We study arithmetic and hyperarithmetic degrees of categoricity. We extend a result of E. Fokina, I. Kalimullin, and R. Miller to show that for every computable ordinal $\alpha$, $\mathbf{0}^{}$ is the degree of categoricity of some computable structure $\mathcal{A}$. We show additionally that for $\alpha$ a computable successor ordinal, every degree $2$-c.e. in and above $\mathbf{0}^{}$ 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 $\Pi_{1}^{1}$-complete. (shrink)

We investigate the complexity of various combinatorial theorems about linear and partial orders, from the points of view of computability theory and reverse mathematics. We focus in particular on the principles ADS (Ascending or Descending Sequence), which states that every infinite linear order has either an infinite descending sequence or an infinite ascending sequence, and CAC (Chain-AntiChain), which states that every infinite partial order has either an infinite chain or an infinite antichain. It is well-known that Ramsey's Theorem for pairs (...) (RT₂²) splits into a stable version (SRT₂²) and a cohesive principle (COH). We show that the same is true of ADS and CAC, and that in their cases the stable versions are strictly weaker than the full ones (which is not known to be the case for RT₂² and SRT₂²). We also analyze the relationships between these principles and other systems and principles previously studied by reverse mathematics, such as WKL₀, DNR, and BΣ₂. We show, for instance, that WKL₀ is incomparable with all of the systems we study. We also prove computability-theoretic and conservation results for them. Among these results are a strengthening of the fact, proved by Cholak, Jockusch, and Slaman, that COH is Π₁¹-conservative over the base system RCA₀. We also prove that CAC does not imply DNR which, combined with a recent result of Hirschfeldt, Jockusch, Kjos-Hanssen, Lempp, and Slaman, shows that CAC does not imply SRT₂² (and so does not imply RT₂²). This answers a question of Cholak, Jockusch, and Slaman. Our proofs suggest that the essential distinction between ADS and CAC on the one hand and RT₂² on the other is that the colorings needed for our analysis are in some way transitive. We formalize this intuition as the notions of transitive and semitransitive colorings and show that the existence of homogeneous sets for such colorings is equivalent to ADS and CAC, respectively. We finish with several open questions. (shrink)

Whenever a structure with a particularly interesting computability-theoretic property is found, it is natural to ask whether similar examples can be found within well-known classes of algebraic structures, such as groups, rings, lattices, and so forth. One way to give positive answers to this question is to adapt the original proof to the new setting. However, this can be an unnecessary duplication of effort, and lacks generality. Another method is to code the original structure into a structure in the given (...) class in a way that is effective enough to preserve the property in which we are interested. In this paper, we show how to transfer a number of computability-theoretic properties from directed graphs to structures in the following classes: symmetric, irreflexive graphs; partial orderings; lattices; rings ; integral domains of arbitrary characteristic; commutative semigroups; and 2-step nilpotent groups. This allows us to show that several theorems about degree spectra of relations on computable structures, nonpreservation of computable categoricity, and degree spectra of structures remain true when we restrict our attention to structures in any of the classes on this list. The codings we present are general enough to be viewed as establishing that the theories mentioned above are computably complete in the sense that, for a wide range of computability-theoretic nonstructure type properties, if there are any examples of structures with such properties then there are such examples that are models of each of these theories. (shrink)

In this paper we investigate computable models of -categorical theories and Ehrenfeucht theories. For instance, we give an example of an -categorical but not -categorical theory such that all the countable models of except its prime model have computable presentations. We also show that there exists an -categorical but not -categorical theory such that all the countable models of except the saturated model, have computable presentations.

The spectrum of a relation on a computable structure is the set of Turing degrees of the image of R under all isomorphisms between and any other computable structure . The relation is intrinsically computably enumerable if its image under all such isomorphisms is c.e. We prove that any computable partially ordered set is isomorphic to the spectrum of an intrinsically c.e. relation on a computable structure. Moreover, the isomorphism can be constructed in such a way that the image of (...) the minimum element of the partially ordered set is computable. This solves the spectrum problem. The theorem and modifications of its proof produce computably categorical structures whose expansions by finite number of constants are not computably categorical and, indeed, ones whose expansions can have any finite number of computable isomorphism types. They also provide examples of computably categorical structures that remain computably categorical under expansions by constants but have no Scott family. (shrink)

Given two incomparable c.e. Turing degrees a and b, we show that there exists a c.e. degree c such that c = (a ⋃ c) ⋂ (b ⋃ c), a ⋃ c | b ⋃ c, and c < a ⋃ b.

We study the Medvedev degrees of mass problems with distinguished topological properties, such as denseness, closedness, or discreteness. We investigate the sublattices generated by these degrees; the prime ideal generated by the dense degrees and its complement, a prime filter; the filter generated by the nonzero closed degrees and the filter generated by the nonzero discrete degrees. We give a complete picture of the relationships of inclusion holding between these sublattices, these filters, and this ideal. We show that the sublattice (...) of the closed Medvedev degrees is not a Brouwer algebra. We investigate the dense degrees of mass problems that are closed under Turing equivalence, and we prove that the dense degrees form an automorphism base for the Medvedev lattice. The results hold for both the Medvedev lattice on the Baire space and the Medvedev lattice on the Cantor space. (shrink)

This paper is essentially the author's Gödel Lecture at the ASL Logic Colloquium '09 in Sofia extended and supplemented by material from some other papers. After a brief description of traditional reverse mathematics, a computational approach to is presented. There are then discussions of some interactions between reverse mathematics and the major branches of mathematical logic in terms of the techniques they supply as well as theorems for analysis. The emphasis here is on ones that lie outside the usual main (...) systems of reverse mathematics. While retaining the usual base theory and working still within second order arithmetic, theorems are described that range from those far below the usual systems to ones far above. (shrink)

We show that the maximal linear extension theorem for well partial orders is equivalent over RCA 0 to ATR 0. Analogously, the maximal chain theorem for well partial orders is equivalent to ATR 0 over RCA 0.

