Existing LMs struggle with proof-oriented programming due to data scarcity, which manifest in two key ways: (1) a lack of sufficient corpora for proof-oriented programming languages such as F*, and (2) the absence of large-scale, project-level proof-oriented implementations that can teach the model the intricate reasoning process when performing proof-oriented programming. We present the first on synthetic data augmentation for project level proof oriented programming for both generation and repair. Our method addresses data scarcity by synthesizing basic proof-oriented programming problems for proficiency in that language; incorporating diverse coding data for reasoning capability elicitation and creating new proofs and repair data within existing repositories. This approach enables language models to both synthesize and repair proofs for function- and repository-level code. We show that our fine-tuned 14B parameter model, PoPilot, can exceed the performance of the models that outperforms GPT-4o in project-level proof-oriented programming by 64% relative margin, and can improve GPT-4o’s performance by 54% by repairing its outputs over GPT-4o’s self-repair.

Instruction-tuned language models excel in knowledge, reasoning, and instruction-following. While knowledge and reasoning are well-explored, the factors enabling generalization to unseen instructions remain underexplored due to challenges in isolating instruction-following dynamics.In this work, we model instruction-following as a computational process and design controlled experiments inspired by the Turing-complete Markov algorithm to disentangle its dynamics. Our findings reveal that the ability to generalize to instructions with unseen semantics emerges only when training data is strategically diversified across rich semantics. This finding gives us the hammer that breaks down the wall separating training instructions from unseen ones encountered in the wild. For specialist models, a balanced mix of in-domain and diverse out-of-domain tasks enhances performance more effectively than simply increasing in-domain data. For generalist models, domain diversification consistently outweighs the costs of reduced task-specific data, regardless of data budgets. Furthermore, we show that proper diversification with a lower data budget can outperform simply scaling up data volume. These findings highlight strategic data diversification as key to optimizing instruction-following and improving model performance across applications.