Finitary Reducibility on Equivalence Relations
Ng, Keng Meng
Date of Issue2016
School of Physical and Mathematical Sciences
We introduce the notion of finitary computable reducibility on equivalence relations on the domain ω. This is a weakening of the usual notion of computable reducibility, and we show it to be distinct in several ways. In particular, whereas no equivalence relation can be Π_(n+2)^0-complete under computable reducibility, we show that, for every n, there does exist a natural equivalence relation which is Π_(n+2)^0-complete under finitary reducibility. We also show that our hierarchy of finitary reducibilities does not collapse, and illustrate how it sharpens certain known results. Along the way, we present several new results which use computable reducibility to establish the complexity of various naturally defined equivalence relations in the arithmetical hierarchy.
The Journal of Symbolic Logic
© 2016 Association for Symbolic Logic. This paper was published in The Journal of Symbolic Logic and is made available as an electronic reprint (preprint) with permission of Association for Symbolic Logic. The published version is available at: [http://dx.doi.org/10.1017/jsl.2016.23]. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper is prohibited and is subject to penalties under law.