We investigate the decidability of model checking logics of time, knowledge, and probability, with respect to two epistemic semantics: the clock and synchronous perfect recall semantics in partially observable discrete-time Markov chains. Decidability results are known for certain restricted logics with respect to these semantics, subject to a variety of restrictions that are either unexplained or involve a longstanding unsolved mathematical problem. We show that mild generalizations of the known decidable cases suffice to render the model checking problem definitively undecidable. In particular, for the synchronous perfect recall semantics, a generalization from temporal operators with finite reach to operators with infinite reach renders model checking undecidable. The case of the clock semantics is closely related to a monadic second-order logic of time and probability that is known to be decidable, except on a set of measure zero. We show that two distinct extensions of this logic make model checking undecidable. One of these involves polynomial combinations of probability terms, the other involves monadic second-order quantification into the scope of probability operators. These results explain some of the restrictions in previous work.

中文翻译：

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模型检验概率时间认知逻辑的不可判定案例

我们研究了模型检查时间、知识和概率逻辑的可判定性，涉及两个认知语义：部分可观察离散时间马尔可夫链中的时钟和同步完美回忆语义。对于与这些语义相关的某些受限逻辑，可判定性结果是已知的，受到各种无法解释或涉及长期未解决的数学问题的限制。我们表明，对已知可判定案例的温和概括足以使模型检查问题最终无法判定。特别是，对于同步完美召回语义，从具有有限范围的时间算子到具有无限范围的算子的泛化使得模型检查无法确定。时钟语义的情况与已知可判定的时间和概率的一元二阶逻辑密切相关，除了在一组测量零上。我们展示了这种逻辑的两个不同的扩展使模型检查无法确定。其中一个涉及概率项的多项式组合，另一个涉及将一元二阶量化纳入概率算子的范围。这些结果解释了以前工作中的一些限制。