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

With Joel Hamkins. Appears in *Effective mathematics of the uncountable*, ASL Lecture Notes in Logic, 2014. (arχiv | journal)

*Abstract*: 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 $\Undertilde\Delta^1_2$ in a satisfying way.

*Further notes*: This article summarizes and contextualizes my previous work here with Joel, as well as other results in the field.