Abstract complexity theory and the degrees of unsolvability
Carl G. Jockusch, Benjamin James Schaeffer · 1998
We focus on the $\Delta\sbsp{2}{0}$ sets and show that abstract complexity theory can be applied to the degrees of unsolvability simply by relativizing the notion of a complexity measure to ${\bf 0\prime}.$ Since the Gap Theorem holds in our context, we need to develop the concept of a $\Delta\sbsp{2}{0}$ honest function just as one needs to define honest functions in computational complexity theory. These functions turn out to be the appropriate complexity bounds, and the concept enables us to prove general hierarchy results for $\Delta\sbsp{2}{0}$ sets. Since we must often deal with noncomputable complexity bounds, we develop a hierarchy of $\Delta\sbsp{2}{0}$ functions, the compositon hierarchy, to classify those functions that have relatively simple computable approximations. The degrees in $L\sb2$ have especially pleasant complexity theoretic properties, and within this context we use the composition hierarchy to formulate and prove hierarchy results for c.e. degrees, generic degrees, and anc degrees. Furthermore, the degrees in $L\sb1$ and those in $L\sb2-L\sb1$ are seen to have many complexity theoretic properties in common. In addition, we develop several variations on the standard notions of genericity, including one, semigenericity, that can be satisfied by sets in $\overline{L\sb1}.$ Finally, we prove results indicating how complexity theoretic considerations can lead to structural consequences in the degrees.