# NormWorlds-CF: Solver-Verified Counterfactual Normative Reasoning with Metamorphic-Relation GRPO

arXiv: https://arxiv.org/abs/2607.03957 · HTML: https://mlsanity.com/paper/2607.03957/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 28 of its 35 references that are in the map.
- First posted on arXiv 4 Jul 2026.

## Placed in

- LLM reasoning / [Mathematical reasoning](https://mlsanity.com/lr/mathematical-reasoning/)

## Builds on (most cited first, 28 of 28)

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- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Deep reinforcement learning from human preferences](https://mlsanity.com/paper/1706.03741/) (2017)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Fine-Tuning Language Models from Human Preferences](https://mlsanity.com/paper/1909.08593/) (2019)
- [CLEVR: A Diagnostic Dataset for Compositional Language and Elementary Visual Reasoning](https://mlsanity.com/paper/1612.06890/) (2016)
- [Learning the Difference that Makes a Difference with Counterfactually-Augmented Data](https://mlsanity.com/paper/1909.12434/) (2019)
- [PICARD: Parsing Incrementally for Constrained Auto-Regressive Decoding from Language Models](https://mlsanity.com/paper/2109.05093/) (2021)
- [Rubrics as Rewards: Reinforcement Learning Beyond Verifiable Domains](https://mlsanity.com/paper/2507.17746/) (2025)
- [LegalBench: A Collaboratively Built Benchmark for Measuring Legal Reasoning in Large Language Models](https://mlsanity.com/paper/2308.11462/) (2023)
- [ProofWriter: Generating Implications, Proofs, and Abductive Statements over Natural Language](https://mlsanity.com/paper/2012.13048/) (2020)
- [Towards AI-Complete Question Answering: A Set of Prerequisite Toy Tasks](https://mlsanity.com/paper/1502.05698/) (2015)
- [LexGLUE: A Benchmark Dataset for Legal Language Understanding in English](https://mlsanity.com/paper/2110.00976/) (2021)
- [Transformers as Soft Reasoners over Language](https://mlsanity.com/paper/2002.05867/) (2020)
- [SaulLM-7B: A pioneering Large Language Model for Law](https://mlsanity.com/paper/2403.03883/) (2024)
- [Chatlaw: A Multi-Agent Legal Assistant based on a Role-Aligned Mixture-of-Experts Architecture](https://mlsanity.com/paper/2306.16092/) (2023)
- [DISC-LawLLM: Fine-tuning Large Language Models for Intelligent Legal Services](https://mlsanity.com/paper/2309.11325/) (2023)
- [LegalBench-RAG: A Benchmark for Retrieval-Augmented Generation in the Legal Domain](https://mlsanity.com/paper/2408.10343/) (2024)
- [LawBench: Benchmarking Legal Knowledge of Large Language Models](https://mlsanity.com/paper/2309.16289/) (2023)
- [Metamorphic Testing: A New Approach for Generating Next Test Cases](https://mlsanity.com/paper/2002.12543/) (2020)
- [RobustLR: Evaluating Robustness to Logical Perturbation in Deductive Reasoning](https://mlsanity.com/paper/2205.12598/) (2022)
- [Metamorphic Relation Generation: State of the Art and Visions for Future Research](https://mlsanity.com/paper/2406.05397/) (2024)
- [Normative Reasoning in Large Language Models: A Comparative Benchmark from Logical and Modal Perspectives](https://mlsanity.com/paper/2510.26606/) (2025)
- [OpenExempt: A Diagnostic Benchmark for Legal Reasoning and a Framework for Creating Custom Benchmarks on Demand](https://mlsanity.com/paper/2601.13183/) (2026)

## Abstract (arXiv)

Language models can reach the right normative verdict for the wrong reason. We introduce NormWorlds-CF, a solver-verified environment for counterfactual normative reasoning in executable rule worlds. Its deterministic solver produces final answers, proof and falsification certificates, argument statuses, support sets, and paired-world change labels, enabling supervision and evaluation without LLM judges. The benchmark contains staged SFT diagnostics and a compact paired-world task with 270 root families and 1080 canonical-to-variant pairs. The SFT diagnostics show that final-answer supervision can saturate verdict accuracy without inducing falsification competence: answer-only SFT reaches perfect answer accuracy but scores zero on joint falsification certificates, while full-mix training with targeted replay reaches strong all-task accuracy (0.99). For the structured-change task, we introduce metamorphic-relation GRPO (MR-GRPO), a class-conditioned reward for GRPO that gives partial credit for relation families and solver-visible change fields. In matched Qwen3-1.7B continuation experiments, MR-GRPO improves held-out relation accuracy and relation-family correctness, and reduces wrong-family error, compared to sparse and answer-only GRPO. In Qwen3-4B three-seed validation, sparse reward preserves coarse relation labels best, answer-only reward improves answer-change but weakens relation-family structure, and MR-GRPO leads on answer-, support-, and status-change fields as well as class-conditioned MR and change-presence. These results show that verified counterfactual structure can shape post-training beyond final answers, while exact full change-record generation, invariant subtype recognition, and out-of-distribution (OOD) transfer remain open problems.