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A game for the Borel functions
Semmes, B.T.
Publication date 2009
Link to publication
Citation for published version (APA):
Semmes, B. T. (2009). A game for the Borel functions. Institute for Logic, Language and Computation.
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Chapter 6
Conclusion
We have seen a number of games in this thesis. In Chapter 2, we saw the tree game and its characterization of the Borel functions. In the second part of the thesis, we saw more games for certain subclasses of Borel functions, and we saw that they can be used to prove decomposition theorems.
The question nautrally arises: can we obtain a result for more general sub-classes of Borel functions? It is hoped that the game-theoretic tools we have developed in this thesis can be generalized to obtain a more elegant characteri-zation theorem. In particular, all of the games we have looked at can be viewed as restricted tree games. The Wadge game can be viewed as the restricted tree game in which Player II is required to produceφ such that dom(φ) is linear; for the eraser game, we require that dom(φ) is finitely branching; for the backtrack game, we require that dom(φ) branches finitely at the root and is linear therafter; for the gameG2,3, we require that dom(φ) may branch infinitely at the root but is finitely branching thereafter; and for the multitape game, we require that dom(φ) may branch infinitely at the root but is linear thereafter.
Thus, it would seem natural to come up with more general restrictions on dom(φ), and work with m’s and n’s or α’s and β’s instead of numbers between 1 and 3. (The author refuses to prove any decomposition theorems with 4’s in them.)
The tree game itself is simple and characterizes a class of functions widely considered in descriptive set theory. Going beyond the Borel functions, one might try to generalize the tree game to characterize classes of projective functions, possibly by allowing Player II to produce multiple infinite branches.
