Sagicc/nanoGentzen-Jev

nanoGentzen Stage-2 Constructive Logic Curriculum

Dataset Summary

The nanoGentzen Stage-2 Constructive Logic Curriculum is a formally verified dataset containing 8,000 Gentzen sequent transitions and epistemic states in Intuitionistic Propositional Logic (Gentzen LJ calculus).

The dataset was engineered specifically to train and calibrate neuro-symbolic reasoning models (such as nanoGentzen-Jev-v3), overcoming the critical distribution biases of naive random formula generators:

1. Modus Ponens (L_IMP) Scarcity: Balances multi-step enterprise rule chains (A => B, B => C, C => D |- D) where classical generators provide < 2% implication rules.
2. Antecedent Pivot Position Skew: Shuffles premises uniformly across positions 0..15, teaching attention pointer heads to avoid positional bias.
3. Epistemic 3-State Grounding: Explicitly models constructive truth (V = 1.0), epistemic open branches/missing facts (V = 0.5), and constructive countermodel refutations (V = 0.0).

Every positive trace in this dataset has been certified by a pure Python constructive Gentzen LJ proof kernel.

Fine-Tuning nanoGentzen-v2 to Jev-v3

This dataset served as the Stage-2 training corpus that transformed nanoGentzen-v2 (4.86M params) into nanoGentzen-Jev-v3 (4.93M params). While v2 had achieved strong foundational proof capabilities over 400,000 pretraining steps, it was hampered by three structural limitations when applied to real-world business policies: a static 16-slot pivot classifier with 96.3% bias toward the first three premises, a severe scarcity of Modus Ponens (L_IMP was only 1.17% of training steps), and a scalar value head that could not distinguish between provably false refutations (0.0) and open/incomplete premises (0.5).

To address this without catastrophic forgetting, the 400k-pretrained transformer backbone was warm-started, and two specialized heads were grafted onto it: a dynamic Span Pointer Head using query-key cross-attention over antecedent tokens (scaling to 32+ premises), and a 3-Class Epistemic Softmax Head predicting [P(Refuted), P(Undecided), P(Provable)]. Fine-tuning on this 8,000-sample balanced curriculum produced nanoGentzen-Jev-v3, achieving a 0.081 Brier score, a -31.3% reduction in calibration error (ECE), and 73.1% accuracy on the LocalLLaMA/typed-decisions benchmark while preserving 100.00% formal proof soundness.

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Dataset Breakdown & Statistics

Curriculum Partition	Target Value	Samples	Share	Description
horn_policy_provable	1.0	5,642	70.5%	Multi-hop Horn policy chains with uniform antecedent pivot distribution (positions 0..15).
propositional_theorem	1.0	1,221	15.3%	Decomposed steps of constructive propositional Gentzen LJ theorems.
horn_policy_refuted	0.0	627	7.8%	Inconsistent premise sets or provably falsified goals with countermodels.
horn_policy_undecided	0.5	510	6.4%	Open branches with missing premises or intuitionistically unprovable sequents.
Total	—	8,000	100%	Formally verified symbolic logic curriculum

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Data Structure

Data Fields

Each record in gentzen_dataset.jsonl contains the following fields:

• sample_id (int): Unique sample sequence identifier (1..8000).
• curriculum (str): Origin curriculum bucket:
  • "horn_policy_provable": Provable multi-hop deduction chain.
  • "propositional_theorem": Standard Gentzen decomposition step.
  • "horn_policy_undecided": Incomplete facts or open branch (V = 0.5).
  • "horn_policy_refuted": Inconsistent or refuted branch (V = 0.0).
• sequent (str): The symbolic Gentzen sequent formatted as Gamma |- Delta.
• rule (str): Target Gentzen calculus rule name ("AXIOM", "L_IMP", "L_AND", "R_OR", "L_OR", "R_NOT", "UNDECIDED", "REFUTED").
• rule_idx (int): Discrete rule index for Cross-Entropy classification. Set to -100 (ignore_index) for undecided and refuted states.
• pivot (int): 0-indexed antecedent position in Gamma targeted by the rule. Set to -100 (ignore_index) for right rules, axioms, or non-theorems.
• target_value (float): Epistemic target utility:
  • 1.0: Provably True / Closed proof tree
  • 0.5: Undecided / Open branch (missing information)
  • 0.0: Provably False / Refuted
• root_sequent (str): The original root goal sequent from which this intermediate derivation step originated.
• trace_step (int): Current derivation step within the proof tree.
• total_trace_steps (int): Total steps in the proof derivation trace.
• input_ids (list[int]): Pre-tokenized sequence using the official nanoGentzen vocabulary.
• token_length (int): Total number of tokens before padding.

