[论文解读] Highly Incremental: A Simple Programmatic Approach for Many Objectives (Extended Version)

Philipp Schröer, Joost-Pieter Katoen|arXiv (Cornell University)|Mar 2, 2026

Formal Methods in Verification被引用 0

一句话总结

本文提出了一种基于奖励的编程扩展和一个单调变换，它将多样的概率程序目标简化为标准的最弱前验推理，从而实现统一分析与自动化（Caesar 验证器）。

ABSTRACT

We present a one-fits-all programmatic approach to reason about a plethora of objectives on probabilistic programs. The first ingredient is to add a reward-statement to the language. We then define a program transformation applying a monotone function to the cumulative reward of the program. The key idea is that this transformation uses incremental differences in the reward. This simple, elegant approach enables to express e.g., higher moments, threshold probabilities of rewards, the expected excess over a budget, and moment-generating functions. All these objectives can now be analyzed using a single existing approach: probabilistic wp-reasoning. We automated verification using the Caesar deductive verifier and report on the application of the transformation to some examples.

研究动机与目标

• Motivate the need to reason about a wide range of quantitative objectives for probabilistic programs beyond simple expectation.
• Introduce a reward statement to collect non-negative rewards during execution as a unifying abstraction.
• Develop a program transformation that uses incremental reward differences to reduce complex transformed rewards to standard expectations.
• Demonstrate how the approach can express higher moments, threshold probabilities, and moment-generating functions within the wp framework.
• Enable automated verification by integrating with existing wp-based tools (Caesar) and applying the transformation to examples.

提出的方法

• Extend probabilistic guarded command language (pGCL) with a reward a statement that accumulates a non-negative reward during execution.
• Define a reward transformation that computes E(f(rew(S))) for a function f, where f can be complex (e.g., squares, thresholds, MGFs).
• Reuse and adapt the weakest pre-expectation (wp) semantics to accommodate reward statements and to reason about transformed rewards.
• Prove soundness by relating the programmatic transformation to an associated Markov-chain transformation.
• Show how the approach can model a variety of objectives (e.g., higher moments) within a single wp-based framework.
• Discuss automation and the application of the Caesar deductive verifier to the extended framework and example programs.

实验结果

研究问题

• RQ1 Can a single wp-based framework, augmented with reward statements, express and analyze a broad set of probabilistic-program objectives?
• RQ2 Is there a sound programmatic transformation that allows reasoning about transformed rewards (e.g., higher moments, thresholded rewards) using standard wp techniques?
• RQ3 How can higher-order objectives like moments or MGFs be modeled and verified within the same formalism?
• RQ4 To what extent can existing verification tools (like Caesar) be applied to the extended reward-enabled wp calculus on real examples?

主要发现

• A unified programming approach uses reward statements to model diverse quantitative objectives within the wp framework.
• A simple, incremental-difference based program transformation reduces complex transformed rewards to standard expected rewards, enabling reuse of wp theory and tools.
• The method can express and analyze higher moments, threshold probabilities, discounted rewards, and moment-generating functions.
• The extended wp semantics with rewards is sound with respect to the associated Markov-chain transformation.
• Automation is feasible via existing deductive tools (Caesar) and applies to example probabilistic programs.