Finitary Reducibility on Equivalence Relations

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 comp...

Full description

Saved in:
Bibliographic Details
Main Authors: Miller, Russell, Ng, Keng Meng
Other Authors: School of Physical and Mathematical Sciences
Format: Article
Language:English
Published: 2016
Subjects:
Online Access:https://hdl.handle.net/10356/84705
http://hdl.handle.net/10220/41955
Tags: Add Tag
No Tags, Be the first to tag this record!
Institution: Nanyang Technological University
Language: English
Description
Summary: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.