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. "
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