anand-bala/automata-embeddings

Automata Embeddings Dataset

Datasets for training and evaluating learned embeddings of AFA (Alternating Finite Automaton) residual languages derived from LTLf (Linear Temporal Logic over finite traces) task specifications.

Each dataset is produced by:

1. Expressing a task specification as an LTLf formula.
2. Compiling it to an AFA.
3. Exhaustively enumerating all reachable residual polynomials via BFS over Brzozowski derivatives.
4. Recording all step transitions, multi-step transition sequences, acceptance labels, and non-equivalence (bisimulation) witness pairs.

All files use the bz-2.0 HDF5 schema. See craftax/README.md for the full schema definition.

Configurations

Config	Files	Tasks	Status
craftax	dataset_v2.h5	67	Current
xland	dataset_d1.h5, dataset_d2.h5, dataset_d4.h5, dataset_d6.h5	27,218	Current

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craftax

Exhaustive Brzozowski derivative enumeration for all 67 achievement tasks in Craftax. Each achievement's prerequisites are expressed as an LTLf formula derived from the Craftax tech-tree dependency graph, compiled to an AFA, and symbolically differentiated to discover all reachable residual polynomials.

Dataset Summary

Quantity	Total
Tasks	67
Reachable residuals	6,966
Step transitions	6,651,472
Multi-step transitions	474,864
Non-equivalence pairs	233,888
File size	48 MB

Per-Task Ranges

Quantity	Min	Max	Median
AFA states	1	20	6
Atomic propositions	1	19	5
Alphabet size	2	524,288	32
Reachable residuals	2	512	17
Step transitions	4	1,048,576	1,056
Multi-step transitions	24	10,000	10,000
Non-equivalence pairs	0	32,896	136

Per-residual polynomials are always rank 1; degree ranges 1--12 (median 10).

Splits

Split	Tasks
Train	47
Validation	10
Test	10

Validation and test splits each contain at least one task with non-equivalence pairs so that the bisimulation loss is measurable during evaluation.

Supported Tasks

• Derivative (step) consistency: single-symbol transitions between residuals.
• Compositionality (multi-step): multi-symbol transition sequences.
• Acceptance: whether a residual accepts the empty word.
• Bisimulation (non-equivalence): pairs of residuals distinguished by a witness word.

See craftax/README.md for the full dataset card, including the HDF5 schema and loading code.

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xland

Brzozowski derivative enumeration for procedurally generated task specifications drawn from XLand-MiniGrid. Tasks are distributed across four files with increasing LTLf formula depth (d=1,2,4,6), producing datasets of very different structural complexity.

Generation

Tasks are generated by scripts/generate_xland_dataset.sh. Each task is a procedurally sampled LTLf formula parameterised by depth d; higher depth produces formulas with more nested temporal operators, larger AFA state spaces, and higher CP ranks in the residual polynomials.

Files

File	Depth	Tasks	Role
dataset_d1.h5	d=1	25,000	Primary training backbone; no zero-neq tasks
dataset_d2.h5	d=2	1,900	Higher-rank, higher-AP extension
dataset_d4.h5	d=4	257	OOD evaluation only
dataset_d6.h5	d=6	61	OOD evaluation only

Multi-step transitions are capped at 10,000 per task; residuals and step transitions are exhaustively enumerated.

Aggregated Statistics (all files)

Quantity	Total
Tasks	27,218
Reachable residuals	137,222
Step transitions	~2,700,000
Multi-step transitions	~69,900,000
Non-equivalence pairs	~966,000

Per-File Ranges

Quantity	d1 (25k tasks)	d2 (1.9k tasks)	d4 (257 tasks)	d6 (61 tasks)
AFA states (max)	5	7	5	5
Atomic propositions (max)	5	7	5	5
Alphabet size (max)	32	128	32	32
Reachable residuals (max / median)	48 / 4	512 / 2	34 / 2	34 / 2
Step transitions (max)	1,536	62,317	1,088	1,088
Non-equivalence pairs (max)	1,122	10,000	558	558
Zero-neq tasks	0	~206	~69	~18

Per-residual CP rank ranges 1--10 (median 2, P99 = 10); degree ranges 1--7. Unlike Craftax (always rank 1), xland exercises the full rank spectrum, making it the primary dataset for testing rank generalisation.

Task-Complexity Bins

Bin	Residual count	Tasks
Tiny	<= 4	23,047
Small	5--32	3,231
Medium	33--128	936
Large	> 128	4

The Tiny bin dominates (85% of tasks). Metrics should always be stratified by complexity bin to avoid hiding failures on the Medium and Large tails.

Recommended Training Corpus

• Train pool: d1 + d2 filtered to tasks with at least one non-equivalence pair (num_neq > 0).
• OOD evaluation slices: d2 zero-neq tasks, all of d4, all of d6, and the named outlier xland_000411 (512 residuals, 62,317 step transitions -- the hardest task in the corpus).

Rationale: d1 is a clean 25k-task backbone. d2_nonzero adds higher-rank, larger-alphabet tasks. d4 and d6 serve as distribution-shift probes. Zero-neq tasks contribute consistency and acceptance rows but no negative supervision, diluting bisimulation training.

Key Differences from Craftax

Property	Craftax	Xland
Tasks	67	27,218
Max AFA states	20	7
Max atomic propositions	19	7
Max alphabet size	524,288	128
CP rank range	1 (always)	1--10
Polynomial degree range	1--12	1--7
Zero-neq tasks	7 of 67	293 of 27,218

Craftax has larger, sparser automata with vast alphabets; xland has many more tasks, smaller automata, and the rank heterogeneity that Craftax cannot test. The two datasets are not directly comparable and should be treated as separate experiment families.

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Citation

@software{automata_embeddings,
  author = {Anand Balakrishnan},
  title  = {Automata Embeddings via Homomorphisms},
  url    = {https://github.com/anand-bala/automata-embeddings},
  year   = {2026}
}