SolidCoder: Bridging the Mental-Reality Gap in LLM Code Generation through Concrete Execution

Unlike natural-language specifications, executable formal specifications provide machine-checkable constraints for verifying, debugging, and repairing code. However, writing such specifications is labor-intensive, and existing LLM-based methods mainly infer whole-program pre/postconditions, missing the intermediate semantic commitments that programmers rely on when reasoning about an algorithm. Our study further shows that prompting current CodeLLMs often produces executable assertions that are syntactically invalid, trivial, or too weak to reject behavior-changing faults. In this paper, we study executable checkpoint specification generation, where assertions are inserted at meaningful internal program points to describe expected intermediate states. We introduce SpecCoder, a verification-guided CodeLLM training framework that learns from validated reference programs, behavior-changing mutants, and multi-turn specification-refinement traces. SpecCoder selects specifications that hold on correct executions while rejecting faulty executions, turning specifications from passive annotations into executable evidence. To evaluate this setting, we introduce HumanExec, a benchmark built from recent Codeforces competitive programming problems with test suites, reference solutions, and human buggy submissions, supporting three tasks: specification generation, program correctness checking, and program repair. Experiments on HumanExec show that SpecCoder substantially improves checkpoint-specification quality over base CodeLLMs. Across Qwen2.5-Coder models, SpecCoder improves inline-specification correctness by up to 55.8%, completeness by up to 358.1%, and executable assertion validity by up to 26.6%. These gains further translate to downstream correctness reasoning and repair, showing that executable checkpoints provide fine-grained evidence for reliable verification.

Effective code generation requires both model capability and a problem representation that carefully structures how models reason and plan. Existing approaches augment reasoning steps or inject specific structure into how models think, but leave scattered problem conditions unchanged. Inspired by the way humans organize fragmented information into coherent explanations, we propose StoryCoder, a narrative reformulation framework that transforms code generation questions into coherent natural language narratives, providing richer contextual structure than simple rephrasings. Each narrative consists of three components: a task overview, constraints, and example test cases, guided by the selected algorithm and genre. Experiments across 11 models on HumanEval, LiveCodeBench, and CodeForces demonstrate consistent improvements, with an average gain of 18.7% in zero-shot pass@10. Beyond accuracy, our analyses reveal that narrative reformulation guides models toward correct algorithmic strategies, reduces implementation errors, and induces a more modular code structure. The analyses further show that these benefits depend on narrative coherence and genre alignment, suggesting that structured problem representation is important for code generation regardless of model scale or architecture. Our code is available at https://github.com/gu-ni/StoryCoder.

Verified code generation asks a large language model (LLM) to generate both an executable program and a machine-checkable proof that the program meets a formal specification, promising software that is correct by construction. The de facto workflow decouples the two halves of the problem: first synthesize a program, then attempt to prove it correct. We observe that this sequential pipeline can be both ineffective and inefficient in practice. A program generated without anticipating its proof can be subtly incorrect or structurally difficult to verify, forcing the LLM into brittle repair loops that alternate between patching the code and patching the proof. Inspired by Dijkstra's view that a program and its correctness argument should be developed hand in hand, we propose $P^3$, an LLM-based agentic workflow that first derives a unified program-and-proof plan from the specification, then elaborates the implementation and proof scaffold under this shared plan. To evaluate verified code generation in realistic settings, we further introduce Lean4Commit0, a repository-derived, library-level benchmark built by extracting core APIs from real-world software repositories and translating their requirements, including relational specifications across APIs, into Lean tasks. Using four frontier LLM backends, we evaluate $P^3$ on Verina, AlgoVeri, and our Lean4Commit0 benchmark, where it achieves the highest solve rate in every benchmark--model setting. Compared with the stronger baseline, it improves solve rates by 4.6--11.2 percentage points and reduces per-task API cost by up to roughly 40% and wall-clock time by up to roughly 37% on the difficult subset of each benchmark. A targeted ablation further shows gains of 3.3--8.3 points over implementation-only planning, isolating the benefit of planning the program and proof jointly.