Ambos-Spies, K. and R.A. Shore, Undecidability and 1-types in the recursively enumerable degrees, Annals of Pure and Applied Logic 63 3–37. We show that the theory of the partial ordering of recursively enumerable Turing degrees is undecidable and has uncountably many 1-types. In contrast to the original proof of the former which used a very complicated O''' argument our proof proceeds by a much simpler infinite injury argument. Moreover, it combines with the permitting technique to get similar results for any (...) ideal of the r.e. degrees. (shrink)

We show that the partial order of Σ0 3-sets under inclusion is elementarily definable with parameters in the semilattice of r.e. wtt-degrees. Using a result of E. Herrmann, we can deduce that this semilattice has an undecidable theory, thereby solving an open problem of P. Odifreddi.

Cholak, Goncharov, Khoussainov, and Shore [1] showed that for each k > 0 there is a computably categorical structure whose expansion by a constant has computable dimension k. We show that the same is true with k replaced by ω. Our proof uses a version of Goncharov's method of left and right operations.

Cenzer, D., R. Downey, C. Jockusch and R.A. Shore, Countable thin Π01 classes, Annals of Pure and Applied Logic 59 79–139. A Π01 class P {0, 1}ω is thin if every Π01 subclass of P is the intersection of P with some clopen set. Countable thin Π01 classes are constructed having arbitrary recursive Cantor- Bendixson rank. A thin Π01 class P is constructed with a unique nonisolated point A and furthermore A is of degree 0’. It is shown that no (...) set of degree ≥0” can be a member of any thin Π01 class. An r.e. degree d is constructed such that no set of degree d can be a member of any thin Π01 class. It is also shown that between any two distinct comparable r.e. degrees, there is a degree that contains a set which is of rank one in some thin Π01 class. It is shown that no maximal set can have rank one in any Π01 class, while there exist maximal sets of rank 2. The connection between Π01 classes, propositional theories and recursive Boolean algebras is explored, producing several corollaries to the results on Π01 classes. For example, call a recursive Boolean algebra thin if it has no proper nonprincipal recursive ideals. Then no thin recursive Boolean algebra can have a maximal ideal of degree ≥0”. (shrink)

iterations of REA operators, as well as extensions, generalizations and other applications are given in [6] while those for the ...

There are $\Pi_5$ formulas in the language of the Turing degrees, D, with ≤, ∨ and $\vedge$ , that define the relations $x" \leq y"$ , x" = y" and so $x \in L_{2}(y)=\{x\geqy|x"=y"\}$ in any jump ideal containing $0^(\omega)$ . There are also $\Sigma_6$ & $\Pi_6$ and $\Pi_8$ formulas that define the relations w = x" and w = x', respectively, in any such ideal I. In the language with just ≤ the quantifier complexity of each of these definitions (...) increases by one. On the other hand, no Π2 or Σ2 formula in the language with just ≤ defines L2 or $x\inL_{2}(y)$ . Our arguments and constructions are purely degree theoretic without any appeals to absoluteness considerations, set theoretic methods or coding of models of arithmetic. As a corollary, we see that every automorphism of I is fixed on every degree above 0" and every relation on I that is invariant under double jump or joining with 0" is definable over I if and only if it is definable in second order arithmetic with set quantification ranging over sets whose degrees are in I. Similar direct coding arguments show that every hyperjump ideal I is rigid and biinterpretable with second order arithmetic with set quantification ranging over sets with hyperdegrees in I. Analogous results hold for various coarser degree structures. (shrink)

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 we 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 earlier works by Selivanov. In a more recent work, he showed that 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. (shrink)

We show that the elementary theory of the recursively enumerable tt-degrees has the same computational complexity as true first-order arithmetic. As auxiliary results, we prove theorems about exact pairs and initial segments in the tt-degrees.

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.

We consider the set of jumps below a Turing degree, given by JB={x':x≤a}, with a focus on the problem: Which recursively enumerable degrees a are uniquely determined by JB? Initially, this is motivated as a strategy to solve the rigidity problem for the partial order R of r.e. degrees. Namely, we show that if every high2 r.e. degree a is determined by JB, then R cannot have a nontrivial automorphism. We then defeat the strategy—at least in the form presented—by constructing (...) pairs a0, a1 of distinct r.e. degrees such that JB=JB within any possible jump class {x:x'=c}. We give some extensions of the construction and suggest ways to salvage the attack on rigidity. (shrink)

If r is a reducibility between sets of numbers, a natural question to ask about the structure ? r of the r-degrees containing computably enumerable sets is whether every element not equal to the greatest one is branching (i.e., the meet of two elements strictly above it). For the commonly studied reducibilities, the answer to this question is known except for the case of truth-table (tt) reducibility. In this paper, we answer the question in the tt case by showing that (...) every tt-incomplete computably enumerable truth-table degree a is branching in ? tt . The fact that every Turing-incomplete computably enumerable truth-table degree is branching has been known for some time. This fact can be shown using a technique of Ambos-Spies and, as noticed by Nies, also follows from a relativization of a result of Degtev. We give a proof here using the Ambos-Spies technique because it has not yet appeared in the literature. The proof uses an infinite injury argument. Our main result is the proof when a is Turing-complete but tt-incomplete. Here we are able to exploit the Turing-completeness of a in a novel way to give a finite injury proof. (shrink)