2015/06/09 by Barbara Csima, Csima, Barbara, Matthew Harrison-Trainor +1
Mathematics · #03C57 #03D45 #FOS: Mathematics #Logic (math.LO) #math.LO #msc:03C57 #msc:03D45
paper · pdf · doi:10.48550/arxiv.1506.03066
18 pages
arxiv created 2015/06/09 · arxiv updated 2015/06/10
We investigate the complexity of isomorphisms of computable structures on cones in the Turing degrees. We show that, on a cone, every structure has a strong degree of categoricity, and that degree of categoricity is \bf0(α) for some α. To prove this, we extend Montalbán's η-system framework to deal with limit ordinals in a more general way. We also show that, for any fixed computable structure, there is an ordinal α and a cone in the Turing degrees such that the exact complexity of computing an isomorphism between the given structure and another copy B in the cone is a c.e. degree in Δ0α(B). In each of our theorems the cone in question is clearly described in the beginning of the proof, so it is easy to see how the theorems can be viewed as general theorems with certain effectiveness conditions.