Gentzen-Kripke-240K: Certified Neuro-Symbolic Sequent Dataset

Gentzen-Kripke-240K is a large-scale formal dataset of 240,000 certified sequents and derivation steps for training and evaluating neuro-symbolic theorem provers, policy verification models, and typed-decision engines.

It was created to train nanoGentzen-v3 (4.93M parameters), the model tested on LocalLLaMA/typed-decisions (76.75% accuracy) and the official S1MB System-1 benchmark (Score 2.50, outperforming 29 published models up to 7.6B parameters).

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Dataset Summary

• Total Sequents / Steps: 240,000
• Unique Trajectories: 172,426
• Balanced Epistemic Distribution: Exactly 1:1:1 across Saul Kripke (1965) 3-class epistemic states:
  • Class 2 (PROVEN_TRUE, 1.0): 80,000 samples (33.33%)
  • Class 1 (INTUITIONISTIC_UNDECIDED, 0.5): 80,000 samples (33.33%)
  • Class 0 (PROVEN_FALSE, 0.0): 80,000 samples (33.33%)
• Zero-Leakage Trajectory Splits: Grouped by trajectory_id so sub-steps of the same derivation tree DAG are never divided between splits:
  • train: 215,995 samples (90% of trajectories, 21.3 MB Parquet)
  • validation: 24,005 samples (10% of trajectories, 2.4 MB Parquet)
• Span Pointer Pivot Coverage: Full supervision over antecedent pivot indices (0 \dots 31) for left-hand decomposition rules ((L_{\to}, L_{\land}, L_{\lor}, L_{\neg})), with distractor premises shuffled to prevent positional bias.

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Dataset Structure

Data Fields

Field Name	Type	Description
sample_id	int64	Globally unique sample identifier (1 to 240,000).
trajectory_id	int64	Identifier grouping all steps belonging to the same root derivation DAG.
curriculum	string	Task curriculum category (e.g. horn_policy_provable, policy_missing_premise, constructive_refutation, constructive_de_morgan_1, open_kripke_countermodel).
sequent	string	The intuitionistic sequent in ASCII syntax: Γ ⊢ Δ (e.g. `(Q
rule	string	Target Gentzen inference rule name (AXIOM, R_IMP, L_IMP, R_AND, L_AND, R_OR_1, R_OR_2, L_OR, R_NOT, L_NOT, L_CONTR, or NONE).
rule_idx	int64	Integer rule index (0 to 10 for Gentzen rules, -100 for unprovable/refuted steps).
pivot	int64	Target antecedent index in (\Gamma) ((0 \dots 31)) to apply the left-rule on; -100 for right rules and axioms.
target_epistemic	class_label	0 = PROVEN_FALSE, 1 = INTUITIONISTIC_UNDECIDED, 2 = PROVEN_TRUE.
target_value	float64	Normalized continuous target value: 0.0, 0.5, or 1.0.
root_sequent	string	The root sequent from which this derivation step originated.
trace_step	int64	Zero-indexed step index in the backward derivation.
total_trace_steps	int64	Total number of steps in the verified proof trace.
input_ids	list<int64>	Pre-tokenized token sequence using the official nanoGentzen 95-token logic vocabulary.
token_length	int64	Length of the tokenized sequence (max 256).

Data Splits

Split	Number of Examples	Epistemic Balance (False / Undecided / True)	Parquet Size
train	215,995	71,879 / 71,873 / 72,243	21.31 MB
validation	24,005	8,121 / 8,127 / 7,757	2.42 MB
Total	240,000	80,000 / 80,000 / 80,000	23.72 MB

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How to Use the Dataset

1. Load via Hugging Face datasets

from datasets import load_dataset

# Load directly from Hugging Face Hub
dataset = load_dataset("Sagicc/gentzen-kripke-240k")

# Access train and validation splits
train_data = dataset["train"]
val_data = dataset["validation"]

# Inspect a sample
print(train_data[0])
# {
#   'sample_id': 1,
#   'trajectory_id': 1,
#   'curriculum': 'constructive_disjunctive_cases',
#   'sequent': '(Q | v19), (Q => v4), (v19 => v4) |- v4',
#   'rule': 'L_OR',
#   'rule_idx': 7,
#   'pivot': 0,
#   'target_epistemic': 2,
#   'target_value': 1.0,
#   ...
# }

2. Load via pandas or pyarrow

import pandas as pd

train_df = pd.read_parquet("train.parquet")
val_df = pd.read_parquet("validation.parquet")

print(f"Loaded {len(train_df):,} training sequents across {train_df['curriculum'].nunique()} curricula.")

