Infinite time decidable equivalence relation theory

Details Infinite time decidable equivalence relation theory Infinite time decidable equivalence relation theory

2009-10-24

arxiv.org/abs/0910.4616v1

Notre Dame J. Formal Logic Volume 52, Number 2 (2011), 203-228

Mathematics

Logic

30 pages, 3 figures

Scientific paper

10.1215/00294527-1306199

We introduce an analog of the theory of Borel equivalence relations in which we study equivalence relations that are decidable by an infinite time Turing machine. The Borel reductions are replaced by the more general class of infinite time computable functions. Many basic aspects of the classical theory remain intact, with the added bonus that it becomes sensible to study some special equivalence relations whose complexity is beyond Borel or even analytic. We also introduce an infinite time generalization of the countable Borel equivalence relations, a key subclass of the Borel equivalence relations, and again show that several key properties carry over to the larger class. Lastly, we collect together several results from the literature regarding Borel reducibility which apply also to absolutely Delta_1^2 reductions, and hence to the infinite time computable reductions.

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