# The Mathematics of Coordinated Inference

## A Study of Generalized Hat Problems

• Christopher S. Hardin
• Alan D. Taylor
Book

Part of the Developments in Mathematics book series (DEVM, volume 33)

1. Front Matter
Pages i-xi
2. Christopher S. Hardin, Alan D. Taylor
Pages 1-9
3. Christopher S. Hardin, Alan D. Taylor
Pages 11-18
4. Christopher S. Hardin, Alan D. Taylor
Pages 19-27
5. Christopher S. Hardin, Alan D. Taylor
Pages 29-47
6. Christopher S. Hardin, Alan D. Taylor
Pages 49-59
7. Christopher S. Hardin, Alan D. Taylor
Pages 61-69
8. Christopher S. Hardin, Alan D. Taylor
Pages 71-82
9. Christopher S. Hardin, Alan D. Taylor
Pages 83-92
10. Christopher S. Hardin, Alan D. Taylor
Pages 93-101
11. Back Matter
Pages 103-109

### Introduction

Two prisoners are told that they will be brought to a room and seated so that each can see the other. Hats will be placed on their heads; each hat is either red or green. The two prisoners must simultaneously submit a guess of their own hat color, and they both go free if at least one of them guesses correctly. While no communication is allowed once the hats have been placed, they will, however, be allowed to have a strategy session before being brought to the room. Is there a strategy ensuring their release? The answer turns out to be yes, and this is the simplest non-trivial example of a “hat problem.”

This book deals with the question of how successfully one can predict the value of an arbitrary function at one or more points of its domain based on some knowledge of its values at other points. Topics range from hat problems that are accessible to everyone willing to think hard, to some advanced topics in set theory and infinitary combinatorics. For example, there is a method of predicting the value f(a) of a function f mapping the reals to the reals, based only on knowledge of f's values on the open interval (a – 1, a), and for every such function the prediction is incorrect only on a countable set that is nowhere dense.

The monograph progresses from topics requiring fewer prerequisites to those requiring more, with most of the text being accessible to any  graduate student in mathematics. The broad range of readership  includes researchers, postdocs, and graduate students in the fields of  set theory, mathematical logic, and combinatorics, The hope is that this book will bring together mathematicians from different areas to  think about set theory via a very broad array of coordinated inference problems.

### Keywords

Gabay-O'Connor Theorem Galvin's setting Tutte-Berge formula generalized hat problems hat problem prisoners hat problem pseudo-scattered sets set theory

#### Authors and affiliations

• Christopher S. Hardin
• 1
• Alan D. Taylor
• 2
1. 1.New YorkUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-319-01333-6
• Copyright Information Springer International Publishing Switzerland 2013
• Publisher Name Springer, Cham
• eBook Packages Mathematics and Statistics
• Print ISBN 978-3-319-01332-9
• Online ISBN 978-3-319-01333-6
• Series Print ISSN 1389-2177
• Series Online ISSN 2197-795X
• Buy this book on publisher's site
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