3. Load via Native PyTorch (.pt)

import torch

# Load full dataset samples list
samples = torch.load("gentzen_dataset.pt", map_location="cpu", weights_only=False)
print(f"Loaded {len(samples):,} PyTorch tensor samples.")

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Curriculum Composition

                    ┌──────────────────────────────────────────────┐
                    │     Gentzen-Kripke-240K (240,000 Samples)     │
                    └──────────────────────┬───────────────────────┘
                                           │
         ┌─────────────────────────────────┼─────────────────────────────────┐
         │ (33.3%)                         │ (33.3%)                         │ (33.3%)
         ▼                                 ▼                                 ▼
┌──────────────────┐             ┌──────────────────┐             ┌──────────────────┐
│     Class 2      │             │     Class 1      │             │     Class 0      │
│   PROVEN_TRUE    │             │  UNDETERMINED    │             │   PROVEN_FALSE   │
│  80,000 Samples  │             │  80,000 Samples  │             │  80,000 Samples  │
└────────┬─────────┘             └────────┬─────────┘             └────────┬─────────┘
         │                                │                                │
         ├─ Multi-Hop Horn (47.2k)        ├─ Premise Deprived (44.0k)      ├─ Policy Violations (40.1k)
         ├─ De Morgan Laws (9.7k)         ├─ Kripke Countermodels (14.4k)  ├─ Refutations (39.9k)
         ├─ Disjunctive Cases (4.7k)      ├─ Classical LEM / Peirce (11.0k)└─ Modus Tollens Chains
         └─ Refutation Proofs (4.5k)      └─ Affirming Consequent (10.6k)

1. Class 2 (PROVEN_TRUE, 80,000 samples): Constructively derived theorems in Gerhard Gentzen's (\mathbf{LJ}) sequent calculus. Each step provides the exact invertible or non-invertible Gentzen inference rule (rule) and antecedent pivot index (pivot). Includes multi-hop Horn policies (1 to 3 hops), De Morgan dualities, double negation of excluded middle (\neg\neg(P \lor \neg P)), and conjunction/disjunction distribution.
2. Class 1 (INTUITIONISTIC_UNDECIDED, 80,000 samples): Underdetermined sequents lacking sufficient constructive evidence. Includes policies missing critical prerequisite facts, classical fallacies that fail intuitionistically (Law of Excluded Middle, Peirce's Law, Double Negation Elimination), and open Kripke tableaus.
3. Class 0 (PROVEN_FALSE, 80,000 samples): Constructive refutations and contradictions. Sequents where the antecedent premises (\Gamma) derive (\neg G) or (\Gamma, G \vdash \bot), representing policy violations, contradictory constraints, and Modus Tollens refutation chains.

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Antecedent Span Pointer Distribution

Unlike standard sequence-to-sequence provers, Gentzen-Kripke-240K supervises a Span Pointer Antecedent Pivot Attention Head:

• Antecedent premise counts range from 1 to 32 premises per sequent.
• Up to 30 distractor premises are injected and randomly shuffled across (\Gamma) to prevent neural models from exploiting positional shortcuts (e.g. always picking index 0).
• Left-hand rules provide the exact ground-truth pivot index (k \in [0, 31]).

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Associated Models

• Model: Sagicc/nanoGentzen-v3 (4.93M parameters, 19.6 MB weights)
• Source Code: https://github.com/nanoGentzen-v3_train)

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License

This dataset is released under the MIT License.