mhieuuu/ac-solver-dataset

AC-Solver Transformer Dataset

Training corpus for a decoder-only Transformer language model on the Andrews-Curtis (AC) conjecture — one of the longest-standing open problems in combinatorial group theory, and a challenging benchmark for reinforcement learning.

Paper: What Makes Math Problems Hard for Reinforcement Learning: A Case Study Code: github.com/shehper/AC-Solver

---

---

Background: The Andrews-Curtis Conjecture

A balanced presentation of a group is a description of the form

⟨ x, y | r₀, r₁ ⟩

where x and y are generators (and their inverses x⁻¹, y⁻¹), and r₀, r₁ are relator words — strings over {x, x⁻¹, y, y⁻¹}. When the relators completely determine that x = y = 1 (the identity), the presentation describes the trivial group, and the simplest such presentation is ⟨ x, y | x, y ⟩.

The Andrews-Curtis (AC) conjecture (1965) asserts:

Every balanced presentation of the trivial group can be reduced to ⟨ x, y | x, y ⟩ by a finite sequence of AC moves.

The AC moves are:

1. Replace rᵢ with rᵢ · rⱼ (concatenate relator j onto relator i)
2. Replace rᵢ with w · rᵢ · w⁻¹ for any word w (conjugate a relator)
3. Replace rᵢ with rᵢ⁻¹ (invert a relator)

Despite being 60 years old, the conjecture remains open. The difficulty is that some presentations may require solution paths of millions or billions of steps — far beyond current search algorithms.

In code, a presentation ⟨ x, y | r₀, r₁ ⟩ is represented as a pair of integer arrays using the encoding x=1, x⁻¹=-1, y=2, y⁻¹=-2, with zeros as right-padding:

# ⟨ x, y | x²y⁻³, xy x⁻¹y⁻¹x⁻¹y⁻¹ ⟩
r0 = [1, 1, -2, -2, -2, 0, 0, ...]   # x x y⁻¹ y⁻¹ y⁻¹
r1 = [1, 2, 1, -2, -1, -2, 0, ...]   # x y x⁻¹ y⁻¹ x⁻¹ y⁻¹
presentation = r0 + r1                # length 2 × lmax

---

The Miller-Schupp Series

The Miller-Schupp series is a parametric family of balanced trivial-group presentations studied extensively as potential hard instances of the AC conjecture:

MS(n, w) = ⟨ x, y | xⁿ y⁻¹ x⁻¹ y x y⁻ⁿ y⁻¹, w x w⁻¹ y ⟩

This dataset is built from the 1,190 Miller-Schupp seed presentations used in the paper, spanning a range of parameters that produce presentations of varying difficulty. They are stored in ac_solver/search/miller_schupp/data/all_presentations.txt.

Of these 1,190 seeds:

• 533 are GS-solved: reachable from the trivial state by greedy search (a polynomial-time heuristic). These are "easy" in the sense that short solution paths exist.
• 657 are GS-unsolved: not solved by greedy search within the allowed budget. Some may be provably AC-trivial via longer paths; others are potential counterexamples to the conjecture.

---

Why This Dataset Exists

The Transformer is trained as a language model over presentation sequences to learn the statistical structure of the AC graph neighbourhood. Its learned representations serve two downstream purposes:

1. Hardness Oracle H(P): The EOS-token embedding of a presentation is fed to a small MLP classifier to predict whether the presentation is GS-solved or GS-unsolved (F1 ≈ 0.962 target). This oracle scores the difficulty of any presentation encountered during RL training.

2. Enriched State Representation: The 512-dimensional Transformer embedding can be concatenated to the raw RL state, giving the PPO policy access to long-range structural information beyond what the raw int8 array conveys.

---

Generation Process

Presentations are generated via Algorithm 6 from the paper: a stratified Markov-Chain Monte Carlo (MCMC) walk in presentation space.

for each of the 1,190 seed presentations p:
    for phase in 0 … 127:                   # gradually increase lmax
        lmax_phase = l_min + phase * step   # presentation space grows
        for chain in 0 … 11:               # 12 independent chains
            start from p
            apply 1,000 random AC′ moves   # free+cyclic reduction after each
            record the resulting presentation

Key design choices:

• Phased lmax: Early phases restrict relator length, producing presentations near the seed. Later phases allow longer relators, exploring harder regions.
• 12 independent chains: Reduces correlation between samples from the same seed.
• AC′ moves (the 12 AC move variants used in training) include concatenation and conjugation, each with all four generator choices — but exclude inversion (which doesn't change relator length and adds little signal).
• Free + cyclic reduction after every move ensures canonical form.

This produces a balanced corpus: every seed contributes exactly 128 × 12 = 1,536 presentations, totalling 1,190 × 1,536 = 1,827,840 rows.

---

Schema

Column	Dtype	Shape	Description
presentation	int8	(256,)	Two relators concatenated and zero-padded to lmax=128 each.
pres_idx	int32	scalar	Index 0–1189 into the 1,190 Miller-Schupp seeds.
phase	int32	scalar	MCMC phase index 0–127.
chain	int32	scalar	Markov chain index 0–11.
gs_solved	bool	scalar	True if the seed (pres_idx) is GS-solved.

---

Encoding Details

PPO / RL encoding (used in this dataset)

Symbol	int8 value
x	1
x⁻¹	-1
y	2
y⁻¹	-2
padding	0

presentation[0:128] = relator r0, right-padded with zeros to length 128. presentation[128:256] = relator r1, right-padded with zeros to length 128.

