We study notions of generic and coarse computability in the context of computable structure theory. Our notions are stratified by the Σβ hierarchy. We focus on linear orderings. We show that at the Σ1 level, all linear orderings have both generically and coarsely computable copies. This behavior changes abruptly at higher levels; we show that at the Σα+2 level for any α∈ω1CK the set of linear orderings with generically or coarsely computable copies is Σ11-complete and therefore maximally complicated. This development is new even in the general analysis of generic and coarse computability of countable structures. In the process of proving these results, we introduce new tools for understanding generically and coarsely computable structures. We are able to give a purely structural statement that is equivalent to having a generically computable copy and show that every relational structure with only finitely many relations has coarsely and generically computable copies at the lowest level of the hierarchy.

Abstract Cohesive powersof computable structures are effective analogs of ultrapowers, where cohesive sets play the role of ultrafilters. Let $\omega $ , $\zeta $ , and $\eta $ denote the respective order-types of the natural numbers, the integers, and the rationals when thought of as linear orders. We investigate the cohesive powers of computable linear orders, with special emphasis on computable copies of $\omega $ . If $\mathcal {L}$ is a computable copy of $\omega $ that is computably isomorphic to the usual presentation of $\omega $ , then every cohesive power of $\mathcal {L}$ has order-type $\omega + \zeta \eta $ . However, there are computable copies of $\omega $ , necessarily not computably isomorphic to the usual presentation, having cohesive powers not elementarily equivalent to $\omega + \zeta \eta $ . For example, we show that there is a computable copy of $\omega $ with a cohesive power of order-type $\omega + \eta $ . Our most general result is that if $X \subseteq \mathbb {N} \setminus \{0\}$ is a Boolean combination of $\Sigma _2$ sets, thought of as a set of finite order-types, then there is a computable copy of $\omega $ with a cohesive power of order-type $\omega + \boldsymbol {\sigma }(X \cup \{\omega + \zeta \eta + \omega ^*\})$ , where $\boldsymbol {\sigma }(X \cup \{\omega + \zeta \eta + \omega ^*\})$ denotes the shuffle of the order-types inXand the order-type $\omega + \zeta \eta + \omega ^*$ . Furthermore, ifXis finite and non-empty, then there is a computable copy of $\omega $ with a cohesive power of order-type $\omega + \boldsymbol {\sigma }(X)$ .

Abstract We improve on and generalize a 1960 result of Maltsev. For a field F, we denote by $H(F)$ the Heisenberg group with entries in F. Maltsev showed that there is a copy of F defined in $H(F)$ , using existential formulas with an arbitrary non-commuting pair of elements as parameters. We show that F is interpreted in $H(F)$ using computable $\Sigma _1$ formulas with no parameters. We give two proofs. The first is an existence proof, relying on a result of Harrison-Trainor, Melnikov, R. Miller, and Montalbán. This proof allows the possibility that the elements of F are represented by tuples in $H(F)$ of no fixed arity. The second proof is direct, giving explicit finitary existential formulas that define the interpretation, with elements of F represented by triples in $H(F)$ . Looking at what was used to arrive at this parameter-free interpretation of F in $H(F)$ , we give general conditions sufficient to eliminate parameters from interpretations.

Abstract In recent years, computability theorists have extensively studied generically and coarsely computable sets. This study of approximate computability was originally motivated by asymptotic density problems in combinatorial group theory. We generalize the notions of generic and coarse computability of sets, introduced by Jockusch and Schupp, to arbitrary structures by defining generically and coarsely computable and computably enumerable structures. There are two directions in which these notions could potentially trivialize: either all structures could have a densely computable copy or only those having a computable (or computably enumerable) copy. We show that some particular classes of structures realize each of these extremal conditions, while other classes realize neither of them. To further explore these concepts, we introduce a graded family of elementarity conditions for substructures, in which we require that the dense sets under consideration be ‘strong’ substructures of the original structure. Here, again, for a given class, the notion could trivialize in the same two directions and we show that both are possible. For each class that we investigate, there is some natural number $n$ such that requiring $\varSigma _{n}$ elementarity of substructures is enough to trivialize the class of generically or densely computable structures, witnessing the essentially structural character of these notions.

