Muddy Waters
Hans van Ditmarsch
TL;DR
The paper presents Mützen, a Christmas-themed hat puzzle extending muddy children, and develops a rigorous epistemic-logical formalization using public announcement logic (PAL) with fixpoints and, optionally, public assignments. It introduces solvable and solvable' notions to capture the informative content of the initial announcement and explores how iterative announcements (the bell) propagate knowledge to eliminate ignorance. Through a detailed model of 126 gnomes and 127 colors, the work demonstrates both a solvability condition and a method to express it with fixpoints, while addressing the complexities of non-monotonicity and color-permutation invariance. The result is a structured framework for encoding complex knowledge puzzles within dynamic epistemic logic, highlighting the interplay between announcements, fixpoints, and assignments and illustrating how such puzzles can be analyzed formally. This contributes to the understanding of how public information interacts with epistemic updates in multi-agent settings and provides a blueprint for similar logico-algebraic encodings of interactive puzzles.
Abstract
In the 2013 Advent calender of the Berlin Mathematics Research Center MATH+, Gerhard Woeginger presents a novel hat problem with an uncommon initial announcement. Although the information given is insufficient for the hat bearers to learn their colour, they are informed that the colours have been chosen so that they can learn their colour. We formalize this announcement in public announcement logic and in an extension of public announcement logic with fixpoints.