# How Does RL Post-training Induce Skill Composition? A Case Study on Countdown

arXiv: https://arxiv.org/abs/2512.01775 · HTML: https://mlsanity.com/paper/2512.01775/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 26 of its 35 references that are in the map.
- First posted on arXiv 1 Dec 2025.

## Placed in

- LLM reasoning / [Reinforcement learning for reasoning](https://mlsanity.com/lr/reinforcement-learning-for-reasoning/)

## Builds on (most cited first, 26 of 26)

- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Tree of Thoughts: Deliberate Problem Solving with Large Language Models](https://mlsanity.com/paper/2305.10601/) (2023)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [HybridFlow: A Flexible and Efficient RLHF Framework](https://mlsanity.com/paper/2409.19256/) (2024)
- [Understanding R1-Zero-Like Training: A Critical Perspective](https://mlsanity.com/paper/2503.20783/) (2025)
- [Does Reinforcement Learning Really Incentivize Reasoning Capacity in LLMs Beyond the Base Model?](https://mlsanity.com/paper/2504.13837/) (2025)
- [Exploring Length Generalization in Large Language Models](https://mlsanity.com/paper/2207.04901/) (2022)
- [REASONING GYM: Reasoning Environments for Reinforcement Learning with Verifiable Rewards](https://mlsanity.com/paper/2505.24760/) (2025)
- [Skill-it! A Data-Driven Skills Framework for Understanding and Training Language Models](https://mlsanity.com/paper/2307.14430/) (2023)
- [A Theory for Emergence of Complex Skills in Language Models](https://mlsanity.com/paper/2307.15936/) (2023)
- [Self-Improvement in Language Models: The Sharpening Mechanism](https://mlsanity.com/paper/2412.01951/) (2024)
- [Transformers Can Achieve Length Generalization But Not Robustly](https://mlsanity.com/paper/2402.09371/) (2024)
- [Sample More to Think Less: Group Filtered Policy Optimization for Concise Reasoning](https://mlsanity.com/paper/2508.09726/) (2025)
- [Skill-Mix: a Flexible and Expandable Family of Evaluations for AI models](https://mlsanity.com/paper/2310.17567/) (2023)
- [e3: Learning to Explore Enables Extrapolation of Test-Time Compute for LLMs](https://mlsanity.com/paper/2506.09026/) (2025)
- [Learning Adaptive Parallel Reasoning with Language Models](https://mlsanity.com/paper/2504.15466/) (2025)
- [OMEGA: Can LLMs Reason Outside the Box in Math? Evaluating Exploratory, Compositional, and Transformative Generalization](https://mlsanity.com/paper/2506.18880/) (2025)
- [From f(x) and g(x) to f(g(x)): LLMs Learn New Skills in RL by Composing Old Ones](https://mlsanity.com/paper/2509.25123/) (2025)
- [Weight Ensembling Improves Reasoning in Language Models](https://mlsanity.com/paper/2504.10478/) (2025)
- [Self-Improving Transformers Overcome Easy-to-Hard and Length Generalization Challenges](https://mlsanity.com/paper/2502.01612/) (2025)
- [How to Reuse and Compose Knowledge for a Lifetime of Tasks: A Survey on Continual Learning and Functional Composition](https://mlsanity.com/paper/2207.07730/) (2022)
- [Can Models Learn Skill Composition from Examples?](https://mlsanity.com/paper/2409.19808/) (2024)
- [LLM-First Search: Self-Guided Exploration of the Solution Space](https://mlsanity.com/paper/2506.05213/) (2025)

## Built on by (most cited first, 6 of 6)

- [The Unlearnability Phenomenon in RLVR for Language Models](https://mlsanity.com/paper/2605.16787/) (2026)
- [New Skills or Sharper Primitives? A Probabilistic Perspective on the Emergence of Reasoning in RLVR](https://mlsanity.com/paper/2602.08281/) (2026)
- [Zipping the Thought: When and How Compressed Reasoning Data Works in LLM Post-Training](https://mlsanity.com/paper/2605.28008/) (2026)
- [Provable Benefits of RLVR over SFT for Reasoning Models: Learning to Backtrack Efficiently](https://mlsanity.com/paper/2606.22938/) (2026)
- [Characterizing Model-Native Skills](https://mlsanity.com/paper/2604.17614/) (2026)
- [Toward Latent Language Model Skills Steering and Optimization: An Empirical Study](https://mlsanity.com/paper/2608.29459/) (2026)

## Abstract (arXiv)

While reinforcement learning (RL) successfully enhances reasoning in large language models, its role in fostering compositional generalization (the ability to synthesize novel skills from known components) is often conflated with mere length generalization. To this end, we study what RL post-training teaches about skill composition and how the structure of the composition affects the skill transfer. We focus on the Countdown task (given n numbers and a target, form an expression that evaluates to the target) and analyze model solutions as expression trees, where each subtree corresponds to a reusable subtask and thus can be viewed as a ``skill.'' Tracking tree shapes and their success rates over training, we find: (i) out-of-distribution (OOD) generalization to larger n and to unseen tree shapes, indicating compositional reuse of subtasks; (ii) a structure-dependent hierarchy of learnability -- models master shallow balanced trees (workload is balanced between subtasks) before deep unbalanced ones, with persistent fragility on right-heavy structures (even when the composition depth is the same as some left-heavy structures). Our diagnostic reveals what is learned, in what order, and where generalization fails, clarifying how RL-only post-training induces OOD generalization beyond what standard metrics such as pass@k reveal.