Ranking and Invariants for Lower-Bound Inference in Quantitative Verification of Probabilistic Programs

2025-04-05

AI Summary

Inferring sound lower bounds for least fixed points in quantitative verification of probabilistic programs remains challenging. Method: We propose the first lower-bound verification framework grounded in uniqueness conditions, generalizing ranking functions from non-probabilistic programs to the probabilistic setting. Our approach establishes a theoretical connection between generalized ranking supermartingales and uniqueness of fixed points, ensuring correctness of inferred lower bounds. The framework unifies verification of diverse quantitative properties—including weak pre-expectations, expected runtime, and higher-order moments—by integrating template-based constraint solving with weakest preexpectation semantics. Results: We implement an automated verification tool based on this framework. Experimental evaluation demonstrates significant improvements in both effectiveness and precision of lower-bound inference across a broad suite of probabilistic programs, including those with complex control flow and stochastic dynamics.

Technology Category

• Reasoning under Uncertainty: Probabilistic Programming
• Machine Learning: Calibration & Uncertainty Quantification
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Quantitative properties of probabilistic programs are often characterised by the least fixed point of a monotone function $K$. Giving lower bounds of the least fixed point is crucial for quantitative verification. We propose a new method for obtaining lower bounds of the least fixed point. Drawing inspiration from the verification of non-probabilistic programs, we explore the relationship between the uniqueness of fixed points and program termination, and then develop a framework for lower-bound verification. We introduce a generalisation of ranking supermartingales, which serves as witnesses to the uniqueness of fixed points. Our method can be applied to a wide range of quantitative properties, including the weakest preexpectation, expected runtime, and higher moments of runtime. We provide a template-based algorithm for the automated verification of lower bounds. Our implementation demonstrates the effectiveness of the proposed method via an experiment.

Problem

• Estimating lower bounds of least fixed points in probabilistic programs
• Relating fixed point uniqueness to program termination analysis
• Automating verification for quantitative properties using ranking supermartingales

Innovation

• Generalized ranking supermartingales for fixed-point uniqueness
• Template-based algorithm for automated lower-bound verification
• Framework for verifying diverse quantitative program properties