COMPLEXITY OF EQUIVALENCE RELATIONS AND PREORDERS FROM COMPUTABILITY THEORY

@article{Ianovski2014COMPLEXITYOE,
  title={COMPLEXITY OF EQUIVALENCE RELATIONS AND PREORDERS FROM COMPUTABILITY THEORY},
  author={Egor Ianovski and Russell G. Miller and Keng Meng Ng and Andr{\'e} Nies},
  journal={The Journal of Symbolic Logic},
  year={2014},
  volume={79},
  pages={859 - 881}
}
Abstract We study the relative complexity of equivalence relations and preorders from computability theory and complexity theory. Given binary relations R, S, a componentwise reducibility is defined by R ≤ S ⇔ ∃f ∀x, y [x R y ↔ f (x) S f (y)]. Here, f is taken from a suitable class of effective functions. For us the relations will be on natural numbers, and f must be computable. We show that there is a ${\rm{\Pi }}_1^0$-complete equivalence relation, but no ${\rm{\Pi }}_k^0$-complete for k ≥ 2… 
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