The tension between deduction and induction is perhaps the most fundamental issue in areas such as philosophy, cognition and artificial intelligence. In an influential paper, Valiant recognised that the challenge of learning should be integrated with deduction. In particular, he proposed a semantics to capture the quality possessed by the output of Probably Approximately Correct (PAC) learning algorithms when formulated in a logic. Although weaker than classical entailment, it allows for a powerful model-theoretic framework for answering queries. In this paper, we provide a new technical foundation to demonstrate PAC learning with multi-agent epistemic logics. To circumvent the negative results in the literature on the difficulty of robust learning with the PAC semantics, we consider so-called implicit learning where we are able to incorporate observations to the background theory in service of deciding the entailment of an epistemic query. We prove correctness of the learning procedure and discuss results on the sample complexity, that is how many observations we will need to provably assert that the query is entailed given a user-specified error bound. Finally, we investigate under what circumstances this algorithm can be made efficient. On the last point, given that reasoning in epistemic logics especially in multi-agent epistemic logics is PSPACE-complete, it might seem like there is no hope for this problem. We leverage some recent results on the so-called Representation Theorem explored for single-agent and multi-agent epistemic logics with the only knowing operator to reduce modal reasoning to propositional reasoning.

翻译：演绎与归纳之间的张力或许是哲学、认知科学和人工智能领域中最根本的问题。在一篇具有影响力的论文中，Valiant认识到学习挑战应与演绎相结合。具体而言，他提出了一种语义来捕捉当以逻辑形式表述时，概率近似正确（PAC）学习算法输出所具备的质量。尽管该语义弱于经典蕴涵，但它为回答查询提供了强大的模型论框架。本文提供了一个新的技术基础，以展示多智能体认知逻辑中的PAC学习。为规避文献中关于PAC语义鲁棒学习困难的负面结果，我们考虑了所谓的隐式学习，即能够将观察结果融入背景理论，以辅助判定认知查询的蕴涵关系。我们证明了学习过程的正确性，并讨论了样本复杂度的结果，即根据用户指定的误差界，我们需要多少观察结果才能可靠地断言查询被蕴涵。最后，我们探讨了在何种条件下该算法能够实现高效运行。针对最后一点，鉴于认知逻辑（尤其是多智能体认知逻辑）中的推理是PSPACE完全的，该问题似乎毫无希望。我们利用了最近关于仅知道算子的单智能体和多智能体认知逻辑中所谓表示定理的一些结果，将模态推理归约为命题推理。