Abstract
Past public announcement operators have been defined in Hoshi and Yap (Synthese 169(2):259–281, 2009) and Yap (Dynamic logic montréal, 2007), to describe an agent’s knowledge before an announcement occurs. These operators rely on branching-time structures that do not mirror the traditional, relativization-based semantics of public announcement logic (PAL), and favor a historical reading of past announcements. In this paper, we introduce reverse public announcement operators that are interpreted on expanded models. Our model expansion adds accessibility links from an epistemic model \(\mathcal {M}\) to a filtrated submodel of the canonical model for \(\mathbf K _g\). Here \(\mathbf K _g\) is the minimal normal modal logic together with \(\mathbf S5 \) axioms for the universal operator U. This yields a highly general pre-announcement version of \(\mathcal {M}\) that makes our operators potentially useful for studying non-standard interpretations of rescinded announcements in PAL. Indeed, we find that our reverse announcement operators cannot be represented by product update, and that they have an intimate connection with the knowledge forgetting of Zhang and Zhou (Artif Intell J 173(16–17):1525–1537, 2009). We show that the logic resulting from adding reverse announcements to PAL is sound and complete.
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Notes
In the definition of \(\mathcal {H} = (H,\sim ,V)\), we have that H is a set of histories, \(\sim :N\rightarrow \mathcal {P}(H\times H)\) and \(V:P\rightarrow \mathcal {P}(H)\), where P is the set of propositional letters and N is the set of agents as before. The models considered in Hoshi and Yap (2009) are special in that they are branching-time structures generated via repeated product update.
Calling \(\psi \) the non-effect clause reflects the fact that \(\psi \) constrains the effect that \(!\theta ^{-1}\) has on the model by fixing the truth of \(U\psi \).
Some restrictions on \(\psi \) will be formulated in the next section.
So for example \(\Diamond _i(p\wedge q)\) is a prime formula of degree 1.
We can replace \(\mathbf K _g\) with \(\mathbf T _g\) (the logic of reflexive frames with U) in the definitions of this section. This would be closer to traditional logics of knowledge, but it is not necessary for our purposes. \(\mathbf S5 _g\) cannot be used, as will become apparent.
Note in particular that we are using one specific filtration of \(\mathcal {K}'\) through \({\varTheta }\). This is in fact the smallest filtration of \(\mathcal {K}'\) through \({\varTheta }\), in the sense that for any other filtration \((W^K_{{\varTheta }},R^f,V_{{\varTheta }}^K)\), we have that \((R_{{\varTheta }}^K)_i\subset R^f_i\) for all \(i\in N\). One can also find a proof of this in Blackburn et al. (2001). We prefer this filtration both because it is easy to use in computations, and (being the smallest filtration) it preserves only the most essential information carried by the canonical model.
It is unreasonable to expect that \([!\theta ^{-1},\psi ][!\theta ]\phi \leftrightarrow \phi \) is valid. For example if \(\theta \) is a propositional letter p, \(\phi \) is Up and \(\psi \) is \(\top \), this formula is clearly invalid.
In the language of Zhang and Zhou (2009), this says that \(\sigma \) is irrelevant to \(\theta \).
This means that there is a binary relation \(\cong \) on \(W^{\mathcal {M}}\times W^{\mathcal {N}}\) which satisfies all of the conditions to be a bisimulation between \(\mathcal {M}\) and \(\mathcal {N}\), except that if \(u\cong u'\), where \(u\in W^{\mathcal {M}}\) and \(u'\in W^{\mathcal {N}}\), then it is not necessarily the case that \(\mathcal {M},u\models p^*\) iff \(\mathcal {N},u'\models p^*\).
A maximally consistent conjunction of literals is a conjunction of literals which contains either p or \(\lnot p\) for all \(p\in P\).
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Acknowledgements
I would like to thank Dr. Tomohiro Hoshi for his extensive guidance and commentary as I wrote this paper. I am grateful to him for introducing me to logic, and I was very lucky to have him as a mentor. I would also like to thank my reviewers for their comments on this paper, which helped to improve it significantly.
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Haney, R.S. Reverse Public Announcement Operators on Expanded Models. J of Log Lang and Inf 27, 205–224 (2018). https://doi.org/10.1007/s10849-018-9265-7
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DOI: https://doi.org/10.1007/s10849-018-9265-7