[论文解读] Logics for Epistemic Actions: Completeness, Decidability, Expressivity

Alexandru Baltag, Lawrence S. Moss+1 more|arXiv (Cornell University)|Mar 13, 2022

Logic, Reasoning, and Knowledge被引用 4

一句话总结

本文研究带有动作签名的动态认识逻辑，证明了${\cal L}_0({\mathbf{\Sigma}})$的强完备性以及${\cal L}_1({\mathbf{\Sigma}})$的弱完备性，同时通过模态过滤和PDL翻译证明了其可判定性。此外，本文还表明，包含私有公告的逻辑比仅包含公开公告的逻辑更具表达力。

ABSTRACT

We consider dynamic versions of epistemic logic as formulated in Baltag and Moss "Logics for epistemic programs" (2004). That paper proposed a logical language (actually families of languages parameterized by action signatures) for dynamic epistemic logic. It had been shown that validity in the language is Pi-1-1-complete, so there are no recursively axiomatized complete logical systems for it. In contrast, this paper proves a weak completeness result for the fragment without action iteration, and a strong completeness result for the fragment without action iteration and common knowledge. Our work involves a detour into term rewriting theory. The argument uses modal filtration, and thus we obtain the finite model property and hence decidability. We also give a translation of our largest language into PDL, thereby obtaining a second proof of decidability. The paper closes with some results on expressive power. These are mostly concerned with comparing the action-iteration-free language with modal logic augmented by transitive closure operators. We answer a natural question about the languages we obtain by varying the action signature: we prove that a logical language with operators for private announcements is more expressive than one for public announcements.

研究动机与目标

• 建立带有动作签名的动态认识逻辑片段的完备性与可判定性结果。
• 分析不同动作类型（如公开公告与私有公告）的逻辑的表达能力。
• 解决动态认识逻辑中关于公理化与模型性质的开放问题。
• 证明私有公告逻辑严格比公开公告逻辑更具表达力。

提出的方法

• 通过标准动作模型与一种新颖的动作规则，证明${\cal L}_0({\mathbf{\Sigma}})$的强完备性。
• 通过公式上的语法正规形式与良好序关系，建立${\cal L}_1({\mathbf{\Sigma}})$的弱完备性。
• 应用模态过滤，推导出有限模型性质，从而证明${\cal L}_1({\mathbf{\Sigma}})$的可判定性。
• 构建${\cal L}_1({\mathbf{\Sigma}})$到命题动态逻辑（PDL）的翻译，提供可判定性的第二种证明。
• 利用双生态与模型构造方法，比较不同动作签名逻辑之间的表达力。
• 分析迭代算子（如$[\text{Pub } \Diamond{\sf true}]^*$）对模型有限性与可满足性的影响。

实验结果

研究问题

• RQ1 逻辑${\cal L}_0({\mathbf{\Sigma}})$在其标准语义下是否具有强完备性？
• RQ2 ${\cal L}_1({\mathbf{\Sigma}})$是否具有弱完备性，且是否可判定？
• RQ3 ${\cal L}_1({\mathbf{\Sigma}})$能否被忠实翻译为PDL，这又对可判定性意味着什么？
• RQ4 包含私有公告的逻辑是否比仅包含公开公告的逻辑更具表达力？
• RQ5 迭代认识动作（如$[\text{Pub } \Diamond{\sf true}]^*$）的存在是否导致有限模型性质失效？

主要发现

• 通过标准动作模型构造与新动作规则，证明了逻辑${\cal L}_0({\mathbf{\Sigma}})$具有强完备性。
• ${\cal L}_1({\mathbf{\Sigma}})$的弱完备性依赖于语法正规形式与公式上的良好序关系。
• 通过模态过滤，${\cal L}_1({\mathbf{\Sigma}})$的可判定性得以确立，从而获得有限模型性质。
• 将${\cal L}_1({\mathbf{\Sigma}})$翻译为PDL，提供了可判定性的第二种证明。
• 带有私有公告的逻辑${\cal L}_1({\mathbf{\Sigma}}{\mbox{\scriptsize pri}})$严格比仅含公开公告的逻辑${\cal L}_1({\mathbf{\Sigma}}{\mbox{\scriptsize pub}})$更具表达力。
• 逻辑${\cal L}({\mathbf{\Sigma}}{\mbox{\scriptsize pub}})$不具有有限模型性质，如句子$[\text{Pub } \Diamond{\sf true}]^*\Diamond\Box{\sf false}$仅在无限模型中可满足。