Notions of Complexity Within Computable Structure Theory

Abstract

This thesis covers multiple areas within computable structure theory, analyzing the complexities of certain aspects of computable structures with respect to different notions of definability.

In chapter 2 we use a new metatheorem of Antonio Montalb\'an's to simplify an otherwise difficult priority construction. We restrict our attention to linear orders, and ask if, given a computable linear order $\A$ with degree of categoricity $\boldsymbol{d}$, it is possible to construct a computable isomorphic copy of $\A$ such that the isomorphism achieves the degree of categoricity and furthermore, that we did not do this coding using a computable set of points chosen in advance. To ensure that there was no computable set of points that could be used to compute the isomorphism we are forced to diagonalize against all possible computable unary relations while we construct our isomorphic copy. This tension between trying to code information into the isomorphism and trying to avoid using computable coding locations, necessitates the use of a metatheorem. This work builds off of results obtained by Csima, Deveau, and Stevenson for the ordinals $\omega$ and $\omega^2$, and extends it to $\omega^\alpha$ for any computable successor ordinal $\alpha$.

In chapter 3, which is joint work with Alvir and Csima, we study the Scott complexity of countable reduced Abelian $p$-groups. We provide Scott sentences for all such groups, and show some cases where this is an optimal upper bound on the Scott complexity. To show this optimality we obtain partial results towards characterizing the back-and-forth relations on these groups.

In chapter 4, which is joint work with Csima and Rossegger, we study structures under enumeration reducibility when restricting oneself to only the positive information about a structure. We find that relations that can be relatively intrinsically enumerated from such information have a definability characterization using a new class of formulas. We then use these formulas to produce a structural jump within the enumeration degrees that admits jump inversion, and compare it to other notions of the structural jump. We finally show that interpretability of one structure in another using these formulas is equivalent to the existence of a positive enumerable functor between the classes of isomorphic copies of the structures.