Antonio Montalbán

From Automatic Structures to Borel Structures

We study the classes of Buchi and Rabin automatic structures. For Buchi (Rabin) automatic structures their domains consist of infinite strings (trees), and the basic relations, including the equality relation, and graphs of operations are recognized by Buchi (Rabin) automata. A Buchi (Rabin) automatic structure is injective if different infinite strings (trees) represent different elements of the structure. The first part of the paper is devoted to understanding the automata- theoretic content of the well-known Lowenheim-Skolem theorem in model theory. We provide automata-theoretic versions of Lowenheim-Skolem theorem for Rabin and Buchi automatic structures. In the second part, we address the following two well-known open problems in the theory of automatic structures: Does every Buchi automatic structure have an injective Buchi presentation? Does every Rabin automatic structure have an injective Rabin presentation? We provide examples of Buchi structures without injective Buchi and Rabin presentations. To answer these questions we introduce Borel structures and use some of the basic properties of Borel sets and isomorphisms. Finally, in the last part of the paper we study the isomorphism problem for Buchi automatic structures.

Open questions in reverse mathematics

We present a list of open questions in reverse mathematics, including some relevant background information for each question. We also mention some of the areas of reverse mathematics that are starting to be developed and where interesting open question may be found.

On the equimorphism types of linear orderings

§1. Introduction . A linear ordering (also known as total ordering) embeds into another linear ordering if it is isomorphic to a subset of it. Two linear orderings are said to be equimorphic if they can be embedded in each other. This is an equivalence relation, and we call the equivalence classes equimorphism types . We analyze the structure of equimorphism types of linear orderings, which is partially ordered by the embeddability relation. Our analysis is mainly fromthe viewpoints of Computability Theory and Reverse Mathematics. But we also obtain results, as the definition of equimorphism invariants for linear orderings, which provide a better understanding of the shape of this structure in general. This study of linear orderings started by analyzing the proof-theoretic strength of a theorem due to Jullien [Jul69]. As is often the case in Reverse Mathematics, to solve this problem it was necessary to develop a deeper understanding of the objects involved. This led to a variety of results on the structure of linear orderings and the embeddability relation on them. These results can be divided into three groups.

Notes on the Jump of a Structure

We introduce the notions of a complete set of computably infinitary

Boolean Algebras, Tarski Invariants, and Index Sets

\Pi^0_n

INDECOMPOSABLE LINEAR ORDERINGS AND HYPERARITHMETIC ANALYSIS

relations on a structure, of the jump of a structure , and of admitting n th jump inversion .

ON THE n-BACK-AND-FORTH TYPES OF BOOLEAN ALGEBRAS

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

A statement of hyperarithmetic analysis is a sentence of second order arithmetic S such that for every Y⊆ω, the minimum ω-model containing Y of RCA0 + S is HYP(Y), the ω-model consisting of the sets hyperarithmetic in Y. We provide an example of a mathematical theorem which is a statement of hyperarithmetic analysis. This statement, that we call INDEC, is due to Jullien [13]. To the authors knowledge, no other already published, purely mathematical statement has been found with this property until now. We also prove that, over RCA0, INDEC is implied by and implies ACA0, but of course, neither ACA0, nor ACA0+ imply it. We introduce five other statements of hyperarithmetic analysis and study the relations among them. Four of them are related to finitely-terminating games. The fifth one, related to iterations of the Turing jump, is strictly weaker than all the other statements that we study in this paper, as we prove using Steels method of forcing with tagged trees.

RELATIVE TO ANY NON-HYPERARITHMETIC SET

The objective of this paper is to uncover the structure of the back-and- forth equivalence classes at the nite levels for the class of Boolean algebras. As an application, we obtain bounds on the computational complexity of determining the back-and-forth equivalence classes of a Boolean algebra for nite levels. This result has implications for characterizing the relatively intrinsically 0 relations of Boolean algebras as existential formulas over a nite set of relations.

Equivalence between Fraïssé’s conjecture and Jullien’s theorem

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).