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

Published online by Cambridge University Press:  01 December 2016

URI ANDREWS  and
ANDREA SORBI
URI ANDREWS
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF WISCONSIN MADISON, WI 53706-1388, USAE-mail: andrews@math.wisc.eduURL: http://www.math.wisc.edu/∼andrews/
ANDREA SORBI
Affiliation:
DIPARTIMENTO DI INGEGNERIA INFORMATICA E SCIENZE MATEMATICHE UNIVERSITÀ DEGLI STUDI DI SIENA I-53100 SIENA, ITALYE-mail: andrea.sorbi@unisi.itURL: http://www3.diism.unisi.it/∼sorbi/
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Abstract

Let $ \le _c $ be computable the reducibility on computably enumerable equivalence relations (or ceers). We show that for every ceer R with infinitely many equivalence classes, the index sets $\left\{ {i:R_i \le _c R} \right\}$ (with R nonuniversal), $\left\{ {i:R_i \ge _c R} \right\}$, and $\left\{ {i:R_i \equiv _c R} \right\}$ are ${\rm{\Sigma }}_3^0$ complete, whereas in case R has only finitely many equivalence classes, we have that $\left\{ {i:R_i \le _c R} \right\}$ is ${\rm{\Pi }}_2^0$ complete, and $\left\{ {i:R \ge _c R} \right\}$ (with R having 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.

Keywords

03D25computably enumerable equivalence relationcomputable reducibility on equivalence relations
Type
Articles
Information
The Journal of Symbolic Logic , Volume 81 , Issue 4 , December 2016 , pp. 1375 - 1395
Copyright
Copyright © The Association for Symbolic Logic 2016 

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