# Infinite time decidable equivalence relation theory

• S. Coskey and J. D. Hamkins, “Infinite time decidable equivalence relation theory,” Notre Dame Journal of Formal Logic, vol. 52, iss. 2, p. 203–228, 2011.
[Bibtex]
@ARTICLE{CoskeyHamkins2011:InfiniteTimeComputableEquivalenceRelations,
AUTHOR = {Coskey, Samuel and Hamkins, Joel David},
TITLE = {Infinite time decidable equivalence relation theory},
JOURNAL = {Notre Dame Journal of Formal Logic},
FJOURNAL = {Notre Dame Journal of Formal Logic},
VOLUME = {52},
YEAR = {2011},
NUMBER = {2},
PAGES = {203--228},
ISSN = {0029-4527},
MRCLASS = {03D65 (03D30 03E15)},
MRNUMBER = {2794652},
DOI = {10.1215/00294527-1306199},
URL = {http://wp.me/p5M0LV-3M},
eprint = "0910.4616",
archivePrefix = {arXiv},
primaryClass = {math.LO},
}

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.