In their seminal paper Artemov and Protopopescu provide Hilbert formal systems, Brower–Heyting–Kolmogorov and Kripke semantics for the logics of intuitionistic belief and knowledge. Subsequently Krupski has proved that the logic of intuitionistic knowledge is PSPACE-complete and Su and Sano have provided calculi enjoying the subformula property. This paper continues the investigations around sequent calculi for Intuitionistic Epistemic Logics by providing sequent calculi that have the subformula property and that are terminating in linear depth. Our calculi allow us to design a procedure that for invalid formulas returns a Kripke model of minimal depth. Finally we also discuss refutational sequent calculi, that is sequent calculi to prove that formulas are invalid.

在他们的开创性论文中，Artemov 和 Protopopescu 为直觉主义信念和知识的逻辑提供了 Hilbert 形式系统、Brower-Heyting-Kolmogorov 和 Kripke 语义。随后Krupski证明了直觉知识的逻辑是PSPACE-complete的，Su和Sano提供了具有子公式性质的演算。本文通过提供具有子公式属性且终止于线性深度的相继演算，继续围绕直觉认识逻辑的相继演算进行研究。我们的演算允许我们设计一个过程，该过程对于无效的公式返回最小深度的 Kripke 模型。最后我们还讨论了反驳的相继演算，即证明公式无效的相继演算。