THE COMPLEXITY OF INDEX SETS OF CLASSES OF COMPUTABLY ENUMERABLE EQUIVALENCE RELATIONS

Abstract

AbstractLet$ \le _c $be computable the reducibility on computably enumerable equivalence relations (or ceers). We show that for every ceerRwith infinitely many equivalence classes, the index sets$\left\{ {i:R_i \le _c R} \right\}$(withRnonuniversal),$\left\{ {i:R_i \ge _c R} \right\}$, and$\left\{ {i:R_i \equiv _c R} \right\}$are${\rm{\Sigma }}_3^0$complete, whereas in caseRhas only finitely many equivalence classes, we have that$\left\{ {i:R_i \le _c R} \right\}$is${\rm{\Pi }}_2^0$complete, and$\left\{ {i:R \ge _c R} \right\}$(withRhaving at least two distinct equivalence classes) is${\rm{\Sigma }}_2^0$complete. Next, solving an open problem from [1], we prove that the index set of the effectively inseparable ceers is${\rm{\Pi }}_4^0$complete. Finally, we prove that the 1-reducibility preordering on c.e. sets is a${\rm{\Sigma }}_3^0$complete preordering relation, a fact that is used to show that the preordering relation$ \le _c $on ceers is a${\rm{\Sigma }}_3^0$complete preordering relation.