Complexity of equivalence relations and preorders from computability theory

TL;DR

This paper investigates the complexity levels of equivalence relations and preorders in computability and complexity theory, establishing completeness results for various classes and relations.

Contribution

It demonstrates the existence of a $\Pi_1$-complete equivalence relation and shows natural preorders are $\Sigma_k$-complete, clarifying their relative computational complexities.

Findings

Existence of a $\Pi_1$-complete equivalence relation.

No $\Pi_k$-complete equivalence relations for $k extgreater 1$.

Natural $\Sigma_k$ preorders are $\Sigma_k$-complete.

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 $\Sigma 4$.