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We prove that the ∆2 Turing degrees have a finite automorphism base. We apply this result to show that the automorphism group of DT (≤ 0′) is countable and that all its members have arithmetic presentations. We prove that every relation on DT (≤ 0′) induced by an arithmetically definable degree invariant relation is definable with finitely many ∆2 parameters and show that rigidity for DT (≤ 0′) is equivalent to its biinterpretability with first order arithmetic.

The $\Delta ^0_2$ Turing degrees: Automorphisms and Definability