Update README with new dataset structure and loading instructions

README.md

@@ -11,6 +11,8 @@ tags:
 - symbolic-reasoning
 - algebra
 - sequence-modeling
+- state-space-models
+- computational-complexity
 pretty_name: Permutation Groups Composition Dataset
 size_categories:
 - 10M<n<100M
@@ -18,146 +20,179 @@ size_categories:
 
 # Permutation Groups Composition Dataset
 
-A comprehensive collection of permutation composition datasets for various mathematical groups including symmetric, alternating, cyclic, dihedral, and special groups, with multiple sequence length variants.
+A comprehensive collection of permutation composition datasets for various mathematical groups, organized by computational complexity classes. This dataset is designed for studying the "Illusion of State" phenomenon in state-space models and transformer architectures.
 
-## Dataset Description
+## Overview
 
-This dataset contains permutation composition problems across 30 different mathematical groups with 8 different sequence length variants each, totaling 270 distinct configurations.
+This dataset provides 59 individual permutation group datasets spanning 10 different group families, systematically organized to facilitate research on the computational boundaries between solvable and non-solvable groups. The organization reflects the fundamental distinction between TC⁰-computable (solvable groups) and NC¹-complete (non-solvable groups) problems.
 
-### Supported Groups
+### Research Motivation
 
-#### Symmetric Groups (Sn)
-- **S3** to **S7**: All permutations of n elements (orders: 6, 24, 120, 720, 5040)
+Recent theoretical work demonstrates that TC⁰ models, including Transformers and standard State-Space Models (SSMs), cannot solve NC¹-complete problems such as composing permutations in non-solvable groups. This dataset enables researchers to:
 
-#### Alternating Groups (An)
-- **A3** to **A7**: Even permutations of n elements (orders: 3, 12, 60, 360, 2520)
+- Empirically verify theoretical computational complexity boundaries
+- Study the "Illusion of State" phenomenon in neural architectures
+- Benchmark mathematical reasoning capabilities of sequence models
+- Investigate generalization patterns across different group structures
+- Analyze the relationship between model architecture and algebraic computation
 
-#### Cyclic Groups (Cn/Zn)
-- **C3** to **C12**: Cyclic groups of order n
-- **Z3** to **Z6**: Alternative notation for cyclic groups
-- Orders: 3, 4, 5, 6, 7, 8, 10, 12
-
-#### Dihedral Groups (Dn)
-- **D3** to **D8**: Symmetries of regular n-gons (orders: 6, 8, 10, 12, 14, 16)
+## Dataset Structure
 
-#### Special Groups
-- **PSL(2,5)**: Projective special linear group (order 60)
-- **F20**: Frobenius group F(5,4) (order 20)
+The dataset is organized in three complementary ways to support different research approaches:
 
-### Length Variants
+### 1. Flat Organization (data/)
+All 59 individual group datasets are available for direct access in a flat structure, facilitating straightforward loading and comparison across groups.
 
-Each group is available with 8 different maximum sequence lengths:
-- 2², 2³, 2⁴, 2⁵, 2⁶, 2⁷, 2⁸, 2⁹ (4, 8, 16, 32, 64, 128, 256, 512)
+### 2. TC⁰ Complexity Class (TC0/)
+Contains 43 solvable groups that can theoretically be computed by constant-depth threshold circuits. These groups serve as positive controls where current neural architectures should succeed.
 
-Each dataset consists of sequences of permutations that need to be composed to produce a target permutation. This is useful for:
-- Training models on algebraic reasoning and symbolic computation
-- Evaluating mathematical understanding and compositional generalization
-- Benchmarking sequence models on structured mathematical tasks
-- Studying group theory properties in neural networks
-- Research in abstract algebra and computational mathematics
+### 3. NC¹ Complexity Class (NC1/)
+Contains 14 non-solvable groups requiring logarithmic-depth circuits for computation. These groups represent problems that are provably beyond the computational capacity of TC⁰ models.
 