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Example Instances

1. Provable Multi-Hop Horn Rule (horn_policy_provable)

{
  "sample_id": 2,
  "curriculum": "horn_policy_provable",
  "sequent": "(v2 => v9), (v6 => v10), v9, v0, P, (v3 => v0), (P => v4), (v4 => S), v10 |- S",
  "rule": "L_IMP",
  "rule_idx": 2,
  "pivot": 7,
  "target_value": 1.0,
  "root_sequent": "(v2 => v9), (v6 => v10), v9, v0, P, (v3 => v0), (P => v4), (v4 => S), v10 |- S",
  "trace_step": 1,
  "total_trace_steps": 5,
  "token_length": 72
}

Notice: The target pivot formula is antecedent #7 ((v4 => S)), demonstrating long-range antecedent indexing.

2. Intuitionistically Undecided / Open Branch (horn_policy_undecided)

{
  "sample_id": 7,
  "curriculum": "horn_policy_undecided",
  "sequent": "(v2 => v9), (v6 => v10), v9, v0, (v3 => v0), (P => v4), (v4 => S), v10 |- S",
  "rule": "UNDECIDED",
  "rule_idx": -100,
  "pivot": -100,
  "target_value": 0.5,
  "root_sequent": "(v2 => v9), (v6 => v10), v9, v0, (v3 => v0), (P => v4), (v4 => S), v10 |- S",
  "trace_step": 1,
  "total_trace_steps": 1,
  "token_length": 69
}

Notice: The essential premise P is missing from the antecedents. The goal S cannot be proved or disproved constructively. The model is supervised with target_value = 0.5 and rule_idx = -100.

3. Constructive Refutation / Contradiction (horn_policy_refuted)

{
  "sample_id": 8,
  "curriculum": "horn_policy_refuted",
  "sequent": "~P, (~P => ~S), (v2 => v9), (v6 => v10), v9, v0, (v3 => v0), (P => v4), (v4 => S), v10 |- S",
  "rule": "REFUTED",
  "rule_idx": -100,
  "pivot": -100,
  "target_value": 0.0,
  "root_sequent": "~P, (~P => ~S), (v2 => v9), (v6 => v10), v9, v0, (v3 => v0), (P => v4), (v4 => S), v10 |- S",
  "trace_step": 1,
  "total_trace_steps": 1,
  "token_length": 84
}

Notice: Antecedents contain ~P and (~P => ~S), constructively proving ~S which contradicts the goal S. Supervised with target_value = 0.0.

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How to Load and Use

With Hugging Face datasets

from datasets import load_dataset

# Load directly from local JSONL or Hugging Face Hub:
dataset = load_dataset("json", data_files="gentzen_dataset.jsonl")
print(dataset["train"][0])

With PyTorch Directly

import torch

# Load the pre-collated tensor dictionary:
data = torch.load("gentzen_dataset.pt", weights_only=False)

input_ids = data["input_ids"]        # Shape: (8000, 256)
target_rules = data["target_rules"]  # Shape: (8000,) with -100 for open/refuted
target_pivots = data["target_pivots"]# Shape: (8000,) with -100 for open/refuted
target_values = data["target_values"]# Shape: (8000,) in {1.0, 0.5, 0.0}

print(f"Loaded {len(input_ids)} tensor samples ready for DataLoader.")

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Training Loss Guidance

When training a neural-symbolic policy-value network on this dataset, implement masked multi-task cross-entropy:

import torch.nn as nn
import torch.nn.functional as F

# Policy heads (ignore -100 on undecided/refuted states)
loss_rule = F.cross_entropy(logits_rule, target_rules, ignore_index=-100)
loss_pivot = F.cross_entropy(logits_pivot, target_pivots, ignore_index=-100)

# Epistemic head (3 classes: 0 = Refuted, 1 = Undecided, 2 = Provable)
# Map target_value {0.0, 0.5, 1.0} to class {0, 1, 2}
class_targets = (target_values * 2).long()
loss_epistemic = F.cross_entropy(logits_epistemic, class_targets)

# Value MSE
pred_value = probs_epistemic[:, 2] + 0.5 * probs_epistemic[:, 1]
loss_value = F.mse_loss(pred_value, target_values)

total_loss = loss_rule + loss_pivot + loss_epistemic + 0.5 * loss_value

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Formal Verification & Soundness

All samples in horn_policy_provable and propositional_theorem are certified by the pure Gentzen LJ calculus kernel:

• Identity Rule (I): Γ, A |- A
• Left Conjunction (L_AND): Γ, A ∧ B |- C ==> Γ, A, B |- C
• Right Conjunction (R_AND): Γ |- A ∧ B ==> Γ |- A and Γ |- B
• Left Implication (L_IMP / Modus Ponens): Γ, A => B |- C ==> Γ, B |- C (when Γ |- A)
• Left Negation (L_NOT): Γ, ¬A |- C ==> Γ |- A
• Intuitionistic Single-Consequent Invariant: |Δ| <= 1 in all sequents.

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Citation & Attribution

@dataset{nanogentzen_stage2_curriculum_2026,
  title={nanoGentzen Stage-2 Constructive Logic Curriculum},
  author={Andrija Sagicc},
  year={2026},
  publisher={Hugging Face},
  howpublished={\url{https://huggingface.co/datasets/andrax/nanogentzen-stage2-curriculum}}
}