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The Topological Setting

  • Christopher S. Hardin
  • Alan D. Taylor
Chapter
Part of the Developments in Mathematics book series (DEVM, volume 33)

Abstract

In this chapter we start to move further away from the hat problem metaphor and think instead of trying to predict a function’s value at a point based on knowing (something about) its values on nearby points. The most natural setting for this is a topological space and if we wanted to only consider continuous colorings, then the limit operator would serve as a unique optimal predictor. But we want to consider arbitrary colorings. Thus we have each point in a topological space representing an agent, and if f and g are two colorings, then f and g are indistinguishable to agent a if f and g agree on some deleted neighborhood of the point a. It turns out that an optimal predictor in this case is wrong only on a set that is “scattered” (a concept with origins going back to Cantor).

To illustrate one corollary of this topological result, consider the hat problem in which the agents are indexed by real numbers, and each agent sees the hats worn by those to his left (that is, those indexed by smaller real numbers). The set of hat colors is some arbitrary set K. The question is whether or not there is a predictor ensuring that the set of agents guessing incorrectly is a small infinite set—e.g., indexed by a set of reals that is countable and nowhere dense. The answer here (again assuming the axiom of choice) is yes. In fact, there is a predictor guaranteeing the set of agent guessing incorrectly is a set of reals that is well ordered by the usual ordering of the reals. Moreover, this is an optimal predictor.

Keywords

Topological Space Binary Relation Inductive Inference Winning Strategy Basic Neighborhood 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Bibliography

  1. [DG76]
    Daviesm, R.O., Galvin, F.: Solution to query 5. Real Anal. Exch. 2, 74–75 (1976)Google Scholar
  2. [FHMV95]
    Fagin, R., Halpern, J.Y., Moses, Y., Vardi, M.Y.: Reasoning About Knowledge. MIT, Cambridge (1995)zbMATHGoogle Scholar
  3. [Fre90]
    Freiling, C.: Symmetric derivates, scattered, and semi-scattered sets. Trans. Am. Math. Soc. 318, 705–720 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [Geo07]
    George, A.: A proof of induction? Philos. Impr. 7(2), 1–5 (2007)Google Scholar
  5. [HT08b]
    Hardin, C.S., Taylor, A.D.: A peculiar connection between the axiom of choice and predicting the future. Am. Math. Mon. 115(2), 91–96 (2008)MathSciNetzbMATHGoogle Scholar
  6. [HT09]
    Hardin, C.S., Taylor, A.D.: Limit-like predictability for discontinuous functions. Proc. AMS 137, 3123–3128 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [HT10]
    Hardin, C.S., Taylor, A.D.: Minimal predictors in hat problems. Fundam. Math. 208, 273–285 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [HW13]
    Horsten, L., Welch, P.: The aftermath. Math. Intell. 35(1), 16–20 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [Pas08]
    Paseau, A.: Justifying induction mathematically: strategies and functions. Log. Anal. 203, 263–269 (2008)MathSciNetGoogle Scholar
  10. [Mor90]
    Morgan II, J.: Point Set Theory. Marcel Dekker, Inc., New York (1990)zbMATHGoogle Scholar
  11. [Yip94]
    Yiparaki, O.: On some tree partitions. PhD thesis, University of Michigan (1994)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Christopher S. Hardin
    • 1
  • Alan D. Taylor
    • 2
  1. 1.New YorkUSA
  2. 2.Department of MathematicsUnion CollegeSchenectadyUSA

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