hugging Face},
 url = {https://huggingface.co/datasets/BeeGass/Group-Theory-Collection}
}
@software{gass2026gdg,
 author = {Gass, Bryan},
 title = {Group Dataset Generator},
 year = {2026},
 url = {https://github.com/BeeGass/Group-Dataset-Generator}
}
@inproceedings{merrill2024illusion,
 title = {The Illusion of State in State-Space Models},
 author = {Merrill, William and Petty, Jackson and Sabharwal, Ashish},
 booktitle = {Proceedings of the 41st International Conference on Machine Learning},
 year = {2024},
 note = {arXiv:2404.08819}
}
@article{barrington1989bounded,
 title = {Bounded-width polynomial-size branching programs recognize exactly
 those languages in {NC}$^1$},
 author = {Barrington, David A.},
 journal = {Journal of Computer and System Sciences},
 volume = {38},
 number = {1},
 pages = {150--164},
 year = {1989}
}
```
## License
MIT.
## Contact
Questions, issues, or contributions: open a discussion on this dataset repository or an issue on the [generator repo](https://github.com/BeeGass/Group-Dataset-Generator).