Julia F. Knight

Degrees Coded in Jumps of Orderings

All structures to be considered here have universe ω, and all languages come equipped with Godel numberings. If is a structure, then D ( ), the open diagram of , can be thought of as a subset of ω , and it makes sense to talk about the Turing degree deg( D ( )). This depends on the presentation as well as the isomorphism type of . For example, consider the ordering = ( ω , B ⊆ ω , it is possible to code B in a copy of as follows: Let π be the permutation of ω such that for each n ∈ ω , π leaves 2 n and 2 n + 1 fixed if n ∈ B and switches 2 n with 2 n + 1 if n ∉ B . Let be the copy of such that ≃ π . Then n ∈ B iff the sentence 2 n n + 1 is in D ( ). In §4, this idea will be used to show that for any structure that is not completely trivial, {deg( D ( )): ≃ } is closed upwards. It would be satisfying to have a way of assigning Turing degrees to structures such that the degree assigned to a given structure measured the recursion-theoretic complexity of the isomorphism type and was independent of the presentation. Jockusch suggested the following.

Enumerations in computable structure theory

Abstract We exploit properties of certain directed graphs, obtained from the families of sets with special effective enumeration properties, to generalize several results in computable model theory to higher levels of the hyperarithmetical hierarchy. Families of sets with such enumeration features were previously built by Selivanov, Goncharov, and Wehner. For a computable successor ordinal α , we transform a countable directed graph G into a structure G ∗ such that G has a Δ α 0 isomorphic copy if and only if G ∗ has a computable isomorphic copy. A computable structure A is Δ α 0 categorical (relatively Δ α 0 categorical, respectively) if for all computable (countable, respectively) isomorphic copies B of A , there is an isomorphism from A onto B , which is Δ α 0 ( Δ α 0 relative to the atomic diagram of B , respectively). We prove that for every computable successor ordinal α , there is a computable, Δ α 0 categorical structure, which is not relatively Δ α 0 categorical. This generalizes the result of Goncharov that there is a computable, computably categorical structure, which is not relatively computably categorical. An additional relation R on the domain of a computable structure A is intrinsically Σ α 0 (relatively intrinsically Σ α 0 , respectively) on A if in all computable (countable, respectively) isomorphic copies B of A , the image of R is Σ α 0 ( Σ α 0 relative to the atomic diagram of B , respectively). We prove that for every computable successor ordinal α , there is an intrinsically Σ α 0 relation on a computable structure, which not relatively intrinsically Σ α 0 . This generalizes the result of Manasse that there is an intrinsically computably enumerable relation on a computable structure, which is not relatively intrinsically computably enumerable. The Δ α 0 dimension of a structure A is the number of computable isomorphic copies, up to Δ α 0 isomorphisms. We prove that for every computable successor ordinal α and every n ≥ 1 , there is a computable structure with Δ α 0 dimension n . This generalizes the result of Goncharov that there is a structure of computable dimension n for every n ≥ 1 . Finally, we prove that for every computable successor ordinal α , there is a countable structure with isomorphic copies in just the Turing degrees of sets X such that Δ α 0 relative to X is not Δ α 0 . In particular, for every finite n , there is a structure with isomorphic copies in exactly the non- low n Turing degrees. This generalizes the result obtained by Wehner, and independently by Slaman, that there is a structure A with isomorphic copies in exactly the nonzero Turing degrees.

Computable Structure and Non-Structure Theorems

In a lecture in Kazan (1977), Goncharov dubbed a number of problems regarding the classification of computable members of various classes of structures. Some of the problems seemed likely to have nice answers, while others did not. At the end of the lecture, Shore asked what would be a convincing negative result. The goal of the present article is to consider some possible answers to Shores question. We consider structures Д of some computable language, whose universes are computable sets of constants. In measuring complexity, we identify Д with its atomic diagram D(Д), which, via the Gödel numbering, may be treated as a subset of ω. In particular, Д is computable if D(Д) is computable. If K is some class, then Kc denotes the set of computable members of K. A computable characterization for K should separate the computable members of K from other structures, that is, those that either are not in K or are not computable. A computable classification (structure theorem) should describe each member of Kc up to isomorphism, or other equivalence, in terms of relatively simple invariants. A computable non-structure theorem would assert that there is no computable structure theorem. We use three approaches. They all give the “correct” answer for vector spaces over Q, and for linear orderings. Under all of the approaches, both classes have a computable characterization, and there is a computable classification for vector spaces, but not for linear orderings. Finally, we formulate some open problems.

Pairs of recursive structures

A number of recursive constructions can be based on the following: given structures U n and B n and for an arbitrary Π 0 α set S construct recursive structures C n , uniformly in n, such that C n ≅{U n if n∈S, B n if n∈S. We concentrate initially on the special case where U n and B n are independent of n. So we ask for which structures U and B of the same type and which α<ω 1 CK the following statement is true. «For every Πα 0 set S there are structures C n , recursive uniformly in n, such that C n ≅{U if n∈S, B if n¬∈». In this paper, we obtain some general recursive-syntactical conditions for such statements

Comparing Classes of Finite Structures

We compare classes of structures using the notion of a computable embedding, which is a partial order on the classes of structures. Our attention is mainly, but not exclusively, focused on classes of finite structures. Also, a number of problems are formulated.

