In M. Kanazawa (ed.), Proceedings of the 12th Meeting on Mathematics of Language, Lecture Notes in Artificial Intelligence 6878. Springer (2011)
|Abstract||The paper presents two case studies of multi-agent information exchange involving generalized quantiﬁers. We focus on scenarios in which agents successfully converge to knowledge on the basis of the information about the knowledge of others, so-called Muddy Children puzzle and Top Hat puzzle. We investigate the relationship between certain invariance properties of quantiﬁers and the successful convergence to knowledge in such situations. We generalize the scenarios to account for public announcements with arbitrary quantiﬁers. We show that the Muddy Children puzzle is solvable for any number of agents if and only if the quantiﬁer in the announcement is positively active (satisﬁes a version of the variety condition). In order to get the characterization result, we propose a new concise logical modeling of the puzzle based on the number triangle representation of generalized quantiﬁers. In a similar vein, we also study the Top Hat puzzle. We observe that in this case an announcement needs to satisfy stronger conditions in order to guarantee solvability. Hence, we introduce a new property, called bounded thickness, and show that the solvability of the Top Hat puzzle for arbitrary number of agents is equivalent to the announcement being 1-thick.|
|Keywords||Muddy Children Puzzle Top Hat Puzzle generalized quantifiers number triangle epistemic logic|
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