AI 中文总结
研究Agotnes等人关于σ有效性的猜想，通过重新表述问题，对多主体K45、单主体KD45和多主体S5进行了分类，证明原猜想错误，揭示真假宣告的不对称性。
AI 中文摘要
2018年Agotnes等人将公共宣告中的成功和自反驳概念扩展到真谎言、不可能谎言及一般的σ有效性（σ为0和1的有限或无限序列）。他们提出关于此类序列按σ有效性分类的猜想。本文反驳该猜想，对多主体K45、单主体KD45和多主体S5重新明确表述后给出修正分类，多主体KD45多主体情况未解决。结果表明真假宣告存在不对称性。
英文摘要
In their 2018 paper, Agotnes, van Ditmarsch, and Wang extended the notions of success and self-refutation in public announcements to true lies, impossible lies, and $σ$-validity in general. Here, $σ$ is a finite or infinite sequence of $0$s and $1$s. For example, successful formulas and self-refuting formulas are $11$-valid and $10$-valid, respectively. They then posed a conjecture on the classification of such sequences in terms of $σ$-validity. In this paper, we disprove the conjecture and give corrected classifications for multi-agent K45, single-agent KD45, multi-agent KD45 with more than one agent, and multi-agent S5 after reformulating the statement more explicitly. The results indicate that there is an asymmetry between truthful announcements and false announcements: the former are stable while the latter are fragile in general. In particular, all successful formulas remain true forever while some impossible lies can be true at some point when repeatedly announced. Also, although some self-refuting formulas can become true again after following the truth pattern $10$, all $100$-valid formulas are destructive in the sense that they remain false forever once they become false. On the other hand, some true lies are fragile in the sense that truths created by lying can become false again.
Comments21 pages, 4 figures. v2: Added proofs of the nonexistence of non-trivially 0k1-valid and 01k0-valid formulas in multi-agent KD45, thereby completing the classification for this class. Revised related statements and proofs throughout. v3: Added a link to the GitHub repository containing the Lean 4 formalization
