Richard A. Shore

Degree spectra and computable dimensions in algebraic structures

Abstract 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 (with zero-divisors); 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.

Pseudo-jump operators. II: Transfinite iterations, hierarchies and minimal covers

In this paper we introduce a new hierarchy of sets and operators which we call the REA hierarchy for “recursively enumerable in and above”. The hierarchy is generated by composing (possibly) transfinite sequences of the pseudo-jump operators considered in Jockusch and Shore [1983]. We there studied pseudo-jump operators defined by analogy with the Turing jump as ones taking a set A to A ⊕ for some index e . We would now call these 1-REA operators and will extend them to α -REA operators for recursive ordinals α in analogy with the iterated Turing jump operators ( A → A ( α ) for α and Kleenes hyperarithmetic hierarchy. The REA sets will then, of course, be the results of applying these operators to the empty set. They will extend and generalize Kleenes H sets but will still be contained in the class of set singletons thus providing us with a new richer subclass of the set singletons which, as we shall see, is related to the work of Harrington [1975] and [1976] on the problems of Friedman [1975] about the arithmetic degrees of such singletons. Their degrees also give a natural class extending the class H of Jockusch and McLaughlin [1969] by closing it off under transfinite iterations as well as the inclusion of [ d , d′ ] for each degree d in the class. The reason for the class being closed under this last operation is that the REA operators include all operators and so give a new hierarchy for them as well as the sets. This hierarchy also turns out to be related to the difference hierarchy of Ershov [1968], [1968a] and [1970]: every α -r.e. set is α -REA but each level of the REA hierarchy after the first extends all the way through the difference hierarchy although never entirely encompassing even the next level of the difference hierarchy.

Interpretability and Definability in the Recursively Enumerable Degrees

We investigate definability in R, the recursively enumerable Turing degrees, using codings of standard models of arithmetic (SMA’s) as a tool. First we show that an SMA can be interpreted in R without parameters. Building on this, we prove that the recursively enumerable T–degrees satisfy a weak form of the biinterpretability conjecture which implies that all jump classes Lown and Highn−1 (n ≥ 2) are definable in R without parameters and, more generally, that all relations on R that are definable in arithmetic and invariant under the double jump are actually definable in R. This partially answers Soare’s Question 3.7 [35, XVI].

Computable Models of Theories with Few Models

In this paper we investigate computable models of א1-categorical theories and Ehrenfeucht theories. For instance, we give an example of an א1categorical but not א0-categorical theory T such that all the countable models of T except its prime model have computable presentations. We also show that there exists an א1-categorical but not א0-categorical theory T such that all the countable models of T except the saturated model, have computable presentations.

Then-rea enumeration degrees are dense

The notion of enumeration (or simply e-) reducibility between sets as formalized by Rogers even in early versions of (1967) [see for example Friedberg and Rogers (1959)] captures the natural concept of computing a set A from another one B with only positive membership information about B being available. One can view the procedure as a computation determined by a finite set of instructions except that from time to time an outside source informs the computing agent that certain numbers are in B. We say that A is e-reducible to B if there is such a procedure which, when supplied with all the members of B (in any order and at any time during its computation) as inputs, correctly enumerates A, i.e. lists precisely the members of A as its outputs. By identifying partial functions with their graphs, this notion includes the basic reducibility on partial functions introduced by Kleene (1952) and studied in Myhill (1961). When further restricted to the (graphs) of total functions, it coincides with the standard notion of Turing reducibility. In its unrestricted form, it is connected with the general problem of modeling computation from partial information, self application and models of the k-calculus. We refer the reader to Cooper (1990) for a general survey of enumeration reducibility and an extensive bibliography of related articles. To fix our notation, we present the formal definitions of enumeration operators, reducibilities and degrees as in Rogers (1967, Sect. 9.7):

The atomic model theorem and type omitting

We investigate the complexity of several classical model theoretic theorems about prime and atomic models and omitting types. Some are provable in RCA0, others are equivalent to ACA0. One, that every atomic theory has an atomic model, is not provable in RCA0 but is incomparable with WKL0, more than 1 conservative over RCA0 and strictly weaker than all the combinatorial principles of Hirschfeldt and Shore [2007] that are not 1 conservative over RCA0. A priority argument with Shore blocking shows that it is also 1 -conservative over B 2. We also provide a theorem provable by a finite injury priority argument that is conservative over I 1 but implies I 2 over B 2, and a type omitting theorem that is equivalent to the principle that for every X there is a set that is hyperimmune relative to X. Finally, we give a version of the atomic model theorem that is equivalent to the principle that for every X there is a set that is not recursive in X, and is thus in a sense the weakest possible natural principle not true in the !-model consisting of the recursive sets.

Second order logic and first order theories of reducibility orderings

We show that the first order theories of many reducibility orderings are recursively isomorphic to second order logic on countable sets (and so to true second order arithmetic). The reduction procedure uses some initial segment results and Spectors theorem on countable ideals in the degrees to code quantification over symmetric irreflexive binary relations. This is known to be enough to obtain full second order logic. Applications to other theories are mentioned as are several to problems of definability in, and automorphisms of, the Turing degrees.

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.

Computable isomorphisms, degree spectra of relations, and Scott families

Abstract The spectrum of a relation R on a computable structure is the set of Turing degrees of the image of R under all isomorphisms between A and any other computable structure B . The relation R is intrinsically computably enumerable (c.e.) 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 (if it exists) 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.

Nowhere Simple Sets and the Lattice of Recursively Enumerable Sets

Ever since Post [4] the structure of recursively enumerable sets and their classification has been an important area in recursion theory. It is also intimately connected with the study of the lattices and of r.e. sets and r.e. sets modulo finite sets respectively. (This lattice theoretic viewpoint was introduced by Myhill [3].) Key roles in both areas have been played by the lattice of r.e. supersets, , of an r.e. set A (along with the corresponding modulo finite sets) and more recently by the group of automorphisms of and . Thus for example we have Lachlans deep result [1] that Posts notion of A being hyperhypersimple is equivalent to (or ) being a Boolean algebra. Indeed Lachlan even tells us which Boolean algebras appear as —precisely those with Σ 3 representations. There are also many other simpler but still illuminating connections between the older typology of r.e. sets and their roles in the lattice . ( r -maximal sets for example are just those with completely uncomplemented.) On the other hand, work on automorphisms by Martin and by Soare [8], [9] has shown that most other Post type conditions on r.e. sets such as hypersimplicity or creativeness which are not obviously lattice theoretic are in fact not invariant properties of . In general the program of analyzing and classifying r.e. sets has been directed at the simple sets. Thus the subtypes of simple sets studied abound — between ten and fifteen are mentioned in [5] and there are others — but there seems to be much less known about the nonsimple sets. The typologies introduced for the nonsimple sets begin with Posts notion of creativeness and add on a few variations. (See [5, §8.7] and the related exercises for some examples.) Although there is a classification scheme for r.e. sets along the simple to creative line (see [5, §8.7]) it is admitted to be somewhat artificial and arbitrary. Moreover there does not seem to have been much recent work on the nonsimple sets.