Supermartingales for Unique Fixed Points: A Unified Approach to Lower Bound Verification

Satoshi Kura; Hiroshi Unno; Takeshi Tsukada

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arXiv · 2504.04132

Supermartingales for Unique Fixed Points: A Unified Approach to Lower Bound Verification

Satoshi Kura·Hiroshi Unno·Takeshi Tsukada

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Abstract

Many quantitative properties of probabilistic programs can be characterized as least fixed points, but verifying their lower bounds remains a challenging problem. We present a new approach to lower-bound verification that exploits and extends the connection between the uniqueness of fixed points and program termination. The core technical tool is a generalization of ranking supermartingales, which serves as witnesses of the uniqueness of fixed points. Our method provides a simple and unified reasoning principle applicable to a wide range of quantitative properties, including termination probability, the weakest preexpectation, expected runtime, higher moments of runtime, and conditional weakest preexpectation. We provide a template-based algorithm for automated verification of lower bounds and demonstrate the effectiveness of the proposed method via experiments.

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Satoshi Kura, Hiroshi Unno, Takeshi Tsukada. 2026-04-19. Supermartingales for Unique Fixed Points: A Unified Approach to Lower Bound Verification. https://arxiv.org/abs/2504.04132

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