Computable Boolean Algebras

Feiner [F] showed that a Boolean algebra need not have a computable copy (see also [T2]). Downey and Jockusch [D-J] showed that every low Boolean algebra does have a computable copy. Thurber [T3], showed that every low2 Boolean algebra has a computable copy. Here we show that every Boolean algebra which is low3, or even low4, has a computable copy.The results of [D-J] and [T3] were obtained by passing to linear orderings. In [D-J], there is an embedding theorem saying that any linear ordering which is with the successor relation as an added predicate can be embedded in a slightly larger linear ordering which is computable. An isomorphism theorem of Remmel [R] is used to show that the interval algebras of the two linear orderings are isomorphic (except in a trivial case). In [T3], there is an embedding theorem saying that any linear ordering which is with certain added predicates can be embedded in one which is with successor. Again the isomorphism theorem of Remmel is used to show that the interval algebras are isomorphic (except in a trivial case).Here, instead of passing to linear orderings, we work directly with Boolean algebras. We begin with a review of the known results. We re-formulate the embedding theorems of Downey-Jockusch and Thurber in terms of Boolean algebras. We extract from Remmels isomorphism theorem some information on complexity. In this way, we show that a low Boolean algebra is isomorphic to a computable one by an isomorphism which is , at worst, and the same is true for a low2 Boolean algebra.

A Metatheorem for Constructions by Finitely Many Workers

The aim of the present paper is to give some general conditions for constructions by finitely many workers. The object of the construction is to attach «labels» to the nodes in a highly nonrecursive path through a tree, while recursively enumerating neighborhoods of an «adherent» point in a metric space. There is a family of relations associated with the labels, and a metatheorem says that the construction will succeed if these relations satisfy a list of properties. A relativized version of the metatheorem will be used to prove a variant of a result on degrees of models of a given theory, «representing» a prescribed Scott set

Recursive Structures and Ershov's Hierarchy

Ash and Nerode [2] gave natural definability conditions under which a relation is intrinsically r. e. Here we generalize this to arbitrary levels in Ershovs hierarchy of Δ sets, giving conditions under which a relation is intrinsically α-r. e. Mathematics Subject Classification: 03C57, 03D55.

Possible Degrees in Recursive Copies II

Abstract Let A be a recursive structure, and let R be a recursive relation on A . Harizanov (1991) isolated a syntactical condition which (with additional effectiveness conditions) is necessary and sufficient for A to have recursive copies in which the image of R is r.e. of arbitrary r.e. degree. We had conjectured that a certain extension of Harizanovs syntactical condition would (with some effectiveness conditions) be necessary and sufficient for A to have recursive copies in which the image of R is ∑α0 of arbitrary ∑α0 degree, but this is not the case. Here we give examples illustrating some restrictions on the possible ∑α0 degrees. In these examples, the image of R cannot be ∑α0 of degree d unless d possesses an “α-table” (a sequence of sets in which each one is r.e. in and above the earlier ones).

Models of Arithmetic and Closed Ideals

A set J of Turing degrees is called an ideal if (1) J ≠ ∅, (2) for any pair of degrees a, , if a, ϵ J , then a ⋃ ϵ J , and (3) for any ⋃ ϵ J and any , if ϵ J . A set J of degrees is said to be closed if for any theory T with a set of axioms of degree in J, T has a completion of degree in J . Closed ideals of degrees arise naturally in the following way. If is a recursively saturated structure, let I ( ) = { for some ā ϵ }. Let D ( ) = { : is recursive in d -saturated}. (Recursive in d -saturation is defined like recursive saturation except that the sets of formulas considered are recursive in d .) These two sets of degrees were investigated in [2]. It was shown that if is a recursively saturated model of P , Pr = Th(ω, +), or Pr′ = Th( Z , +, 1), then I ( ) = D ( ), and this set is a closed ideal. Any closed ideal J can be represented as I ( ) = D ( ) for some recursively saturated model of Pr′. For sets J of power at most ℵ 1 , Pr′ can be replaced by P . Assuming CH, all closed ideals have power at most ℵ 1 , but if CH fails, there are closed ideals of power greater than ℵ 1 , and it is not known whether these can be represented as I ( ) = D ( ) for a recursively saturated model of P . In the present paper, it will first be shown that information about representation of closed ideals provides new information about an old problem of MacDowell and Specker [6] and extends an old result of Scott [8] in a natural way. It will also be shown that the representation results from [2] answer a problem of Friedman [1]. This part of the paper is aimed at convincing the reader that representation problems are worth investigating.