We prove that for every Borel equivalence relation E, either E is Borel reducible to E0, or the family of Borel equivalence relations incompatible with E has cofinal essential complexity. It follows that if F is a Borel equivalence relation and F is a family of Borel equivalence relations of non-cofinal essential complexity which together satisfy the dichotomy that for every Borel equivalence r...

A formalisation of soundness of the notion of α-equivalence in nominal abstract syntax modulo associative (A) and associative-commutative (AC) equational theories is described. Initially, the notion of α-equivalence is specified based on a so called “weak” nominal relation as suggested by Urban in his nominal development in Isabelle/HOL. Then, it is formalised in Coq that this equality is indee...

We study the notion of polynomial-time relation reducibility among computable equivalence relations. We identify some benchmark equivalence relations and show that the reducibility hierarchy has a rich structure. Specifically, we embed the partial order of all polynomial-time computable sets into the polynomial-time relation reducibility hierarchy between two benchmark equivalence relations Eλ ...

See [Oli04]. In [AdKech00], Adams and Kechris showed that the relation of equality on Borel sets (and therefore, any Borel equivalence relation whatsoever) is Borel reducible to the equivalence relation of Borel bireducibility. (In somewhat finer terms, they showed that the partial order of inclusion on Borel sets is Borel reducible to the quasi-order of Borel reducibility.) Their technique was...