We obtain modal completeness of the interpretability logics IL $\!\!\textsf {P}_{\textsf {0}}$ and ILR w.r.t. generalised Veltman semantics. Our proofs are based on the notion of full labels [2]. We also give shorter proofs of completeness w.r.t. the generalised semantics for many classical interpretability logics. We obtain decidability and finite model property w.r.t. the generalised semantics for IL $\textsf {P}_{\textsf {0}}$ and ILR. Finally, we develop a construction that might be useful for proofs of completeness of extensions of ILW w.r.t. the generalised semantics in the future, and demonstrate its usage with $\textbf {IL}\textsf {W}^\ast = \textbf {IL}\textsf {WM}_{\textsf {0}}$ .

中文翻译：

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可解释性逻辑和广义 VELTMAN 语义

我们获得了可解释性逻辑的模态完整性伊利诺伊州$\!\!\textsf {P}_{\textsf {0}}$和伊利诺伊州Rwrt 广义 Veltman 语义。我们的证明基于完整标签的概念[2]。我们还针对许多经典可解释性逻辑的广义语义给出了更短的完整性证明。我们通过广义语义获得可判定性和有限模型属性伊利诺伊州$\textsf {P}_{\textsf {0}}$和伊利诺伊州R. 最后，我们开发了一个可能对证明扩展完整性有用的结构伊利诺伊州Wwrt 未来的广义语义，并展示它的用法$\textbf {IL}\textsf {W}^\ast = \textbf {IL}\textsf {WM}_{\textsf {0}}$.