Developments in Mathematics

The Mathematics of Coordinated Inference

A Study of Generalized Hat Problems

Authors: Hardin, Christopher S., Taylor, Alan D.

  • ​Presents a comprehensive treatment of material previously available in journals only
  • Contains a number of new results and extensions of known results
  • States a number of open and accessible problems  
  • Unified notation is used for a cohesive presentation
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About this book

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.

Reviews

From the book reviews:

“The book presents, in a unified way, attractive topics in graph theory, topology, and set theory that all relate to the dilemma faced by Alice and Bob and others in hat problems. The first few chapters are of great general interest as they summarize hat problems that any mathematician can understand. The later chapters will be of interest to those well versed in set theory or certain aspects of point-set topology.” (Stan Wagon, Mathematical Reviews, October, 2014)

Table of contents (9 chapters)

  • Introduction

    Hardin, Christopher S. (et al.)

    Pages 1-9

  • The Finite Setting

    Hardin, Christopher S. (et al.)

    Pages 11-18

  • The Denumerable Setting: Full Visibility

    Hardin, Christopher S. (et al.)

    Pages 19-27

  • The Denumerable Setting: One-Way Visibility

    Hardin, Christopher S. (et al.)

    Pages 29-47

  • Dual Hat Problems, Ideals, and the Uncountable

    Hardin, Christopher S. (et al.)

    Pages 49-59

Buy this book

eBook $79.99
price for USA (gross)
  • ISBN 978-3-319-01333-6
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $109.00
price for USA
  • ISBN 978-3-319-01332-9
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $109.00
price for USA
  • ISBN 978-3-319-37605-9
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
The Mathematics of Coordinated Inference
Book Subtitle
A Study of Generalized Hat Problems
Authors
Series Title
Developments in Mathematics
Series Volume
33
Copyright
2013
Publisher
Springer International Publishing
Copyright Holder
Springer International Publishing Switzerland
eBook ISBN
978-3-319-01333-6
DOI
10.1007/978-3-319-01333-6
Hardcover ISBN
978-3-319-01332-9
Softcover ISBN
978-3-319-37605-9
Series ISSN
1389-2177
Edition Number
1
Number of Pages
XI, 109
Topics