Breaking the Mold: Nonlinear Ranking Function Synthesis Without Templates
<|reference_start|>Breaking the Mold: Nonlinear Ranking Function Synthesis Without Templates: This paper studies the problem of synthesizing (lexicographic) polynomial ranking functions for loops that can be described in polynomial arithmetic over integers and reals. While the analogous ranking function synthesis problem for linear arithmetic is decidable, even checking whether a given function ranks an integer loop is undecidable in the nonlinear setting. We side-step the decidability barrier by working within the theory of linear integer/real rings (LIRR) rather than the standard model of arithmetic. We develop a termination analysis that is guaranteed to succeed if a loop (expressed as a formula) admits a (lexicographic) polynomial ranking function. In contrast to template-based ranking function synthesis in real arithmetic, our completeness result holds for lexicographic ranking functions of unbounded dimension and degree, and effectively subsumes linear lexicographic ranking function synthesis for linear integer loops.<|reference_end|>
@article{zhu2024breaking,
    title={Breaking the Mold: Nonlinear Ranking Function Synthesis Without
  Templates},
    author={Shaowei Zhu and Zachary Kincaid},
    journal={CAV 2024. Lecture Notes in Computer Science, vol 14681. Springer,
  Cham},
    year={2024},
    doi={10.1007/978-3-031-65627-9_21},
    archivePrefix={arXiv},
    eprint={2409.18063},
    primaryClass={cs.PL cs.LO}
}