nanoGentzen-v2 Synthetic Deduction Dataset (400k Transitions)

The nanoGentzen-v2 Dataset is a formal synthetic corpus designed to train Policy-Value Transformers for automated theorem proving in Intuitionistic Propositional Logic (LI) using Gentzen Sequent Calculus.

Scaled to 400,000 certified state-action derivation transitions, each record represents a discrete backward proof step along an AND-OR search tree. The dataset provides balanced multi-task supervision for inference rule selection, antecedent premise targeting, and branch provability value estimation.

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Code & Generation Pipeline: GitHub - nanoGentzen

Model Checkpoint: Hugging Face - nanoGentzen-v2

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Dataset Balance & Key Statistics

The dataset is constructed with a balanced 50/50 binary provability distribution to eliminate value-head bias during proof search:

Provability Target (target_value)	Class Meaning	Transition Count	Percentage
1.0 (True)	Constructively Provable Subgoal in LI	200,004	50.001%
0.0 (False)	Refuted / Unprovable Counter-Model	199,996	49.999%
Total	Full Derivation Corpus	400,000	100.0%

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File Formats & Artifacts

File	Format	Scale / Size	Description
gentzen_dataset.pt	PyTorch Binary	400,000 rows (~410 MB)	Tensorized training tensors (input_ids, target_rule, target_pivot, target_value).
gentzen_dataset.jsonl	JSON Lines	400,000 transitions (~95 MB)	Complete derivation trace records with AST sequents and token sequences.

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Data Schema & Field Definitions

Each record in gentzen_dataset.jsonl contains structured metadata for backward Gentzen proof step supervision:

{
  "sample_id": 10482,
  "sequent": "(P => Q), (Q => R) ⟶ (P => R)",
  "rule": "R_IMP",
  "rule_idx": 1,
  "pivot": 0,
  "target_value": 1.0,
  "root_sequent": "(P => Q), (Q => R) ⟶ (P => R)",
  "trace_step": 1,
  "total_trace_steps": 5,
  "input_ids": [2, 64, 22, 6, 22, 65, 23, 24, 22, 65, 22, 6, 22, 66, 23, 22, 5, 22, 64, 22, 6, 22, 66, 23, 4],
  "token_length": 25
}

Field Descriptions:

• sample_id (int): Unique sequential index for the derivation transition.

• sequent (str): The active Gentzen sequent in string notation (Γ ⟶ Δ or Gamma |- Delta).

• rule (str): Target deduction rule to apply (AXIOM, R_IMP, L_IMP, R_AND, L_AND, R_OR_1, R_OR_2, L_OR, R_NOT, L_NOT, L_CONTR).

• rule_idx (int): Discrete integer class label for the Rule Policy Head (0 to 10; -100 for unprovable negative samples).

• pivot (int): Index of the targeted antecedent premise in Γ (0 to 15; -100 if right-side rule or unprovable).

• target_value (float): Branch provability ground truth in [0.0, 1.0] (1.0 = provable in LI, 0.0 = unprovable counter-model or fallacy).

• root_sequent (str): The top-level goal theorem from which this sub-goal was derived.

• trace_step (int): Step index along the active backward derivation path.

• total_trace_steps (int): Total steps required to close the full proof tree.

• input_ids (List[int]): Tokenized integer sequence mapped via vocab.json (95-token vocabulary).

• token_length (int): Active sequence length before batch padding.

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Action Space & Rule Classification (rule_idx)

rule_idx	Rule Symbol	Name	Formal Sequent Reduction
0	AXIOM	Identity / Ex Falso Axiom	Γ, A ⊢ A or 0, Γ ⊢ Δ
1	R_IMP	Right Implication (⟶ ⇒)	Γ ⊢ (A ⇒ B) ⟹ A, Γ ⊢ B
2	L_IMP	Left Implication (⇒ ⟶)	(A ⇒ B), Γ ⊢ Δ ⟹ Γ ⊢ A and B, Γ ⊢ Δ
3	R_AND	Right Conjunction (⟶ &)	Γ ⊢ (A & B) ⟹ Γ ⊢ A and Γ ⊢ B
4	L_AND	Left Conjunction (& ⟶)	(A & B), Γ ⊢ Δ ⟹ A, B, Γ ⊢ Δ
5	R_OR_1	Right Disjunction 1 (⟶	₁)
6	R_OR_2	Right Disjunction 2 (⟶	₂)
7	L_OR	Left Disjunction (	⟶)
8	R_NOT	Right Negation (⟶ ~)	Γ ⊢ ~A ⟹ A, Γ ⊢ 0
9	L_NOT	Left Negation (~ ⟶)	~A, Γ ⊢ Δ ⟹ Γ ⊢ A
10	L_CONTR	Left Contraction (contr ⟶)	Duplicate premise Γ[i] for multi-use reductions

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Generation Methodology & Sampling Distribution

The 400,000 transitions were synthesized using parallel CPU worker pools across three generative distributions:

1. Hard Theorem Schemas (35% Distribution Weight):

• Multi-step constructive schemas (Transitivity, Modus Tollens, Constructive De Morgan, Glivenko Double Negation theorems).

2. Random Propositional Syntax Trees (50% Distribution Weight):

• Recursively sampled propositional trees across depths 1 to 4 with 0 to 6 antecedent premises in Γ.

• Solved and certified via deterministic backward Gentzen solver with depth budget ≤ 10.

3. Adversarial & Unprovable Counter-Models (15% Distribution Weight):

• 1-token near-miss fallacies (Affirming the Consequent, Denying the Antecedent, broken implication chains) and classical non-theorems (LEM, Peirce's Law).
• Supervised with target_value = 0.0 and masked policy targets (-100) to train value-head pruning.

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How to Load the Dataset

1. PyTorch Binary Tensor Loader (.pt)

Directly matches the DataLoader pipeline used during training:

import torch

data = torch.load("gentzen_dataset.pt", weights_only=False)

input_ids = data["input_ids"]        # Shape: (400000, 256)
target_rule = data["target_rule"]    # Shape: (400000,)
target_pivot = data["target_pivot"]  # Shape: (400000,)
target_value = data["target_value"]  # Shape: (400000,)

print(f"Loaded {input_ids.shape[0]:,} training steps.")
print("Sample Sequent Tensor:", input_ids[0][:12])
print("Target Rule Label    :", target_rule[0].item())
print("Target Value Label   :", target_value[0].item())

2. JSON Lines Loader (.jsonl)

import json

samples = []
with open("gentzen_dataset.jsonl", "r", encoding="utf-8") as f:
    for line in f:
        samples.append(json.loads(line))

print(f"Total parsed records: {len(samples):,}")
print("Transition 0:", samples[0]["sequent"], "⟶ Target Rule:", samples[0]["rule"])

3. Hugging Face Datasets Hub

from datasets import load_dataset

dataset = load_dataset("Sagicc/nanoGentzen-v2", split="train")
print(dataset[0])

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License

This dataset is released under the MIT License.