Complexity of equivalence relations and preorders from computability theory
Ng, Keng Meng
Date of Issue2014
School of Physical and Mathematical Sciences
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 de ned by R ≤ S ⇔ ∃f ∀x, y [xRy ↔ 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 ∏ 0 1-complete equivalence relation, but no ∏ 0 k-complete for k≥2. We show that ∑ 0 k preorders arising naturally in the abovementioned areas are ∑ 0 k-complete. This includes polynomial time m-reducibility on exponential time sets, which is ∑ 0 2, almost inclusion on r.e. sets, which is ∑ 0 3, and Turing reducibility on r.e. sets, which is ∑ 0 4 .
The journal of symbolic logic
© 2014 Association for Symbolic Logic. This is the author created version of a work that has been peer reviewed and accepted for publication by The Journal of Symbolic Logic, Association for Symbolic Logic. It incorporates referee’s comments but changes resulting from the publishing process, such as copyediting, structural formatting, may not be reflected in this document. The published version is available at: [http://dx.doi.org/10.1017/jsl.2013.33].