Separating the degree spectra of structures

Date of Completion

January 2009






In computable model theory, mathematical structures are studied on the basis of their computability or computational complexity. The degree spectrum DgSp( A ) of a countable structure A is one way to measure the computability of the structure. Given various classes of countable structures, such as linear orders, groups, and graphs, we separate two classes K1 and K2 in the following way: we say that K1 is distinguished from K2 with respect to degree spectrum if there is an A ∈ K1 such that for all B ∈ K2 , DgSp( A ) ≠ DgSp( B ). In the dissertation, we will investigate this separation idea. We look at specific choices for K1 and K2 —for example, we show that linear orders are distinguished from finite-components graphs, equivalence structures, rank-1 torsion-free abelian groups, and daisy graphs with respect to degree spectrum. Out of these proofs, there comes a general pattern for the kinds of structures from which linear orders are distinguished with respect to degree spectrum. In the future, we may also replace linear orders with possibly more general kinds of structures. ^