Infinite Time Decidable Equivalence Relation Theory

Notre Dame Journal of Formal Logic 52 (2):203-228 (2011)
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Abstract

We introduce an analogue 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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Joel David Hamkins
Oxford University

Citations of this work

Infinite Computations with Random Oracles.Merlin Carl & Philipp Schlicht - 2017 - Notre Dame Journal of Formal Logic 58 (2):249-270.

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References found in this work

The Higher Infinite.Akihiro Kanamori - 2000 - Studia Logica 65 (3):443-446.
Infinite time Turing machines.Joel David Hamkins & Andy Lewis - 2000 - Journal of Symbolic Logic 65 (2):567-604.
A simple maximality principle.Joel David Hamkins - 2003 - Journal of Symbolic Logic 68 (2):527-550.
Countable borel equivalence relations.S. Jackson, A. S. Kechris & A. Louveau - 2002 - Journal of Mathematical Logic 2 (01):1-80.

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