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arXiv · 1302.0580

Complexity of equivalence relations and preorders from computability theory

Egor Ianovski·Keng Meng Ng·Russell Miller·Andre Nies
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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\le S \iff \ex f \, \forall x, y \, [xRy \lra f(x) Sf(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 $\Pi_1$-complete equivalence relation, but no $\Pi k$-complete for $k \ge 2$. We show that $\Sigma k$ preorders arising naturally in the above-mentioned areas are $\Sigma k$-complete. This includes polynomial time $m$-reducibility on exponential time sets, which is $\Sigma 2$, almost inclusion on r.e.\ sets, which is $\Sigma 3$, and Turing reducibility on r.e.\ sets, which is $\Sigma 4$.

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Egor Ianovski, Keng Meng Ng, Russell Miller, Andre Nies. 2013-02-04. Complexity of equivalence relations and preorders from computability theory. https://arxiv.org/abs/1302.0580

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