We provide a categorical notion called uncertain bisimilarity, which allows to reason about bisimilarity in combination with a lack of knowledge about the involved systems. Such uncertainty arises naturally in automata learning algorithms, where one investigates whether two observed behaviours come from the same internal state of a black-box system that can not be transparently inspected. We model this uncertainty as a set functor equipped with a partial order which describes possible future developments of the learning game. On such a functor, we provide a lifting-based definition of uncertain bisimilarity and verify basic properties. Beside its applications to Mealy machines, a natural model for automata learning, our framework also instantiates to an existing compatibility relation on suspension automata, which are used in model-based testing. We show that uncertain bisimilarity is a necessary but not sufficient condition for two states being implementable by the same state in the black-box system. To remedy the failure of the one direction, we characterize uncertain bisimilarity in terms of coalgebraic simulations.

翻译：我们提出了一种称为不确定双相似性的范畴论概念，该概念允许在结合对涉及系统缺乏认知的情况下推理双相似性。这种不确定性自然地出现在自动机学习算法中，其中需要探究两个观察到的行为是否来自同一无法透明检查的黑箱系统的内部状态。我们将这种不确定性建模为配备一个偏序的集合函子，该偏序描述了学习博弈中可能的未来发展。在此类函子上，我们给出了基于提升的不确定双相似性定义，并验证了其基本性质。除了应用于Mealy机器（一种自动机学习的自然模型）外，我们的框架还可实例化为悬挂自动机上的一种现有兼容性关系，而悬挂自动机常用于基于模型的测试。我们证明，不确定双相似性是黑箱系统中两个状态能否由同一状态实现的一个必要但不充分条件。针对该单向失效问题，我们通过余代数模拟来刻画不确定双相似性。