Degrees of categoricity of computable structures

Defining the degree of categoricity of a computable structure ℳ to be the least degree d for which ℳ is d -computably categorical, we investigate which Turing degrees can be realized as degrees of categoricity. We show that for all n , degrees d.c.e. in and above 0 ( n ) can be so realized, as can the degree 0 ( ω ) .