# Infinite time Turing machines and an application to the hierarchy of equivalence relations on the reals

• S. Coskey and J. D. Hamkins, “Infinite time Turing machines and an application to the hierarchy of equivalence relations on the reals,” in Effective mathematics of the uncountable, Assoc. Symbol. Logic, La Jolla, CA, 2013, vol. 41, pp. 33-49.
@incollection {CoskeyHamkins2013:ITTMandApplicationsToEquivRelations,
AUTHOR = {Coskey, Samuel and Hamkins, Joel David},
TITLE = {Infinite time {T}uring machines and an application to the
hierarchy of equivalence relations on the reals},
BOOKTITLE = {Effective mathematics of the uncountable},
SERIES = {Lect. Notes Log.},
VOLUME = {41},
PAGES = {33--49},
PUBLISHER = {Assoc. Symbol. Logic, La Jolla, CA},
YEAR = {2013},
MRCLASS = {03D30 (03D60 03E15)},
MRNUMBER = {3205053},
eprint = {1101.1864},
archivePrefix = {arXiv},
primaryClass = {math.LO},
url = {http://jdh.hamkins.org/ittms-and-applications/},
}

We describe the basic theory of infinite time Turing machines and some recent developments, including the infinite time degree theory, infinite time complexity theory, and infinite time computable model theory. We focus particularly on the application of infinite time Turing machines to the analysis of the hierarchy of equivalence relations on the reals, in analogy with the theory arising from Borel reducibility. We define a notion of infinite time reducibility, which lifts much of the Borel theory into the class ${\Delta}^1_2$ in a satisfying way.