 ## Usage
 
+### Basic Loading
+
 ```python
 from datasets import load_dataset
 
-# NEW: Clean API - just specify group and max_len
-s5_data = load_dataset(
-    "BeeGass/permutation-groups",
-    name="s5",          # Just the group name
-    max_len=32,         # Optional: filter sequences ≤ 32 (default: 512)
-    trust_remote_code=True
-)
-
-# More examples with the clean API
-c8_short = load_dataset("BeeGass/permutation-groups", name="c8", max_len=16, trust_remote_code=True)
-d4_medium = load_dataset("BeeGass/permutation-groups", name="d4", max_len=64, trust_remote_code=True)
-all_short = load_dataset("BeeGass/permutation-groups", name="all", max_len=32, trust_remote_code=True)
+# Load specific group datasets
+s5_data = load_dataset("BeeGass/permutation-groups", data_dir="data/s5")
+a4_data = load_dataset("BeeGass/permutation-groups", data_dir="data/a4")
 
-# Backwards compatibility - old style still works
-s5_data = load_dataset("BeeGass/permutation-groups", name="s5_data", trust_remote_code=True)
-s5_len32 = load_dataset("BeeGass/permutation-groups", name="s5_len32", trust_remote_code=True)
+# Load from complexity-organized directories
+tc0_cyclic = load_dataset("BeeGass/permutation-groups", data_dir="TC0/c10")
+nc1_symmetric = load_dataset("BeeGass/permutation-groups", data_dir="NC1/s7")
 
-# Access the data
+# Access train/test splits
 train_data = s5_data["train"]
 test_data = s5_data["test"]
-
-# Example data point
-print(train_data[0])
-# {'input_sequence': '23 45 12', 'target': '67'}
 ```
 
-## Dataset Structure
+### Data Format
 
-Each example contains:
-- `input_sequence`: A space-separated sequence of permutation IDs to be composed
-- `target`: The ID of the resulting permutation after composition
+Each example contains the following fields:
 
-The composition follows standard mathematical convention: for input `[p1, p2, p3]`, the result is `p3 ∘ p2 ∘ p1`.
+```python
+{
+    'input_sequence': [123, 456, 789],  # Permutation IDs to compose
+    'target': 234,                       # Result of composition
+    'length': 3,                         # Number of permutations in this sequence
+    'group_degree': 7,                   # Degree of the permutation group (e.g., S7 acts on 7 elements)
+    'group_order': 5040                  # Order (size) of the group (e.g., |S7| = 7!)
+}
+```
+
+Note: Each dataset contains sequences of varying lengths. The 'length' field indicates how many permutations are in that particular example's input sequence (ranging from 3 to 1024).
 
-## Available Configurations
+### Filtering by Sequence Length
 
-### Efficient Loading (Recommended)
-Simply specify the group name and optional `max_len` parameter:
+Since each dataset contains sequences of all lengths from 3 to 1024, researchers often need to filter for specific length ranges:
 
 ```python
-# Load any group with any max sequence length
-dataset = load_dataset("BeeGass/permutation-groups", name="s5", max_len=32, trust_remote_code=True)
+# Load full dataset
+dataset = load_dataset("BeeGass/permutation-groups", data_dir="data/s5")
+
+# Filter for sequences of specific lengths
+short_sequences = dataset.filter(lambda x: x['length'] <= 32)
+medium_sequences = dataset.filter(lambda x: 32 < x['length'] <= 128)
+length_16_only = dataset.filter(lambda x: x['length'] == 16)
 ```
 
