Ranking Function Generation with LLMs for Termination

Ranking functions are a fundamental technique for proving loop termination, but constructing them manually is challenging and requires substantial expertise. We propose RAINFALL, a neuro-symbolic framework which follows the generate-verify-refine paradigm and tightly integrates LLMs with symbolic verification. RAINFALL formulates ranking function synthesis as a code-completion task by inserting placeholders into the input program and constraining the LLM to fill four predefined slots for each loop. The completed program is then checked by a symbolic verifier. If verification succeeds, a valid ranking function is obtained; otherwise, the verifier's diagnostics are translated into natural-language feedback that guides iterative refinement of the LLM's output. We evaluate RAINFALL on two public benchmark suites comprising 358 terminating C programs. Using MiniMax V2.7, RAINFALL synthesizes verifiable ranking functions for 323 programs (90.22%), outperforming all promising baselines. In particular, it solves 12 more programs than PROTON 2.1, the best-performing baseline and winner of the SV-COMP 2026 Termination category. Furthermore, combining RAINFALL with PROTON 2.1 increases the success rate to 96.65% (346/358 programs). We also conduct ablation study confirming individual contributions of placeholder contracts, few-shot examples, symbolic verification, and diagnostic information feedback. Experiments across LLMs further confirm the effectiveness of RAINFALL. Our results suggest a generic, sound framework for ranking-function synthesis and, beyond termination, LLM-assisted synthesis of formally checkable liveness certificates, both of which represent new directions of software engineering research.