We say that a structure $\mathcal{A}$ admits \emph{strong jump inversion} provided that for every oracle $X$, if $X'$ computes $D(\mathcal{C})'$ for some $\mathcal{C}\cong\mathcal{A}$, then $X$ computes $D(\mathcal{B})$ for some $\mathcal{B}\cong\mathcal{A}$. Jockusch and Soare \cite{JS} showed that there are low linear orderings without computable copies, but Downey and Jockusch \cite{DJ} showed that every Boolean algebra admits strong jump inversion. More recently, D.\ Marker and R.\ Miller \cite{MM} have shown that all countable models of $DCF_0$ (the theory of differentially closed fields of characteristic $0$) admit strong jump inversion. We establish a general result with sufficient conditions for a structure $\mathcal{A}$ to admit strong jump inversion. Our conditions involve an enumeration of $B_1$-types, where these are made up of formulas that are Boolean combinations of existential formulas. Our general result applies to some familiar kinds of structures, including some classes of linear orderings and trees. We do not get the result of Downey and Jockusch for arbitrary Boolean algebras, but we do get a result for Boolean algebras with no $1$-atom, with some extra information on the complexity of the isomorphism. Our general result gives the result of Marker and Miller. In order to apply our general result, we produce a computable enumeration of the types realized in models of $DCF_0$. This also yields the fact that the saturated model of $DCF_0$ has a decidable copy.

We investigate computability-theoretic properties of computable structures with single unary functions f such that, for every x in the image, f−1(x) has exactly two elements, which we call 2:1 structures. We also investigate structures for which f−1(x) has either exactly two or zero elements, which we call (2,0):1 structures. In particular, we are interested in the complexity of isomorphisms between these structures. We prove that a computable 2:1 structure A is computably categorical if and only if A has only finitely many ℤ-chains. We show that every computable 2:1 structure is Δ20-categorical. We further investigate computable and higher level categoricity of various natural subclasses of (2,0):1 structures, including highly computable and locally finite strufctures.

Abstract We study the complexity of the isomorphism relation on classes of computable structures. We use the notion of FF-reducibility introduced in [9] to show completeness of the isomorphism relation on many familiar classes in the context of all equivalence relations on hyperarithmetical subsets of ω.

Journal Article Spectra of high n and non-low n degrees Get access Andrey Frolov, Andrey Frolov N. G. Chebotarev Research Inst. of Mechanics and Mathematics, Kazan Federal University, Universitetskaya St., 17, Kazan 420008, Russia.E-mail: [email protected]; [email protected] Search for other works by this author on: Oxford Academic Google Scholar Iskander Kalimullin, Iskander Kalimullin N. G. Chebotarev Research Inst. of Mechanics and Mathematics, Kazan Federal University, Universitetskaya St., 17, Kazan 420008, Russia.E-mail: [email protected]; [email protected] Search for other works by this author on: Oxford Academic Google Scholar Valentina Harizanov, Valentina Harizanov Department of Mathematics, George Washington University, Washington, DC 20052, USA. E-mail: [email protected] Search for other works by this author on: Oxford Academic Google Scholar Oleg Kudinov, Oleg Kudinov Sobolev Institute of Mathematics, Russian Academy of Sciences, Siberian Branch, 630090 Novosibirsk Russia. E-mail: [email protected] Search for other works by this author on: Oxford Academic Google Scholar Russell Miller Russell Miller Department of Mathematics, Queens College & C.U.N.Y. Graduate Center, 365 Fifth Avenue, New York, New York 10016, USA. E-mail: [email protected] Search for other works by this author on: Oxford Academic Google Scholar Journal of Logic and Computation, Volume 22, Issue 4, August 2012, Pages 755–777, https://doi.org/10.1093/logcom/exq041 Published: 30 November 2010 Article history Received: 16 October 2009 Published: 30 November 2010

Abstract Fraïssé studied countable structures through analysis of the age of , i.e., the set of all finitely generated substructures of . We investigate the effectiveness of his analysis, considering effectively presented lists of finitely generated structures and asking when such a list is the age of a computable structure. We focus particularly on the Fraïssé limit. We also show that degree spectra of relations on a sufficiently nice Fraïssé limit are always upward closed unless the relation is definable by a quantifier-free formula. We give some sufficient or necessary conditions for a Fraïssé limit to be spectrally universal. As an application, we prove that the computable atomless Boolean algebra is spectrally universal.

Abstract We show that for every computable limit ordinal α, there is a computable structure that is categorical, but not relatively categorical (equivalently, it does not have a formally Scott family). We also show that for every computable limit ordinal α, there is a computable structure with an additional relation R that is intrinsically on , but not relatively intrinsically on (equivalently, it is not definable by a computable Σα formula with finitely many parameters). Earlier results in [7], [10], and [8] establish the same facts for computable successor ordinals α.