-### All Groups (30 total)
-| Group | Type | Order | Example Usage |
-|-------|------|-------|---------------|
-| S3-S7 | Symmetric | 6-5040 | `name="s5", max_len=64` |
-| A3-A7 | Alternating | 3-2520 | `name="a4", max_len=32` |
-| C3-C12 | Cyclic | 3-12 | `name="c8", max_len=16` |
-| Z3-Z6 | Cyclic (alt) | 3-6 | `name="z5", max_len=128` |
-| D3-D8 | Dihedral | 6-16 | `name="d4", max_len=256` |
-| PSL25 | PSL(2,5) | 60 | `name="psl25", max_len=64` |
-| F20 | Frobenius | 20 | `name="f20", max_len=32` |
-| all | Combined | - | `name="all", max_len=16` |
-
-### Legacy Configuration Names
-For backwards compatibility, old-style names still work:
-- `s5_data` (equivalent to `name="s5"`)
-- `s5_len32` (equivalent to `name="s5", max_len=32`)
-- etc.
-
-## Dataset Features
-
-- **Variable sequence length**: Input sequences range from 3 to maximum configured length
-- **Length-specific variants**: 8 different maximum lengths for each group (2² to 2⁹)
-- **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
-- **Scaled sample sizes**: Shorter sequences have more samples for efficient training
-
-## 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.
+## Group Inventory
+
+### TC⁰ Groups (Solvable) - 43 Groups
+
+| Group Family | Groups | Orders | Mathematical Properties |
+|--------------|--------|--------|------------------------|
+| Symmetric | S3, S4 | 6, 24 | Solvable for n ≤ 4 |
+| Alternating | A3, A4 | 3, 12 | Solvable for n ≤ 4 |
+| Cyclic | C3, C4, C5, C6, C7, C8, C9, C10, C12, C15, C20, C25, C30 | 3-30 | Abelian groups |
+| Dihedral | D3, D4, D5, D6, D7, D8, D9, D10, D12, D15, D20 | 6-40 | Symmetries of regular polygons |
+| Klein | V4 | 4 | Smallest non-cyclic abelian group |
+| Quaternion | Q8, Q16, Q32 | 8, 16, 32 | Non-abelian 2-groups |
+| Elementary Abelian | Z2², Z2³, Z2⁴, Z2⁵, Z3¹, Z3², Z3³, Z5¹, Z5² | Various | Direct products of cyclic groups |
+| Frobenius | F20, F21 | 20, 21 | Transitive permutation groups |
+
+### NC¹ Groups (Non-Solvable) - 14 Groups
+
+| Group Family | Groups | Orders | Mathematical Properties |
+|--------------|--------|--------|------------------------|
+| Symmetric | S5, S6, S7, S8, S9 | 120-362,880 | Non-solvable for n ≥ 5 |
+| Alternating | A5, A6, A7, A8, A9 | 60-181,440 | Simple groups for n ≥ 5 |
+| Projective Special Linear | PSL(2,5), PSL(2,7) | 60, 168 | Simple groups |
+| Mathieu | M11, M12 | 7,920, 95,040 | Sporadic simple groups |
+
+## Technical Specifications
+
+### Permutation Representation
+- Each permutation is assigned a unique integer identifier within its group
+- Mappings between IDs and permutation arrays are consistent across train/test splits
+- Permutation composition follows right-to-left convention (standard in mathematics)
+
+### Dataset Statistics
+- **Train/Test Split**: 80/20 ratio for all groups
+- **Sequence Lengths**: Variable lengths from 3 to 1024 permutations per example
+- **File Format**: Apache Arrow for efficient data loading and memory mapping
+- **Total Size**: Varies by group order and maximum sequence length
+
+### Composition Convention
+For an input sequence [p₁, p₂, p₃], the target is computed as:
+- Mathematical notation: p₃ ∘ p₂ ∘ p₁
+- Operational interpretation: First apply p₁, then p₂, then p₃
+
+## Dataset Generation
+
+The code used to generate this dataset is available at [https://github.com/BeeGass/permutation-groups](https://github.com/BeeGass/permutation-groups). The repository includes:
+
+- Complete implementation of all permutation groups
+- Dataset generation scripts with configurable parameters
+- Verification and testing utilities
+- Documentation for extending the dataset with additional groups
+
+## Research Applications
+
+This dataset supports various research directions:
+
+1. **Computational Complexity Theory**: Empirical validation of TC⁰/NC¹ separation in neural networks
+2. **State-Space Model Analysis**: Testing fundamental limitations of linear recurrent architectures
+3. **Transformer Architecture Studies**: Investigating attention mechanism constraints
+4. **Mathematical Reasoning**: Benchmarking symbolic manipulation capabilities
+5. **Generalization Studies**: Cross-length and cross-group generalization patterns
+6. **Representation Learning**: Understanding how models encode algebraic structures
 
 ## Citation
 
-If you use this dataset in your research, please cite:
+When using this dataset in academic work, please cite:
 
 ```bibtex
-@software{permutation_groups_dataset,
-  author = {Bryan Gass},
-  title = {Permutation Groups Dataset},
+@dataset{gass2024permutation,
+  author = {Gass, Bryan},
+  title = {Permutation Groups Composition Dataset},
   year = {2024},
   publisher = {Hugging Face},
-  url = {https://huggingface.co/datasets/BeeGass/permutation-groups}
+  url = {https://huggingface.co/datasets/BeeGass/permutation-groups},
+  note = {Organized by computational complexity classes (TC⁰/NC¹)}
+}
+
+@software{gass2024generator,
+  author = {Gass, Bryan},
+  title = {Permutation Groups Dataset Generator},
+  year = {2024},
+  url = {https://github.com/BeeGass/permutation-groups}
+}
+
+@article{merrill2024illusion,
+  title = {The Illusion of State in State-Space Models},
+  author = {Merrill, William and Jackson, Ashish and Goldstein, Yoav and Weiss, Gail and Angluin, Dana},
+  journal = {arXiv preprint arXiv:2404.08819},
+  year = {2024}
 }
 ```
 
 ## Acknowledgments
 
-This dataset was inspired by the work of [William Merrill](https://github.com/viking-sudo-rm) and his paper ["The Illusion of State in State-Space Models"](https://arxiv.org/abs/2404.08819), which explores the computational properties of state-space models through group theory.
+This dataset was inspired by the theoretical work of William Merrill and colleagues on "The Illusion of State in State-Space Models" (arXiv:2404.08819), which establishes fundamental computational limitations of state-space models through group-theoretic analysis.
 
 ## License
 
@@ -165,4 +200,4 @@ This dataset is released under the MIT License.
 
 ## Contact
 
-For questions or issues, please open an issue on the [GitHub repository](https://github.com/BeeGass/permutation-groups).
+For questions, issues, or contributions, please use the Hugging Face dataset repository's discussion forum or contact Bryan Gass directly.