Abstract
We give a definition of bisimulation for conditional modalities interpreted on selection functions and prove the correspondence between bisimilarity and modal equivalence, generalizing the Hennessy–Milner Theorem to a wide class of conditional operators. We further investigate the operators and semantics to which these results apply. First, we show how to derive a solid notion of bisimulation for conditional belief, behaving as desired both on plausibility models and on evidence models. These novel definitions of bisimulations are exploited in a series of undefinability results. Second, we treat relativized common knowledge, underlining how the same results still hold for a different modality in a different semantics. Third, we show the flexibility of the approach by generalizing it to multi-agent systems, encompassing the case of multi-agent plausibility models.
Highlights
The Modal Logic literature offers a number of examples of conditional modalities, developed for a variety of reasons: conditionals from conditional logic, conditional belief, relativized common knowledge, to name a few
The definition of such bisimulation should be in principle independent from additional parts of the structure that do not appear in the semantics of the conditional modality: two states should be indistinguishable only if they behave in the same way with respect to the features that the conditional modality can “detect”
The formula α in the language of conditional belief will be invariant between states that are bisimilar according to a CB-bisimulation; Sbp is true in the first model at 1 but false in the second model at 4, α will be true in one world and not in the other: contradiction
Summary
The Modal Logic literature offers a number of examples of conditional modalities, developed for a variety of reasons: conditionals from conditional logic, conditional belief, relativized common knowledge, to name a few. One wants to characterize exactly when two models are indistinguishable by means of a conditional modality Such result is not the end of the story, a well behaved notion of bisimulation should satisfy the following list of desiderata: 1. The definition of such bisimulation should be in principle independent from additional parts of the structure that do not appear in the semantics of the conditional modality: two states should be indistinguishable only if they behave in the same way with respect to the features that the conditional modality can “detect”. We demonstrate that it applies to the same operator interpreted on different semantics (as for point 4 in our list), discussing how this approach provides a solid notion of bisimulation for conditional belief. We explain how the central definition and results are amenable for a multi-agent generalization
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