Abstract We study the weak truth-table and truth-table degrees of the images of subsets of computable structures under isomorphisms between computable structures. In particular, we show that there is a low c.e. set that is not weak truth-table reducible to any initial segment of any scattered computable linear ordering. Countable subsets of 2ω and Kolmogorov complexity play a major role in the proof.

Abstract A Turing degree d is homogeneous bounding if every complete decidable (CD) theory has a d-decidable homogeneous model , i.e., the elementary diagram De ( ) has degree d. It follows from results of Macintyre and Marker that every PA degree (i.e., every degree of a complete extension of Peano Arithmetic) is homogeneous bounding. We prove that in fact a degree is homogeneous bounding if and only if it is a PA degree. We do this by showing that there is a single CD theory T such that every homogeneous model of T has a PA degree.

Abstract We consider embeddings of structures which preserve spectra: if g : ℳ → with computable, then ℳ should have the same Turing degree spectrum (as a structure) that g(ℳ) has (as a relation on ). We show that the computable dense linear order ℒ is universal for all countable linear orders under this notion of embedding, and we establish a similar result for the computable random graph Such structures are said to be spectrally universal. We use our results to answer a question of Goncharov, and also to characterize the possible spectra of structures as precisely the spectra of unary relations on . Finally, we consider the extent to which all spectra of unary relations on the structure ℒ may be realized by such embeddings, offering partial results and building the first known example of a structure whose spectrum contains precisely those degrees c with c′ ≥ τ 0″.

When bounds on complexity of some aspect of a structure are preserved under isomorphism, we refer to them as intrinsic. Here, building on work of Soskov [34], [33], we give syntactical conditions necessary and sufficient for a relation to be intrinsically on a structure. We consider some examples of computable structures and intrinsically relations R. We also consider a general family of examples of intrinsically relations arising in computable structures of maximum Scott rank. For three of the examples, the maximal well-ordered initial segment in a Harrison ordering, the superatomic part of a Harrison Boolean algebra, and the height-possessing part of a Harrison p-group, we show that the Turing degrees of images of the relation in computable copies of the structure are the same as the Turing degrees of paths through Kleene's . With this as motivation, we investigate the possible degrees of these paths. We show that there is a path in which ∅′ is not computable. In fact, there is one in which no noncomputable hyperarithmetical set is computable. There are paths that are Turing incomparable, or Turing incomparable over a given hyperarithmetical set. There is a pair of paths whose degrees form a minimal pair. However, there is no path of minimal degree.

We consider only computable languages, and countable structures, with universe a subset of ω, which we think of as a set of constants. We identify sentences with their Gödel numbers. Thus, for a structure , the complete (elementary) diagram, Dc( ), and the atomic diagram, D( ), are subsets of ω. We classify formulas as usual. A formula is both Σ0 and Π0 if it is open. For n > 0, a formula, in prenex normal form, is Σn, or Πn, if it has n blocks of like quantifiers, beginning with ∃, or ∀. For a formula θ, in prenex normal form, we let neg(θ) denote the dual formula that is logically equivalent to ¬θ—if θ is Σn, then neg(θ) is Πn, and vice versa.

In 1960 G. F. Rose [R] made the following definition: A function f: ω → ω is (m, n)-computable, where 1 ≤ m ≤ n, iff there exists a recursive function R: ωn → ωn such that, for all n-tuples (x1,…, xn) of distinct natural numbers, J. Myhill (see [McN, p. 393]) asked if f had to be recursive if m was close to n; B. A. Trakhtenbrot [T] responded by showing in 1963 that f is recursive whenever 2m > n. This result is optimal, because, for example, the characteristic function of any semirecursive set is (1,2)-computable. Trakhtenbrot's work was extended by E. B. Kinber [Ki1], using similar techniques. In 1986 R. Beigel [B] made a powerful conjecture, much more general than the above results. Partial verification, falling short of a full proof, appeared in [O]. Using new techniques, M. Kummer has recently established the conjecture, which will henceforth be referred to as the cardinality theorem (CT). It is the goal of this paper to show the connections between these various theorems, to review the methods used by Trakhtenbrot, and to use them to prove a special case of CT strong enough to imply Kinber's theorem (see §3). We thus have a hierarchy of results, with CT at the top. We will also include a discussion of Kummer's methods, but not a proof of CT.