Transformer token encoding (for language model training)

The companion tokenizer (ac_solver/transformer/tokenizer.py) converts the PPO encoding to a 6-token vocabulary for the Transformer:

Token ID	Symbol	PPO value
0	x	1
1	x⁻¹	-1
2	y	2
3	y⁻¹	-2
4	SEP	(separator between r₀ and r₁)
5	EOS	(end of sequence)

A tokenized presentation looks like:

[r0_tokens...] SEP [r1_tokens...] EOS

Zeros (padding) are stripped before tokenization. The EOS token's hidden state is used as the presentation's embedding vector.

---

Dataset Statistics

Statistic	Value
Total presentations	1,827,840
Seed presentations (Miller-Schupp)	1,190
GS-solved seeds	533 (44.8%)
GS-unsolved seeds	657 (55.2%)
Presentations per seed	1,536 (128 phases × 12 chains)
Max relator length (lmax)	128
Presentation array length	256 (2 × 128)
AC′ moves per chain	1,000
Random seed	42
Generation time	~3.5 hours (CPU)
Raw shard size	~38 MB/shard × 12 shards

---

Usage

Basic loading

from datasets import load_dataset
import numpy as np

ds = load_dataset("mhieuuu/ac-solver-dataset", split="train")
print(ds)
# Dataset({features: ['presentation', 'pres_idx', 'phase', 'chain', 'gs_solved'], num_rows: 1827840})

Decode a single presentation

row = ds[42]

presentation = np.array(row["presentation"], dtype=np.int8)  # (256,)
r0 = presentation[:128]
r1 = presentation[128:]

# Strip padding (zeros) to get actual words
r0_word = r0[r0 != 0].tolist()
r1_word = r1[r1 != 0].tolist()

print(f"Seed index : {row['pres_idx']}")
print(f"Phase      : {row['phase']}")
print(f"Chain      : {row['chain']}")
print(f"GS-solved  : {row['gs_solved']}")
print(f"r0 word    : {r0_word}")   # e.g. [1, 1, -2, -2, -2]
print(f"r1 word    : {r1_word}")   # e.g. [1, 2, 1, -2, -1, -2]

Filter by difficulty label

# Only presentations from GS-solved seeds
easy = ds.filter(lambda row: row["gs_solved"])

# Only presentations from GS-unsolved seeds
hard = ds.filter(lambda row: not row["gs_solved"])

print(f"Easy: {len(easy):,}  Hard: {len(hard):,}")
# Easy: 818,688  Hard: 1,009,152

Group by seed and iterate over chains

import pandas as pd

df = ds.to_pandas()

# All 1536 presentations derived from seed 0
seed0 = df[df["pres_idx"] == 0].sort_values(["phase", "chain"])
print(seed0.shape)  # (1536, 5)

Streaming (memory-efficient for large jobs)

from datasets import load_dataset

ds = load_dataset("mhieuuu/ac-solver-dataset", split="train", streaming=True)

for row in ds.take(3):
    pres = row["presentation"]
    print(f"pres_idx={row['pres_idx']}  gs_solved={row['gs_solved']}  "
          f"r0_len={sum(1 for v in pres[:128] if v != 0)}")

Use with the AC-Solver Transformer

import torch
import numpy as np
from ac_solver.transformer.model import ACTransformer
from ac_solver.transformer.tokenizer import presentation_to_tokens

# Load model (after training)
model = ACTransformer()
model.load_state_dict(torch.load("ac_solver/transformer/checkpoints/lm_checkpoint.pt")["model"])
model.eval()

# Get 512-dim embedding for a presentation from this dataset
from datasets import load_dataset
ds = load_dataset("mhieuuu/ac-solver-dataset", split="train", streaming=True)
row = next(iter(ds))

pres = np.array(row["presentation"], dtype=np.int8)
tokens = presentation_to_tokens(pres, max_relator_length=128)
token_ids = torch.tensor(tokens, dtype=torch.long).unsqueeze(0)  # (1, T)

with torch.no_grad():
    hidden = model.get_hidden_states(token_ids)   # (1, T, 512)
    embedding = hidden[0, -1, :]                  # EOS token → (512,)

print(embedding.shape)  # torch.Size([512])

---

Intended Use & Limitations

Intended uses:

• Training the ACTransformer language model as a prior over AC presentation space
• Learning a hardness oracle H(P) to predict GS-solved / GS-unsolved
• Providing enriched state representations for PPO-based RL agents
• Benchmarking representation learning on combinatorial group theory objects

Limitations:

• All presentations are generated from the Miller-Schupp family only; the distribution may not generalise to other families of balanced presentations.
• The gs_solved label reflects greedy search reachability, not AC-triviality. A gs_solved=False seed may still be AC-trivial via a longer path.
• The dataset does not include solution paths — only the presentations themselves.
• Presentations from the same pres_idx are correlated (same seed, nearby MCMC walks). Stratify splits by pres_idx to avoid leakage.

---

Related Resources

Resource	Link
Paper (arXiv)	arxiv.org/abs/2408.15332
Code (GitHub)	github.com/shehper/AC-Solver
W&B Training Runs	wandb.ai/shehper/AC-Solver-PPO
What is AC? (primer)	What_is_AC.md

---

Citation

If you use this dataset, please cite the accompanying paper:

@misc{shehper2024makesmathproblemshard,
      title={What makes math problems hard for reinforcement learning: a case study},
      author={Ali Shehper and Anibal M. Medina-Mardones and Bartłomiej Lewandowski
              and Angus Gruen and Piotr Kucharski and Sergei Gukov},
      year={2024},
      eprint={2408.15332},
      archivePrefix={arXiv},
      primaryClass={cs.LG},
      url={https://arxiv.org/abs/2408.15332},
}