Barbara F. Csima

Degrees of Categoricity and the Hyperarithmetic Hierarchy

We study arithmetic and hyperarithmetic degrees of categoricity. We extend a result of Fokina, Kalimullin, and R. Miller to show that for every computable ordinal α, 0 is the degree of categoricity of some computable structure A. We show additionally that for α a computable successor ordinal, every degree 2-c.e. in and above 0 is a degree of categoricity. We further prove that every degree of categoricity is hyperarithmetic and show that the index set of structures with degrees of categoricity is Π1 complete.

Degrees That Are Not Degrees of Categoricity

A computable structure A is x-computably categorical for some Turing degree x, if for every computable structure B isomorphic to A there is an isomorphism f:B -> A with f computable in x. A degree x is a degree of categoricity if there is a computable A such that A is x-computably categorical, and for all y, if A is y-computably categorical then y computes x. We construct a Sigma_2 set whose degree is not a degree of categoricity. We also demonstrate a large class of degrees that are not degrees of categoricity by showing that every degree of a set which is 2-generic relative to some perfect tree is not a degree of categoricity. Finally, we prove that every noncomputable hyperimmune-free degree is not a degree of categoricity.

Boolean Algebras, Tarski Invariants, and Index Sets

Tarski defined a way of assigning to each boolean algebra, B, an invariant inv(B) ∈ In, where In is a set of triples from N, such that two boolean algebras have the same invariant if and only if they are elementarily equivalent. Moreover, given the invariant of a boolean algebra, there is a computable procedure that decides its elementary theory. If we restrict our attention to dense Boolean algebras, these invariants determine the algebra up to isomorphism. In this paper we analyze the complexity of the question “Does B have invariant x?”. For each x ∈ In we define a complexity class Γx, that could be either Σn, Πn, Σn∧Πn, or Πω+1 depending on x, and prove that the set of indices for computable boolean algebras with invariant x is complete for the class Γx. Analogs of many of these results for computably enumerable Boolean algebras were proven in [Sel90] and [Sel91]. According to [Sel03] similar methods can be used to obtain the results for computable ones. Our methods are quite different and give new results as well. As the algebras we construct to witness hardness are all dense, we establish new similar results for the complexity of various isomorphism problems for dense Boolean algebras.

A minimal pair of K-degrees

We construct a minimal pair of K-degrees. We do this by showing the existence of an unbounded nondecreasing function f which forces K-triviality in the sense that γ ∈ 2 ω is K-trivial if and only if for all n, K(γ | n) ≤ K(n) + f(n) + O(1).

Degree spectra and immunity properties

We analyze the degree spectra of structures in which different types of immunity conditions are encoded. In particular, we give an example of a structure whose degree spectrum coincides with the hyperimmune degrees. As a corollary, this shows the existence of an almost computable structure of which the complement of the degree spectrum is uncountable (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Computable Categoricity of Graphs with Finite Components

A computable graph is computably categorical if any two computable presentations of the graph are computably isomorphic. In this paper we investigate the class of computably categorical graphs. We restrict ourselves to strongly locally finite graphs; these are the graphs all of whose components are finite. We present a necessary and sufficient condition for certain classes of strongly locally finite graphs to be computably categorical. We prove that if there exists an infinite

The complexity of central series in nilpotent computable groups

\Delta_2^0

Comparing C.E. Sets Based on Their Settling Times

-set of components that can be properly embedded into infinitely many components of the graph then the graph is not computably categorical. We outline the construction of a strongly locally finite computably categorical graph with an infinite chain of properly embedded components.

Degrees of categoricity on a Cone via η-systems

Abstract The terms of the upper and lower central series of a nilpotent computable group have computably enumerable Turing degree. We show that the Turing degrees of these terms are independent even when restricted to groups which admit computable orders.

The Settling Time Reducibility Ordering and Δ02 Sets

To each computable enumerable (c.e.) set Awith aparticular enumeration {As}seω,there is associated a settling function mA(x), where mA(x) is the last stage when a numberless than or equal to xwas enumerated into A. In[7], R.W. Robinson classified the complexity of c.e. sets into twogroups of complexity based on whether or not the settling functionwas dominant. An extension of this idea to a more refined orderingof c.e. sets was first introduced by Nabutovsky and Weinberger in[6] and Soare [9], for application to differential geometry. Therethey defined one c.e. set Ato settling time dominateanother c.e. set B(B> stA) if for every computable function f, for allbut finitely many x, mB(x) > f(mA(x)). In [4] Csima and Soareintroduced a stronger ordering, where B>sstAif for all computable fandg, for almost all x, mB(x) > f(mA(g(x))). We give a survey ofthe known results about these orderings, make some observations,and outline the open questions.