Probabilistic hyperproperties specify quantitative relations between the
probabilities of reaching different target sets of states from different
initial sets of states. This class of behavioral properties is suitable for
capturing important security, privacy, and system-level requirements. We
propose a new approach to solve the controller synthesis problem for Markov
decision processes (MDPs) and probabilistic hyperproperties. Our specification
language builds on top of the logic HyperPCTL and enhances it with structural
constraints over the synthesized controllers. Our approach starts from a family
of controllers represented symbolically and defined over the same copy of an
MDP. We then introduce an abstraction refinement strategy that can relate
multiple computation trees and that we employ to prune the search space
deductively. The experimental evaluation demonstrates that the proposed
approach considerably outperforms HyperProb, a state-of-the-art SMT-based model
checking tool for HyperPCTL. Moreover, our approach is the first one that is
able to effectively combine probabilistic hyperproperties with additional
intra-controller constraints (e.g. partial observability) as well as
inter-controller constraints (e.g. agreements on a common action).

提出了一种新的方法来解决马尔可夫决策过程（MDPs）和概率超性质的控制器合成问题，该方法在超级概率计时逻辑（HyperPCTL）的基础上增加了合成控制器的结构约束，采用逐步修正策略修剪搜索空间并与先前的 SMT 模型检查工具相比取得了显著的性能改进。

概率超性质的演绎式控制合成

Deductive Controller Synthesis for Probabilistic Hyperproperties

In this paper, we present a Bayesian method for statistical model checking
(SMC) of probabilistic hyperproperties specified in the logic HyperPCTL* on
discrete-time Markov chains (DTMCs). While SMC of HyperPCTL* using sequential
probability ratio test (SPRT) has been explored before, we develop an
alternative SMC algorithm based on Bayesian hypothesis testing. In comparison
to PCTL*, verifying HyperPCTL* formulae is complex owing to their simultaneous
interpretation on multiple paths of the DTMC. In addition, extending the
bottom-up model-checking algorithm of the non-probabilistic setting is not
straight forward due to the fact that SMC does not return exact answers to the
satisfiability problems of subformulae, instead, it only returns correct
answers with high-confidence. We propose a recursive algorithm for SMC of
HyperPCTL* based on a modified Bayes' test that factors in the uncertainty in
the recursive satisfiability results. We have implemented our algorithm in a
Python toolbox, HyProVer, and compared our approach with the SPRT based SMC.
Our experimental evaluation demonstrates that our Bayesian SMC algorithm
performs better both in terms of the verification time and the number of
samples required to deduce satisfiability of a given HyperPCTL* formula.

本文介绍了基于贝叶斯方法的统计模型检查（SMC）算法，用于在离散时间马尔可夫链（DTMC）上检查指定于逻辑 HyperPCTL * 的概率超性质；我们提出了一种基于递归算法的 SMC 超算法，该算法基于修改后的贝叶斯测试，考虑了递归可满足结果的不确定性，并在一个 Python 工具箱 HyProVer 中实现；我们的实验评估表明，与基于 SPRT 的 SMC 相比，我们的贝叶斯 SMC 算法在验证时间和推断给定 HyperPCTL * 公式的可满足条件所需的样本数量方面表现更好。